EXTREMES OF THRESHOLD-DEPENDENT GAUSSIAN PROCESSES
LONG BAI, KRZYSZTOF DȨBICKI, ENKELEJD HASHORVA, AND LANPENG JI

arXiv:1701.05387v1 [math.PR] 19 Jan 2017

n
o
Abstract: In this contribution we are concerned with the asymptotic behaviour, as u → ∞, of P supt∈[0,T ] Xu (t) > u ,

where Xu (t), t ∈ [0, T ], u > 0 is na family of centered Gaussian
o processes with continuous trajectories. A key application of our findings concerns P supt∈[0,T ] (X(t) + g(t)) > u , as u → ∞, for X a centered Gaussian process and g

some measurable trend function. Further applications include the approximation of both the ruin time and the ruin
probability of the Brownian motion risk model with constant force of interest.
Key Words: Extremes; Gaussian processes; fractional Brownian motion; ruin probability; ruin time.
AMS Classification: Primary 60G15; secondary 60G70
1. Introduction

Let X(t), t ≥ 0 be a centered Gaussian process with continuous trajectories. An important problem in applied and
theoretical probability is the determination of the asymptotic behavior of
(
)
(1)

p(u) = P

u→∞

sup (X(t) + g(t)) > u ,

t∈[0,T ]

for some T > 0 and g(t), t ∈ [0, T ] a bounded measurable function. For instance, if g(t) = −ct, then in the context of

risk theory p(u) has interpretation as the ruin probability over the finite-time horizon [0, T ]. Dually, in the context of
queueing theory, p(u) is related to the buffer overload problem; see e.g., [1–5].

For the special case that g(t) = 0, t ∈ [0, T ] the exact asymptotics of (1) is well-known for both locally stationary
and general non-stationary Gaussian processes, see e.g., [6–18]. Commonly, for X a centered non-stationary Gaussian
process it is assumed that the standard deviation function σ is such that t0 = argmaxt∈[0,T ] σ(t) is unique and
σ(t0 ) = 1. Additionally, if the correlation function r and the standard deviation function σ satisfy (hereafter ∼ means
asymptotic equivalence)
(2)

α

β

1 − r(s, t) ∼ a |t − s| ,

1 − σ(t0 + t) ∼ b |t| ,

s, t → t0

for some a, b, β positive and α ∈ (0, 2], then we have (see [10][Theorem D.3])
(3)

2

2

p(u) ∼ C0 u( α − β )+ P {X(t0 ) > u} ,

where (x)+ = max(0, x) and

C0 =

Here Γ(·) is the gamma function, and
(
)
1
W (t)
Hα = lim
E sup e
,
T →∞ T
t∈[0,T ]

u → ∞,


1/α −1/β

Γ(1/β + 1)Hα , if α < β,

 a b
b/a

Pα ,


 1,
Pαb/a

=E

if α = β,

if α > β.

(

sup e
t∈[0,∞)

W (t)−b/a|t|α

)

, with W (t) =

√
α
2Bα (t) − |t| ,

are the Pickands and Piterbarg constants, respectively, where Bα is a standard fractional Brownian motion (fBm)
with self-similarity index α/2 ∈ (0, 1], see [19–25] for properties of both constants.

The more general case with non-zero g has also been considered in the literature; see, e.g., [1, 26–30]. However, most

of the aforementioned contributions treat only restrictive trend functions g. For instance, in [26][Theorem 3] a Höldertype condition for g is assumed, which excludes important cases of g that appear in applications. The restrictions are
Date: January 20, 2017.
1

2

LONG BAI, KRZYSZTOF DȨBICKI, ENKELEJD HASHORVA, AND LANPENG JI

often so severe that simple cases such as the Brownian bridge with drift considered in Example 3.11 below cannot be
covered.
A key difficulty when dealing with p(u) is that X + g is not a centered Gaussian process. It is however possible to get
rid of the trend function g since for any bounded function g and all u large (1) can be re-written as
(
)
X(t)
, t ∈ [0, T ].
pT (u) = P sup Xu (t) > u , Xu (t) =
1
−
g(t)/u
t∈[0,T ]
Here Xu is centered, however it depends on the threshold u, which complicates the analysis.
Extremes of threshold-dependent Gaussian processes Xu (t), t ∈ R have been already dealt with in several contributions,

see e.g., [2, 3, 30–32]. Our principal result in Theorem 2.4 derives the asymptotics of pT (u) for quite general families

of centered Gaussian processes Xu under tractable assumptions on the variance and correlation functions of Xu . To
this end, in Theorem 2.2 we first derive the asymptotics of
(
p∆ (u) = P

)

sup Xu (t) > u ,

t∈∆(u)

u→∞

for some short compact intervals ∆(u).
Applications of our main results include derivation of Proposition 3.1 for a class of locally stationary Gaussian processes
with trend and that of Proposition 3.6 for a class of non-stationary Gaussian processes with trend, as well as those of
their corollaries. For instance, a direct application of Proposition 3.6 yields the asymptotics of (1) for a non-stationary
X with standard deviation function σ and correlation function r satisfying (2) with t0 = argmaxt∈[0,T ] σ(t). If further
the trend function g is continuous in a neighborhood of t0 , g(t0 ) = maxt∈[0,T ] g(t) and
g(t) ∼ g(t0 ) − c|t − t0 |γ ,

(4)

t → t0

for some positive constants c, γ, then (3) holds with C0 specified in Proposition 3.9 and β, u being substituted by
min(β, 2γ) and u − g(t0 ) respectivelly.

Complementary, we investigate asymptotic properties of the first passage time (ruin time) of X(t) + g(t) to u on the
finite-time interval [0, T ], given the process has ever exceeded u during [0, T ]. In particular, for
τu = inf{t ≥ 0 : X(t) > u − g(t)},

(5)

with inf{∅} = ∞, we are interested in the approximate distribution of τu |τu ≤ T , as u → ∞. Normal and exponential
approximations of various Gaussian models have been discussed in [30, 32–35]. In this paper, we derive general results

for the approximations of the conditional passage time in Propositions 3.3, 3.10. The asymptotics of p∆ (u) for a short
compact intervals ∆(u) displayed in Theorem 2.2 plays a key role in the derivation of these results.
Organisation of the rest of the paper: In Section 2, the tail asymptotics of the supremum of a family of centered
Gaussian processes indexed by u are given. Several applications and examples are displayed in Section 3. Finally, we
present all the proofs in Section 4 and Section 5.
2. Main Results
Let Xu (t), t ∈ R, u > 0 be a family of threshold-dependent centered Gaussian processes with continuous trajectories,

variance functions σu2 and correlation functions ru . Our main results concern the asymptotics of slight generalization
of p∆ (u) and pT (u) for families of centered Gaussian processes Xu satisfying some regularity conditions for variance
and coavariance respectivelly.
Let C0∗ (E) be the set of continuous real-valued functions defined on the interval E such that f (0) = 0 and for some
ǫ2 > ǫ1 > 0
(6)

lim

|t|→∞,t∈E

f (t)/|t|ǫ1 = ∞,

if sup{x : x ∈ E} = ∞ or inf{x : x ∈ E} = −∞.

lim

|t|→∞,t∈E

f (t)/|t|ǫ2 = 0,

In the following Rα denotes the set of regularly varying functions at 0 with index α ∈ R, see [36–38] for details.

EXTREMES OF THRESHOLD-DEPENDENT GAUSSIAN PROCESSES

3

We shall impose the following assumptions where ∆(u) is a compact interval:
A1: For any large u, there exists a point tu ∈ R such that σu (tu ) = 1.

A2: There exists some λ > 0 such that

lim

(7)



sup

u→∞ t∈∆(u)

1
σu (tu +t)


− 1 u2 − f (uλ t)

f (uλ t) + 1

=0

holds for some non-negative continuous function f with f (0) = 0.
A3: There exists ρ ∈ Rα/2 , α ∈ (0, 2] such that
lim

sup

u→∞ s,t∈∆(u)
t6=s

and η := lims→0

2

ρ (s)
s2/λ

1 − ru (tu + s, tu + t)
−1 =0
ρ2 (|t − s|)

∈ [0, ∞], with λ given in A2.

Remark 2.1. If f satisfies f (0) = 0 and f (t) > 0, t 6= 0, then
lim

sup

u→∞ t∈∆(u),t6=0

1
σu (tu +t) − 1
u−2 f (uλ t)

−1 =0

for some λ > 0 implies that (7) is valid.
Next we introduce some further notation, starting with the Pickands-type constant defined by
)
(
Hα [0, T ] = E

sup e

√
2Bα (t)−|t|α

, T > 0,

t∈[0,T ]

where Bα is an fBm. Further, define for f ∈ C0∗ ([S, T ]) with S, T ∈ R, S < T and a positive constant a
)
(
f
Pα,a
[S, T ] = E

sup e

√
2aBα (t)−a|t|α −f (t)

,

t∈[S,T ]

and set
f
f
Pα,a
[0, ∞) = lim Pα,a
[0, T ],
T →∞

f
Pα,a
(−∞, ∞) =

lim

S→−∞,T →∞

f
Pα,a
[S, T ].

f
f
The finiteness of Pα,a
[0, ∞) and Pα,a
(−∞, ∞) is guaranteed under weak assumptions on f , which will be shown in

f
[0, ∞).
the proof of Theorem 2.2, see [2, 3, 5, 7, 15, 25, 39–43] for various properties of Hα and Pα,a
−
Denote by I{·} the indicator function. For the regularly varying function ρ(·), we denote by ←
ρ (·) its asymptotic

inverse (which is asymptotically unique). Throughout this paper, we set 0 · ∞ = 0 and u−∞ = 0 if u > 0. Let
Ψ(u) := P {N > u}, with N a standard normal random variable.

In the next theorem we shall consider two functions x1 (u), x2 (u), u ∈ R such that x1 ( 1t ) ∈ Rµ1 , x2 ( 1t ) ∈ Rµ2 with

µ1 , µ2 ≥ λ, and

lim uλ xi (u) = xi ∈ [−∞, ∞], i = 1, 2,

(8)

u→∞

with x1 < x2 .

Theorem 2.2. Let Xu (t), t ∈ R be a family of centered Gaussian processes with variance functions σu2 and correlation

functions ru . If A1-A3 are satisfied with ∆(u) = [x1 (u), x2 (u)], and f ∈ C0∗ ([x1 , x2 ]), then for Mu satisfying Mu ∼

u, u → ∞, we have
(9)

P

(

sup Xu (tu + t) > Mu
t∈∆(u)

where
(10)

f
and Pα,η
(−∞, ∞) ∈ (0, ∞).

)

−I{η=∞}
−
ρ (u−1 )
Ψ(Mu ),
∼ C uλ ←


R x2 −f (t)

dt,
if η = ∞,

 Hα x1 e
f
C=
Pα,η [x1 , x2 ],
if η ∈ (0, ∞),


 sup
−f (t)
, if η = 0,
t∈[x1 ,x2 ] e

u → ∞,

4

LONG BAI, KRZYSZTOF DȨBICKI, ENKELEJD HASHORVA, AND LANPENG JI

Remark 2.3. Let α ∈ (0, 2], a > 0 be given. If f ∈ C0∗ ([x1 , x2 ]) for x1 , x2 , y ∈ R, x1 < x2 , as shown in Appendix, we

have, with fy (t) := f (y + t), t ∈ R

f
fy
[x1 − y, x2 − y],
Pα,a
[x1 , x2 ] = Pα,a

(11)

f
fy
[x1 − y, ∞).
Pα,a
[x1 , ∞) = Pα,a

In particular, if f (t) = ct, c > 0, then for any x ∈ R
ct
cx+ct
ct
Pα,a
[x, ∞) = Pα,a
[0, ∞) = e−cx Pα,a
[0, ∞).

Next, for any fixed T ∈ (0, ∞), in order to analyse pT (u) we shall suppose that:

A1’: For all large u, σu (t) attains its maximum over [0, T ] at a unique point tu such that
σu (tu ) = 1
A4: For all u large enough

and

lim tu = t0 ∈ [0, T ].

u→∞

1
p(ln u)q
≥1+
u2
t∈[0,T ]\(tu +∆(u)) σu (t)
inf

holds for some constants p > 0, q > 1.
A5: For some positive constants G, ς > 0

E (X u (t) − X u (s))2 ≤ G|t − s|ς

holds for all s, t ∈ {x ∈ [0, T ] : σ(x) 6= 0} and X u (t) =

Xu (t)
σu (t) .

Below we define for λ given in A2 and ν, d positve


[0, δu ]
if tu ≡ 0,





if tu ∼ du−ν and ν ≥ λ,

 [−tu , δu ],

(12)

where δu =

∆(u) =



(ln u)
u

λ
q

[−δu , δu ],




[−δu , T − tu ],




[−δu , 0]

if tu ∼ du−ν or T − tu ∼ du−ν when ν < λ, or t0 ∈ (0, T ),

if T − tu ∼ du−ν and ν ≥ λ,

if tu = T,

with q given in A4.

Theorem 2.4. Let Xu (t), t ∈ [0, T ] be a family of centered Gaussian processes with variance functions σu2 and

correlation functions ru . Assume that A1’,A2-A5 are satisfied with ∆(u) = [c1 (u), c2 (u)] given in (12) and
lim ci (u)uλ = xi ∈ [−∞, ∞], i = 1, 2,

u→∞

x1 < x2 .

If f ∈ C0∗ ([x1 , x2 ]), then for Mu suc that limu→∞ Mu /u = 1 we have
)
(
−I{η=∞}
−
ρ (u−1 )
(13)
Ψ(Mu ),
P sup Xu (t) > Mu ∼ C uλ ←
t∈[0,T ]

u → ∞,

where C is the same as in (10) if η ∈ (0, ∞] and C = 1 if η = 0.

Remark 2.5. Theorem 2.4 generalises both [26][Theorem 1] and [32][Theorem 4.1].
3. Applications
3.1. Locally stationary Gaussian processes with trend. In this section we consider the asymptotics of (1) for
X(t), t ∈ [0, T ] a centered locally stationary Gaussian process with unit variance and correlation function r satisfying
(14)

lim sup

h→0 t∈[0,T ]

1 − r(t, t + h)
=1
a(t) |h|α

with α ∈ (0, 2], a(·) a positive continuous function on [0, T ] and further
(15)

r(s, t) < 1, ∀s, t ∈ [0, T ] and s 6= t.

We refer to e.g., [9, 10, 44–46] for results on locally stationary Gaussian processes. Extensions of this class to α(t)locally stationary processes are discussed in [13, 47, 48].

EXTREMES OF THRESHOLD-DEPENDENT GAUSSIAN PROCESSES

5

Regarding the continuous trend function g, we define gm = maxt∈[0,T ] g(t) and set
H := {s ∈ [0, T ] : g(s) = gm } .
Set below, for any t0 ∈ [0, T ]
Qt0 = 1 + I{t0 ∈(0,T )} ,

(16)

wt0 =

(

−∞, if t0 ∈ (0, T ),
0,

if t0 = 0 or t0 = T.

Proposition 3.1. Suppose that (14) and (15) hold for a centered locally stationary Gaussian process X(t), t ∈ [0, T ]
and let g : [0, T ] → R be a continuous function.

i) If H = {t0 } and (4) holds, then as u → ∞
(

(17)

sup (X(t) + g(t)) > u

P

t∈[0,T ]

where (set with a = a(t0 ))
Ct0

)

2

1

∼ Ct0 u( α − γ )+ Ψ (u − gm ) ,


1/α −1/γ

Γ(1/γ + 1)Hα , if α < 2γ,

 Qt0 aγ c
c|t|
=
if α = 2γ,
Pα,a [wt0 , ∞),


 1,
if α > 2γ.

ii) If H = [A, B] ⊂ [0, T ] with 0 ≤ A < B ≤ T , then as u → ∞
(
)
Z

B

P

sup (X(t) + g(t)) > u

t∈[0,T ]

∼ Hα

A

2

(a(t))1/α dtu α Ψ (u − gm ) .

Remarks 3.2. i) If H = {t1 , . . . , tn }, then as mentioned in [10], the tail distribution of the corresponding supremum

is easily obtained assuming that for each ti the assumptions of Proposition 3.1 statement i) hold, implying that
(
)
n
 2 1
X
Ctj u( α − γ )+ Ψ (u − gm ) , u → ∞.
P sup (X(t) + g(t)) > u ∼
t∈[0,T ]

j=1

ii) The novelty of Proposition 3.1 statement i) is that for the trend function g only a polynomial local behavior around
t0 is assumed. In the literature so far only the case that (4) holds with γ = 2 has been considered (see [28]).
iii) By the proof of Proposition 3.1 statement i), if g(t) is a measurable function which is continuous in a neighborhood
of t0 and smaller than gm − ε for some ε > 0 in the rest part over [0, T ], then the results still hold.
We present below the approximation of the conditional passage time τu |τu ≤ T with τu defined in (5).
Proposition 3.3. Suppose that (14) and (15) hold for a centered locally stationary Gaussian process X(t), t ∈ [0, T ].
Let g : [0, T ] → R be a continuous function, H = {t0 } and (4) holds.

i) If t0 ∈ [0, T ), then for any x ∈ (wt0 , ∞)

n
o
P u1/γ (τu − t0 ) ≤ x τu ≤ T ∼

ii) If t0 = T , then for any x ∈ (−∞, 0)











γc1/γ

Rx

wt
0

γ

e−c|t| dt

Qt0 Γ(1/γ)

,

if α < 2γ,

γ

P c|t| [wt ,x]

α,a
0
,
if α = 2γ,
c|t|γ

Pα,a [wt0 ,∞)






−c|t|γ
 sup
, if α > 2γ,
t∈[wt0 ,x] e

n
o
P u1/γ (τu − t0 ) ≤ x τu ≤ T ∼











γc1/γ

R∞

γ

e−c|t| dt
,
Γ(1/γ)
−x

if α < 2γ,

γ

P c|t| [−x,∞)

α,a
,
c|t|γ

Pα,a [0,∞)






 e−c|x|γ ,

if α = 2γ,
if α > 2γ.

6

LONG BAI, KRZYSZTOF DȨBICKI, ENKELEJD HASHORVA, AND LANPENG JI

Example 3.4. Let X(t), t ∈ [0, T ] be a centered stationary Gaussian process with unit variance and correlation

function r that satisfies r(t) = 1 − a|t|α (1 + o(1)), t → 0 for some a > 0, α ∈ (0, 2], and r(t) < 1, for all t ∈ (0, T ].
Let τu be defined as in (5) with g(t) = −ct, c > 0. Then we have
(


c−1 a1/α Hα ,
2
−1)+
(α
Ψ(u)
P max (X(t) − ct) > u ∼ u
ct
t∈[0,T ]
Pα,a
[0, ∞),

and for any x positive


n
o  1 − e−cx ,
ct
P uτu ≤ x τu ≤ T ∼
 Pctα,a [0,x] ,
Pα,a [0,∞)

α ∈ (0, 2),

α = 2,

α ∈ (0, 2),
α = 2.

Example 3.5. Let X(t), t > 0 be a standardized fBm, i.e., X(t) = Bα (t)/tα/2 with Bα an fBm. Let c, T be positive
constants. Then for any n ∈ N, we have







n
X
1
1
2
2πt
T
X(t) + c sin
u α − 2 Ψ(u − c),
P
max
>u ∼
ajα  Hα √
t∈[T,(n+1)T ]
T
2cπ
j=1
where aj =

1
2



(4j+1)T
4

−α

, j = 1, . . . , n.

3.2. Non-stationary Gaussian processes with trend. In this section we consider the asymptotics of (1) for
X(t), t ∈ [0, T ] a centered Gaussian process with non-constant variance function σ 2 . Define below whenever σ(t) 6= 0
X(t) :=

X(t)
,
σ(t)

t ∈ [0, T ],

and set for a continuous function g
mu (t) :=

(18)

σ(t)
,
1 − g(t)/u

t ∈ [0, T ],

u > 0.

Proposition 3.6. Let X and g be as above. Assume that tu = argmaxt∈[0,T ] mu (t) is unique with limu→∞ tu = t0
and σ(t0 ) = 1. Further, we suppose that A2-A5 are satisfied with σu (t) =
and ∆(u) = [c1 (u), c2 (u)] given in (12). If in A2 f ∈

C0∗ ([x1 , x2 ])

lim ci (u)uλ = xi ∈ [−∞, ∞], i = 1, 2,

(19)

P

(

sup (X(t) + g(t)) > u

t∈[0,T ]

)

−
λ←

∼ C u ρ (u

−1

ru (s, t) = r(s, t), X u (t) = X(t)

and
x1 < x2 ,

u→∞

then we have

mu (t)
mu (tu ) ,

−I{η=∞}
)
Ψ



u − g(tu )
σ(tu )



,

u → ∞,

where C is the same as in (10) when η ∈ (0, ∞] and C = 1 when η = 0.

Remarks 3.7. i) Proposition 3.6 extends [26][Theorem 3] and the results of [1] where (1) was analyzed for special X
with stationary increments and special trend function g.
ii) The assumption that σ(t0 ) = 1 is not essential in the proof. In fact, for the general case where σ(t0 ) 6= 1 we have
that (19) holds with

 −2
R x2 −σ−2 f (t)

e 0
dt, if η = ∞,
σ αH

 0 −2 α x1
σ0 f
C=
if η ∈ (0, ∞),
Pα,σ−2 η [x1 , x2 ],

0


1,
if η = 0,

σ0 = σ(t0 ).

Proposition 3.8. Under the notation and assumptions of Proposition 3.6 without assuming A3,A5, if X is differentiable in the mean square sense such that
r(s, t) < 1, s 6= t,


E X ′2 (t0 ) > σ ′2 (t0 ),


and E X ′2 (t) − σ ′2 (t) is continuous in a neighborhood of t0 , then (19) holds with
o

1  n ′2
α = 2, ρ2 (t) =
E X (t0 ) − σ ′2 (t0 ) t2 .
2

EXTREMES OF THRESHOLD-DEPENDENT GAUSSIAN PROCESSES

7

The next result is an extension of a classical theorem concerning the extremes of non-stationary Gaussian processes
discussed in the Introduction, see [10][Theorem D.3].
Proposition 3.9. Let X(t), t ∈ [0, T ] be a centered Gaussian process with correlation function r and variance function
σ 2 such that t0 = argmaxt∈[0,T ] σ(t) is unique with σ(t0 ) = σ > 0. Suppose that g is a bounded measurable function

being continuous in a neighborhood of t0 such that (4) holds. If further (2) is satisfied, then
(
)


2
u − g(t0 )
(α
− β2∗ )+
(20)
,
P sup (X(t) + g(t)) > u ∼ C0 u
Ψ
σ
t∈[0,T ]
where β ∗ = min(β, 2γ),
C0 =

with f (t) =

b
β
∗
σ3 |t| I{β=β }

+


R∞
−2/α 1/α

a Hα wt e−f (t) dt, if α < β ∗ ,

 σ
0
f
Pα,σ
−2 a [wt0 , ∞),


 1,

c
γ
∗
σ2 |t| I{2γ=β }

if α = β ∗ ,

if α > β ∗ ,

and wt0 defined in (16).

Proposition 3.10. i) Under the conditions and notation of Proposition 3.6,
 R
x

e−f (t) dt


R xx1
if

2 e−f (t) dt ,

x1




f
Pα,η
[x1 ,x]
(21)
P uλ (τu − tu ) ≤ x τu ≤ T ∼
,
if
f

P
α,η [x1 ,x2 ]






−f (t)
 sup
, if
t∈[x1 ,x] e

for any x ∈ [x1 , x2 ] we have
η = ∞,
η ∈ (0, ∞),
η = 0.

ii) Under the conditions and notation of Proposition 3.9, if t0 ∈ [0, T ), then for
 Rx
e−f (t) dt

wt

0
R

,
∞

e−f (t) dt

wt

0

n
o 
∗
f
Pα,a
[wt0 ,x]
P u2/β (τu − t0 ) ≤ x τu ≤ T ∼
,
f

Pα,a [wt0 ,∞)






−f (t)
 sup
,
t∈[wt0 ,x] e
and if t0 = T , then for x ∈ (−∞, 0)

n
o
∗
P u2/β (τu − t0 ) ≤ x τu ≤ T ∼











x ∈ (wt0 , ∞)
if α < β ∗ ,
if α = β ∗ ,
if α > β ∗ ,

R ∞ −f (t)
e
dt
R−x
,
∞ −f (t)
e
dt
0

if α < β ∗ ,

Pf

if α = β ∗ ,

[−x,∞)

α,a
,
f

[0,∞)
Pα,a






 e−f (x) ,

if α > β ∗ .

Example 3.11. Let X(t) = B(t) − tB(1), t ∈ [0, 1], where B(t) is a standard Brownian motion and suppose that τu

is defined by (5) with g(t) = −ct. Then
(
(22)

P

sup (X(t) − ct) > u

t∈[0,1]

 
P u τu −

u
c + 2u



≤ x τu ≤ 1



)

2

∼ e−2(u

+cu)

,

∼ Φ(4x), x ∈ (−∞, ∞).

We
2.7], the result in (22) is actually exact, i.e. for any u > 0,
n note that according to [49][Lemma
o
−2(u2 +cu)
P supt∈[0,1] (X(t) − ct) > u = e
.
Now, let T = 1/2. It appears that the asymptotics in this case is different, i.e.,
(
)

(23)

P

sup (X(t) − ct) > u

t∈[0,1/2]

2

∼ Φ(c)e−2(u

+cu)

,

8

LONG BAI, KRZYSZTOF DȨBICKI, ENKELEJD HASHORVA, AND LANPENG JI

and

 
P u τu −

u
c + 2u



≤ x τu ≤

1
2



∼

Φ(4x)
, x ∈ (−∞, c/4].
Φ(c)

Similarly, we have
(

)


2
c
1
P sup X(t) + − c t −
> u ∼ 2Ψ(c)e−2(u −cu)
2
2
t∈[0,1]

(24)
and

2

 R 4x − (|t|+c)
 
2
dt
1
−∞ e
√
, x ∈ (−∞, ∞).
≤ x τu ≤ 1 ∼
P u τu −
2
2 2πΨ(c)

We conclude this section with an application of Proposition 3.6 to the calculation of the ruin probability of a Brownian
motion risk model with constant force of interest over infinite-time horizon.
3.3. Ruin probability in Gaussian risk model. Consider risk reserve process U (t), with interest rate δ, modeled
by
δt

U (t) = ue + c

Z

t

e

δ(t−v)

0

dv − σ

Z

t

eδ(t−v) dB(v),

0

t ≥ 0,

where c, δ, σ are some positive constants and B is a standard Brownian motion. The corresponding ruin probability
over infinite-time horizon is defined as
p(u) = P



inf

t∈[0,∞)


U (t) < 0 .

For this model we also define the ruin time τu = inf{t ≥ 0 : U (t) < 0}. Set below
2
δ p
c
t + r2 − r , t ∈ [0, ∞), r = .
h(t) = 2
σ
δ

We present next approximations of the ruin probability and the conditional ruin time τu |τu < ∞ as u → ∞.
Proposition 3.12. As u → ∞

 2 
h
p(u) ∼ P1,δ/σ
2 −r , ∞ Ψ

(25)
and for x ∈ (−r2 , ∞)

(

P u

2

e

−2δτu

−



c
δu + c

2 !

 p

1
2δu2 + 4cu
σ
)

≤ x τu < ∞

∼

 2 
h
P1,δ/σ
2 −r , x

h
2
P1,δ/σ
2 [−r , ∞)

Remark 3.13. According to [50] (see also [51]) we have
!
√


.
2δ
(u + r)
(26)
Ψ
P
inf U (t) < 0 = Ψ
σ
t∈[0,∞)
By (25) and (11)
(


P
inf U (t) < 0
∼ E
t∈[0,∞]

.

√ !
2c
√
.
σ δ

!) 

2
2δ
δ p
1p
δ
2
2
sup exp
t + r − r − 2 |t|
2δu + 4cu
B(t) − 2
Ψ
σ2
σ
σ
σ
t∈[−r 2 ,∞)

!
!
r
√




√
2δ
c2
c2
2c
2B(t) − t + 2
t + 2 − |t|
exp
+ √
(u + r)
Ψ
∼ E
sup

t∈[− c2 ,∞)
σ δ
σ δ
σ
σ δ
2
σ δ
!
(
√
)

√
2δ
2c √
(u + r) ,
2B (t) − 2t + √ t
Ψ
= E
sup exp
σ
σ δ
t∈[0,∞)
r

which combined with (26) implies that
(

)
√
2c √
(27)
E
sup exp
2B (t) − 2t + √ t
=
σ δ
t∈[0,∞)

Ψ

√ !!−1
2c
√
.
σ δ

EXTREMES OF THRESHOLD-DEPENDENT GAUSSIAN PROCESSES

9

4. Proofs
In the proofs presented in this section Ci , i ∈ N are some positive constants which may be different from line to line.
We first give two preliminary lemmas, which play an important role in the proof of Theorem 2.2.

Lemma 4.1. Let ξ(t), t ∈ R be a centered stationary Gaussian process with unit variance and correlation function r
satisfying

1 − r(t) ∼ aρ2 (|t|),

(28)

t → 0,

with a > 0, and ρ ∈ Rα/2 , α ∈ (0, 2]. Let f be a continuous function, Ku be a family of index sets and
Zu (t) :=

−
ξ(←
ρ (u−1 )t)
,
−
−2
1 + u f (←
ρ (u−1 )uλ t)

t ∈ [S1 , S2 ],

where λ > 0 and −∞ < S1 < S2 < ∞. If Mk (u), k ∈ Ku is such that
(29)

lim sup

u→∞ k∈Ku

then we have
1
lim sup
P
u→∞ k∈Ku Ψ(Mk (u))

(30)
where

Rfη [S1 , S2 ]
with η := limt↓0

ρ2 (t)
t2/λ

:= E

(

sup

e

(

Mk (u)
− 1 = 0,
u

sup

)

Zu (t) > Mk (u)

t∈[S1 ,S2 ]

√
2aBα (t)−a|t|α −f (η −1/α t)

t∈[S1 ,S2 ]

)

=

(

− Rfη [S1 , S2 ] = 0,

Hα [a1/α S1 , a1/α S2 ]

h
Pα,a
[S1 , S2 ]

f (·) ≡ 0,

otherwise,

∈ (0, ∞] and h(t) = f (η −1/α t) for η ∈ (0, ∞), h(t) = f (0) for η = ∞.

Proof of Lemma 4.1: We set η −1/α = 0 if η = ∞. The proof follows by checking the conditions of [52][Theorem 2.1]
where the results still holds if we omit the requirements f (0) = 0 and [S1 , S2 ] ∋ 0. By (29)
lim inf Mk (u) = ∞.

u→∞ k∈Ku

By continuity of f we have
(31)

lim

sup

u→∞ k∈K ,t∈[S ,S ]
u
1
2

−
Mk2 (u)u−2 f (←
ρ (u−1 )uλ t) − f (η −1/α t) = 0.

Moreover, (28) implies


−
−
−
−
ρ (u−1 )(t − t′ ) , u → ∞,
Var(ξ(←
ρ (u−1 )t) − ξ(←
ρ (u−1 )t′ )) = 2 − 2r ←
ρ (u−1 )(t − t′ ) ∼ 2aρ2 ←

holds for t, t′ ∈ [S1 , S2 ]. Thus
(32)

lim sup

sup

u→∞ k∈Ku t6=t′ ∈[S ,S ]
1
2

Mk2 (u)

−
−
Var(ξ(←
ρ (u−1 )t) − ξ(←
ρ (u−1 )t′ ))
− 1 = 0.
←
−
2
2
−1
2au ρ (| ρ (u )(t − t′ )|)

Since ρ2 ∈ Rα which satisfies the uniform convergence theorem (UCT) for regularly varying function, see, e.g., [53],
i.e.,

(33)

lim

sup

u→∞ t,t′ ∈[S ,S ]
1
2


α
−
ρ (u−1 )(t − t′ ) − |t − t′ | = 0,
u 2 ρ2 ←

and further by the Potter’s bound for ρ2 , see [53] we have

−


u 2 ρ2 ←
ρ (u−1 )(t − t′ )
α−ε1
α+ε1
(34)
< ∞,
≤
C
max
|S
−
S
|
,
|S
−
S
|
lim sup sup
1
1
2
1
2
α−ε
1
u→∞ t,t′ ∈[S1 ,S2 ]
|t − t′ |
t6=t′

where ε1 ∈ (0, min(1, α)). We know that for α ∈ (0, 2]
(35)

|t|α − |t′ |

α

≤ C2 |t − t′ |

α∧1

, t, t′ ∈ [S1 , S2 ].

10

LONG BAI, KRZYSZTOF DȨBICKI, ENKELEJD HASHORVA, AND LANPENG JI

By (28) for any small ǫ > 0, when u large enough
−
−
r(←
ρ (u−1 )t) ≤ 1 − ρ2 (←
ρ (u−1 ) |t|)(1 − ǫ),

(36)

−
−
r(←
ρ (u−1 )t) ≥ 1 − ρ2 (←
ρ (u−1 ) |t|)(1 + ǫ)

hold for t ∈ [S1 , S2 ], then by (29) for u large enough
 − −1
−
sup
Mk2 (u)E [ξ(←
ρ (u )t) − ξ(←
ρ (u−1 )t′ )]ξ(0)
sup
k∈Ku |t−t′ |<ε,t,t′ ∈[S1 ,S2 ]

≤ C3 u 2

≤ C3
≤ C3
(37)
(38)

sup

|t−t′ |<ε,t,t′ ∈[S1 ,S2 ]

sup
|t−t′ |<ε,t,t′ ∈[S1 ,S2 ]

−
−
r(←
ρ (u−1 )t) − r(←
ρ (u−1 )t′ )


−
−
−
−
u2 ρ2 (←
ρ (u−1 ) |t|) − u2 ρ2 (←
ρ (u−1 ) |t′ |) + ǫ u2 ρ2 (←
ρ (u−1 ) |t|) + ǫ u2 ρ2 (←
ρ (u−1 ) |t′ |)

u 2 ρ2

sup
|t−t′ |<ε,t,t′ ∈[S1 ,S2 ]



α−ε1
α−ε1
+C4 ǫ |t|
+ |t′ |


←
−
ρ (u−1 )(t) − |t|α + u2 ρ2


α
α
←
−
ρ (u−1 )(t′ ) − |t′ | + |t|α − |t′ |

≤ C5 εα∧1 + C6 ǫ, u → ∞
→ 0, ε → 0, ǫ → 0,

where in (37) we use (34) and (38) follows from (33) and (35).
Hence the proof follows from [52][Theorem 2.1].



Lemma 4.2. Let Zu (s, t), (s, t) ∈ R2 be a centered stationary Gaussian field with unit variance and correlation function

rZu (·, ·) satisfying

1 − rZu (s, t) = au

(39)

−2



s
←
−
ρ (u−1 )

α/2

t
+ ←
−
ρ (u−1 )

α/2



,

(s, t) ∈ R2 ,

with a > 0, ρ2 ∈ Rα and α ∈ (0, 2]. Let Ku be some index sets. Then, for Mk (u), k ∈ Ku satisfying (29) and for any

S1 , S2 , T1 , T2 ≥ 0 such that max(S1 , S2 ) > 0, max(T1 , T2 ) > 0, we have
(
)
1
P
sup Zu (s, t) > Mk (u) − F (S1 , S2 , T1 , T2 ) = 0,
lim sup
u→∞ k∈Ku Ψ(Mk (u))
(s,t)∈D(u)
−
−
−
−
where D(u) = [−←
ρ (u−1 )S1 , ←
ρ (u−1 )S2 ] × [−←
ρ (u−1 )T1 , ←
ρ (u−1 )T2 ] and
F (S1 , S2 , T1 , T2 ) = Hα/2 [−a2/α S1 , a2/α S2 ]Hα/2 [−a2/α T1 , a2/α T2 ].
Proof of Lemma 4.2: The proof follows by checking the conditions of [35][Lemma 5.3].
For D = [−S1 , S2 ] × [−T1 , T2 ] we have
(
P

sup

)

Zu (s, t) > Mk (u)

(s,t)∈Du

=P

(

)

−
−
sup Zu (←
ρ (u−1 )s, ←
ρ (u−1 )t) > Mk (u) .

(s,t)∈D

Since by (39)
−
−
−
−
Var(Zu (←
ρ (u−1 )s, ←
ρ (u−1 )t) − Zu (←
ρ (u−1 )s′ , ←
ρ (u−1 )t′ ))

we obtain
(40)

lim sup

sup

u→∞ k∈Ku (s,t)6=(s′ ,t′ )∈D

Mk2 (u)


−
−
= 2 − 2rZu ←
ρ (u−1 )(s − s′ ), ←
ρ (u−1 )(t − t′ )


α/2
α/2
+ |t − t′ |
= au−2 |s − s′ |

−
−
−
−
Var(Zu (←
ρ (u−1 )s, ←
ρ (u−1 )t) − Zu (←
ρ (u−1 )s′ , ←
ρ (u−1 )t′ ))
− 1 = 0.
′
α/2
′
α/2
2a(|s − s |
+ |t − t | )

Further, since for α/2 ∈ (0, 1]
|t|α/2 − |t′ |

α/2

≤ C1 |t − t′ |

α/2

,

|s|α/2 − |s′ |

α/2

≤ C2 |s − s′ |

α/2

holds for t, t′ ∈ [−T1 , T2 ], s, s′ ∈ [−S1 , S2 ], we have by (39)

−
−
−
−
Mk2 (u)E [Zu (←
ρ (u−1 )s, ←
ρ (u−1 )t) − Zu (←
ρ (u−1 )s′ , ←
ρ (u−1 )t′ )]Zu (0, 0)
sup
sup
k∈Ku |(s,t)−(s′ ,t′ )|<ε
(s,t),(s′ ,t′ )∈D

EXTREMES OF THRESHOLD-DEPENDENT GAUSSIAN PROCESSES

≤ C3 u 2

11

−
−
−
−
ρ (u−1 )s′ , ←
ρ (u−1 )t′ )
ρ (u−1 )s, ←
ρ (u−1 )t) − rZu (←
rZu (←

sup
|(s,t)−(s′ ,t′ )|<ε
(s,t),(s′ ,t′ )∈D

= C3 a

≤ C3 a

sup
|(s,t)−(s′ ,t′ )|<ε
(s,t),(s′ ,t′ )∈D

sup
|(s,t)−(s′ ,t′ )|<ε
(s,t),(s′ ,t′ )∈D

|s|α/2 + |t|α/2 − |s′ |


|s|

α/2

− |s′ |

α/2

α/2

+ |t|

− |t′ |

α/2

α/2

− |t′ |



α/2

≤ C4 εα/2 → 0, u → ∞, ε → 0.
Hence the claim follows from [35][Lemma 5.3].



Proof of Theorem 2.2: We have from A3
ρ2 (t)
−
ρ (u−1 ) = η −λ/2 .
= η ∈ [0, ∞],
lim uλ ←
u→∞
t→0 t2/λ
Without loss of generality, we consider only the case tu = 0 for u large enough.
lim

By A2 for t ∈ ∆(u), for sufficiently large u,
1

(41)

Fu,+ε (t)

≤ σu (t) ≤

1

Fu,−ε (t)



Fu,±ε (t) = 1 + u−2 (1 ± ε)f (uλ t) ± ε

,

for small constant ε ∈ (0, 1). Since further
(

(42)

π(u) := P

sup Xu (t) > Mu

t∈∆(u)

we have

(

Xu (t)
π(u) ≤ P
sup
> Mu
F
t∈∆(u) u,−ε (t)

)

)

,

(

=P

sup Xu (t)σu (t) > Mu

t∈∆(u)

)

(

Xu (t)
π(u) ≥ P
sup
> Mu
F
t∈∆(u) u,+ε (t)

)

.

Set for some positive constant S
−
−
Ik (u) = [k ←
ρ (u−1 )S, (k + 1)←
ρ (u−1 )S], k ∈ Z.
Further, define


x1 (u)
− I{x1 ≤0} ,
Gu,+ε (k) = Mu sup Fu,+ε (s), N1 (u) =
−
S←
ρ (u−1 )
s∈Ik (u)


x2 (u)
Gu,−ε (k) = Mu inf Fu,−ε (s), N2 (u) =
+ I{x2 ≤0} .
−
S←
ρ (u−1 )
s∈Ik (u)


In view of [54], we can find centered stationary Gaussian processes Y±ε (t), t ∈ R with continuous trajectories, unit
variance and correlation function satisfying

r±ε (t) = 1 − (1 ± ε)ρ2 (|t|)(1 + o(1)), t → 0.
Case 1) η = ∞:

For any u positive
N2 (u)−1

(43)

X

k=N1 (u)+1

P

(

sup Xu (t) > Mu
t∈Ik (u)

where

)

N2 (u)

Λ1 (u) =

X

k=N1 (u)

and
Λ2 (u) =

X

P

−

(

2
X
i=1

Λi (u) ≤ π(u) ≤

sup Xu (t) > Mu ,
t∈Ik (u)

N1 (u)≤k,l≤N2 (u),l≥k+2

Set below

N2 (u)

P

X

P

k=N1 (u)

sup

(

Xu (t) > Mu

t∈Ik+1 (u)

(

sup Xu (t) > Mu

t∈Ik (u)

)

,

sup Xu (t) > Mu , sup Xu (t) > Mu

t∈Ik (u)

Hα
, Θ(u) = λ ←
u −
ρ (u−1 )

Z

x2
x1

t∈Il (u)

e−f (t) dtΨ(Mu ).

)

.

)

,

12

LONG BAI, KRZYSZTOF DȨBICKI, ENKELEJD HASHORVA, AND LANPENG JI

which is well-defined since

R x2
x1

e−f (t) dt < ∞ follows by the assumption f ∈ C0∗ ([x1 , x2 ]). By Slepian inequality (see

e.g., [55]), (42) and Lemma 4.1
)
(
N2 (u)
X
P
sup Xu (t) > Mu
k=N1 (u)

N2 (u)

≤

t∈Ik (u)

X

P

N2 (u)

≤

P

N2 (u)

=

P

k=N1 (u)

sup Xu (t) > Gu,−ε (k)

(

)

sup Y+ε (t) > Gu,−ε (k)

t∈Ik (u)

k=N1 (u)

X

)

t∈Ik (u)

k=N1 (u)

X

(

(

)

sup Y+ε (t) > Gu,−ε (k)

t∈I0 (u)

N2 (u)

∼

X

k=N1 (u)

Hα [0, (1 + ε)1/α S]Ψ(Gu,−ε (k))
N2 (u)

X

∼ Hα [0, (1 + ε)1/α S]Ψ(Mu )
Z

1/α

∼

Hα [0, (1 + ε) S]
−
Suλ ←
ρ (u−1 )

∼ Θ(u),

(44)

x2

2

−2

e−Mu u

inf s∈Ik (u) [(1−ε)f (uλ s)−ε]

k=N1 (u)

e−(1−ε)f (t)+ε dtΨ(Mu )

x1

u → ∞, S → ∞, ε → 0.

Similarly, we derive that
N2 (u)−1

X

(45)

P

k=N1 (u)+1

(

sup Xu (t) > u
t∈Ik (u)

)

≥ (1 + o(1))Θ(u), u → ∞, S → ∞, ε → 0.

Moreover,
N2 (u)

Λ1 (u) ≤

X

P

(

≤
(46)

X

sup

k=N1 (u)



sup Y+ε (t) > Gbu,−ε (k)

t∈Ik (u)∪Ik+1 (u)

N2 (u)

≤

)

t∈Ik (u)

k=N1 (u)

−P

(



+P

(

sup
t∈Ik+1 (u)

)!

)

Y+ε (t) > Gbu,−ε (k)

Y−ε (t) > G u,+ε (k)


2Hα [0, (1 + ε)1/α S] − Hα [0, 2(1 − ε)1/α S] Ψ(Gbu,−ε (k))

2Hα [0, (1 + ε)1/α S] − Hα [0, 2(1 − ε)1/α S]



= o(Θ(u)), u → ∞, S → ∞, ε → 0,

N2 (u)

X

k=N1 (u)

Ψ(Gbu,−ε (k))

where
Gbu,−ε (k) = min(Gu,−ε (k), Gu,−ε (k + 1)),

G u,+ε (k) = max(Gu,+ε (k), Gu,+ε (k + 1)).

By A3 for any (s, t) ∈ Ik (u) × Il (u) with N1 (u) ≤ k, l ≤ N2 (u), l ≥ k + 2 we have


2 ≤ V ar Xu (s) + Xu (t) = 4 − 2(1 − ru (s, t)) ≤ 4 − ρ2 (|t − s|) ≤ 4 − C1 u−2 |(l − k − 1)S|α/2

and for (s, t), (s′ , t′ ) ∈ Ik (u) × Il (u) with N1 (u) ≤ k, l ≤ N2 (u)


′
′
Xu (s) + Xu (t)
Xu (s ) + Xu (t )

1 − Cov  q
, q

V ar Xu (s) + Xu (t)
V ar Xu (s′ ) + Xu (t′ )

2 




′
′
Xu (s) + Xu (t)
Xu (s ) + Xu (t )
1


q
−q
= E



2 


V ar X (s) + X (t)
V ar X (s′ ) + X (t′ )
u

u

u

u

EXTREMES OF THRESHOLD-DEPENDENT GAUSSIAN PROCESSES

13

n
2 o
1
 E Xu (s) − Xu (s′ ) + Xu (t) − Xu (t′ )
V ar Xu (s) + Xu (t)

2

1
1

+V ar Xu (s′ ) + Xu (t′ )  q
−q

′
′
V ar Xu (s) + Xu (t)
V ar Xu (s ) + Xu (t )
o
n
o
n
n


2 o
2
2
+ 2E Xu (t) − Xu (t′ )
+ E Xu (s) − Xu (s′ ) + Xu (t) − Xu (t′ )
≤ 2E Xu (s) − Xu (s′ )

=

≤ 8(1 − ru (s, s′ ) + 1 − ru (t, t′ ))

t − t′
s − s′ α/2
−2
+
= 16u
←
−
←
−
ρ (u−1 )
ρ (u−1 )

α/2



.

In view of our assumptions, we can find centered homogeneous Gaussian random fields Zu (s, t) with correlation
!!
α/2
α/2
t
s
−2
.
+ ←
rZu (s, t) = exp −32u
←
−
−
ρ (u−1 )
ρ (u−1 )
Slepian inequality, Lemma 4.2 and (44) imply
Λ2 (u) ≤
≤
≤
≤
≤

X

P

X

P

N1 (u)≤k,l≤N2 (u),l≥k+2

N1 (u)≤k,l≤N2 (u),l≥k+2

X

(
(

sup Xu (s) > Mu , sup Xu (t) > Mu
s∈Ik (u)

t∈Il (u)

sup
(s,t)∈Ik (u)×Il (u)

)

)
e
(Xu (s) + Xu (t)) > 2Gu,−ε (k, l)
2Geu,−ε (k, l)

Zu (s, t) > p
4 − C1 u−2 |(l − k − 1)S|α/2
N1 (u)≤k,l≤N2 (u),l≥k+2
!

2
X
2Geu,−ε (k, l)
2/α
Hα/2 [0, 32 S] Ψ p
4 − C1 u−2 |(l − k − 1)S|α/2
N1 (u)≤k,l≤N2 (u),l≥k+2
!
N2 (u) N2 (u)−N1 (u) 
2
X
X
2Gu,−ε (k)
2/α
Hα/2 [0, 32 S] Ψ p
2
4 − C1 u−2 (lS)α/2
l=1
k=N1 (u)
N2 (u)

≤

(

2

X

k=N1 (u)

P

sup

(s,t)∈I0 (u)×I0 (u)

)

∞

2
X
α/2
Hα/2 [0, 322/α S] Ψ (Gu,−ε (k))
e−C2 (lS)
l=1

N2 (u)

(47)

≤

2Hα/2 322/α Se−C3 S

=

o(Θ(u)),

α/2

X

k=N1 (u)

Hα/2 [0, 322/α S]Ψ (Gu,−ε (k))

u → ∞, S → ∞, ε → 0,

where Geu,−ε (k, l) = min(Gu,−ε (k), Gu,−ε (l)). Combing (43)-(46) with (47), we obtain
π(u) ∼ Θ(u),

u → ∞.

Case 2) η ∈ (0, ∞): This implies λ = 2/α.

Set for any small constant θ ∈ (0, 1) and any constant S1 > 0
(
(
(x2 − θ)η 1/α ,
−S
,
if
x
=
−∞;
1
1
(48)
S2∗ =
S1∗ =
S1 ,
(x1 + θ)η 1/α , if x1 ∈ (−∞, ∞),
(49)

S1∗∗

=

(

−S,

(x1 − θ)η 1/α ,

if x1 = −∞;

if x1 ∈ (−∞, ∞),

S2∗∗

=

(

(x2 + θ)η 1/α ,
S,

if x2 ∈ (−∞, ∞);
if x2 = ∞,

if x2 ∈ (−∞, ∞);
if x2 = ∞.

−
−
−
−
With K ∗ = [←
ρ (u−1 )S1∗ , ←
ρ (u−1 )S2∗ ] and K ∗∗ = [←
ρ (u−1 )S1∗∗ , ←
ρ (u−1 )S2∗∗ ] we have for any S1 > 0 and u large enough


(50)
π(u) ≥ P sup Xu (t) > Mu ,
t∈K ∗

14

LONG BAI, KRZYSZTOF DȨBICKI, ENKELEJD HASHORVA, AND LANPENG JI

π(u) ≤ P

(51)



sup Xu (t) > Mu

t∈K ∗∗



N2 (u)

+

X

P

k=N1 (u)
k6=0,−1

(

sup Xu (t) > Mu

t∈Ik (u)

)

.

Using Slepian inequality and Lemma 4.1, we have that




Y−ε (t)
≥ P sup
P sup Xu (t) > Mu
> Mu
t∈K ∗ Fu,+ε (t)
t∈K ∗
h

+ε
∼ Pα,1
[S1∗ , S2∗ ]Ψ(Mu ), u → ∞,

where h±ε (t) = (1 ± ε)f (η −1/α t) ± ε, and similarly


P sup Xu (t) > Mu



Y+ε (t)
> Mu
≤ P sup
t∈K ∗∗ Fu,−ε (t)

t∈K ∗∗

h



−ε
∼ Pα,1
[S1∗∗ , S2∗∗ ]Ψ(Mu ), u → ∞.

(52)

Moreover, in light of (6), the Slepian inequality and Lemma 4.1
(
)
(
)
N2 (u)
N2 (u)
X
X
Y+ε (t)
P
sup
≤
P
sup Xu (t) > Mu
> Mu
t∈Ik (u) Fu,−ε (t)
t∈Ik (u)
k=N1 (u)
k6=−1,0

k=N1 (u)
k6=−1,0

N2 (u)

≤

X

P

(

)

sup Y+ε (t) > Gu,−ε (k)

t∈I0 (u)

k=N1 (u)
k6=−1,0
N2 (u)

∼

X

k=N1 (u)
k6=−1,0

Hα [0, (1 + ε)1/α S]Ψ (Gu,−ε (k))
N2 (u)

(53)

X

e− inf s∈[k,k+1] ((1−ε)f (sη

∼

Hα [0, (1 + ε)1/α S]Ψ(Mu )

∼

C4 Hα Ψ(Mu )Se−C5 (η

=

o (Ψ(Mu )) , u → ∞, S → ∞, ε → 0.

−1/α

S)−ε)

k=N1 (u)
k6=−1,0

−1/α

S)ǫ1 /2 ε

e

Letting ε → 0, S1 → ∞, S → ∞, and θ → 0 we obtain
f
π(u) ∼ Pα,η
[x1 , x2 ]Ψ(Mu ),
λ
λ
Next, if we set x1 (u) = − lnuu , x2 (u) = lnuu , then

x1 = −∞,

x2 = ∞,

S1∗ = −S1 ,

u → ∞.

S2∗ = S1 ,

S1∗∗ = −S,

S2∗∗ = S.

Inserting (52), (53) into (51) and letting ε → 0 leads to
lim

u→∞

−1/α
π(u)
f
S)ǫ1 /2
≤ Pα,η
[−S, S] + C4 Hα Se−C5 (η
< ∞.
Ψ(Mu )

By (50), we have
lim

u→∞

π(u)
f
≥ Pα,η
[−S1 , S1 ] > 0.
Ψ(Mu )

Letting S1 → ∞, S → ∞ we obtain
f
Pα,η
(−∞, ∞) ∈ (0, ∞),

Case 3) η = 0: Note that
(
π(u) ≤ P

sup

t∈((I−1 (u)∪I0 (u))∩∆(u))

f
π(u) ∼ Pα,η
(−∞, ∞)Ψ(Mu ),

Xu (t)σu (t) > Mu

)

N2 (u)

+

X

k=N1 (u)
k6=−1,0

P

(

u → ∞.

sup Xu (t)σu (t) > Mu
t∈Ik (u)

)

=: J1 (u) + J2 (u).

EXTREMES OF THRESHOLD-DEPENDENT GAUSSIAN PROCESSES

15

By (41)
1
1
1
≤ σu (t) ≤
≤
Fu,+ε (t)
Fu,−ε (t)
1 + u−2 inf s∈∆(u) [(1 − ε)f (uλ s) − ε]

(54)

holds for all t ∈ ∆(u). Hence Lemma 4.1 implies
(
J1 (u) ≤

sup

P

−
−
t∈[−←
ρ (u−1 )S,←
ρ (u−1 )S]

(


)
−2
λ
inf [(1 − ε)f (u s) − ε]
Xu (t) > Mu 1 + u
s∈∆(u)


)
Y+ε (t) > Mu 1 + u−2 inf [(1 − ε)f (uλ s) − ε]

≤

P

∼




−2
λ
1/α
inf [(1 − ε)f (u s) − ε]
Hα [0, 2(1 + ε) S]Ψ Mu 1 + u

sup

−
−
t∈[−←
ρ (u−1 )S,←
ρ (u−1 )S]

s∈∆(u)

s∈∆(u)

∗

∼

Hα [0, 2(1 + ε)1/α S]Ψ (Mu ) e−(1−ε)ω

+ε

∼

Ψ (Mu ) e−ω , u → ∞, S → 0, ε → 0,

∗

where ω ∗ = inf t∈[x1 ,x2 ] f (t). Furthermore, by Lemma 4.1, for any x > 0
(
)
N2 (u)
N2 (u)
X
X
Hα [0, (1 + ε)1/α S]Ψ (Gu,−ε (k))
P
sup Y+ε (t) > Gu,−ε (k) ∼
J2 (u) ≤
k=N1 (u)
k6=−1,0

(55)

t∈I0 (u)

≤

2Hα [0, (1 + ε)1/α S]Ψ(Mu )

≤

(xS)ǫ1 /2

∞
X

k=N1 (u)
k6=−1,0

e−(1−2ε)(kxS)

ǫ1 /2
+2ε

k=1

C6 Hα Ψ(Mu )Se−C7

= o (Ψ(Mu )) , u → ∞, x → ∞, S → 0,

hence
lim

u→∞

∗
π(u)
≤ e−ω ,
Ψ(Mu )

u → ∞.

Next, since f ∈ C0∗ ([x1 , x2 ]) there exists y(u) ∈ ∆(u) satisfying
lim y(u)uλ = y ∈ {z ∈ [x1 , x2 ] : f (z) = ω ∗ }.

u→∞

Consequently, in view of (54)
π(u)

≥ P {Xu (y(u)) > Mu }

≥ P X u (y(u)) > Mu (1 + [(1 + ε)f (uλ y(u)) + ε]u−2 )

= Ψ Mu (1 + (1 + ε)[f (uλ y(u)) + ε]u−2 )
∼ Ψ (Mu ) e−f (y) , u → ∞, ε → 0,

which implies that
∗

π(u) ∼ Ψ (Mu ) e−ω ,

u→∞

establishing the proof.



Proof of Theorem 2.4: Clearly, for any u > 0
(
π(u) ≤ P

where with D(u) := [0, T ] \ (tu + ∆(u)),
(
π(u) := P

sup Xu (t) > Mu

t∈[0,T ]

sup Xu (tu + t) > Mu

t∈∆(u)

)

)

≤ π(u) + π1 (u),

, π1 (u) := P

(

Next, we derive an upper bound for π1 (u) which will finally imply that
(56)

π1 (u) = o(π(u)),

sup Xu (t) > Mu

t∈D(u)

u → ∞.

)

.

16

LONG BAI, KRZYSZTOF DȨBICKI, ENKELEJD HASHORVA, AND LANPENG JI

Thus by A4, A5 and Piterbarg inequality (see e.g., [10][Theorem 8.1], [56][Theorem 3] and [35][Lemma 5.1])
(
)
π1 (u) = P

sup X u (t)σu (t) > Mu

(

t∈D(u)

p(ln u)q
≤ P
sup X u (t) > Mu + C1
u
t∈D(u)


p(ln u)q
≤ C2 T Mu2/ς Ψ Mu + C1
u

(57)

)

u → ∞.

= o (Ψ (Mu )) ,

Since A1’ implies A1, by Theorem 2.2 and A2, A3, we have

R x2 −f (t)
Hα

e
dt, if η = ∞,
−

ρ (u−1 ) x1
 uλ ←
f
π(u) ∼ Ψ (Mu )
(58)
Pα,η
[x1 , x2 ],
if η ∈ (0, ∞),


 1,
if η = 0,

u → ∞,

where the result of case η = 0 comes from the fact that f (t) ≥ 0 for t ∈ [x1 , x2 ], f (0) = 0 and 0 ∈ [x1 , x2 ].

Consequently, it follows from (57) and (58) that (56) holds, and thus the proof is complete.



Proof of Proposition 3.1: Without loss of generality we assume that gm = g(t0 ) = 0.
1/γ

q
with some large q > 1.
i) We present first the proof for t0 ∈ (0, T ). Let ∆(u) = [−δ(u), δ(u)], where δ(u) = (lnuu)
By (4) for u large enough and some small ε ∈ (0, 1)
γ

(59)

1+

1
u − g(t + t0 )
g(t + t0 )
(1 + ε)c |t|
(1 − ε)c |t|
≤
:=
=1−
≤1+
u
σu (t + t0 )
u
u
u

holds for all t ∈ [−θ, θ], θ > 0. It follows that
(
Π(u) ≤ P

sup (X(t) + g(t)) > u

t∈[0,T ]

)

γ

≤ Π(u) + Π1 (u),

with
Π1 (u) := P

(

sup

)

(X(t) + g(t)) > u ,

t∈([0,T ]\[t0 −θ,t0 +θ]

and
Π(u) := P

(

sup

(X(t) + g(t)) > u

t∈[t0 −θ,t0 +θ]

)

(

)
u
=P
sup
X(t)
>u .
u − g(t)
t∈[t0 −θ,t0 +θ]

By (59), we may further write
(60)

lim

1
σu (t0 +t) −
cu−1 |t|γ

sup

u→∞ t∈∆(u),t6=0

1

− 1 = lim

sup

u→∞ t∈∆(u),t6=0

1
σu (t0 +t) − 1
cu−2 |u1/γ t|γ

− 1 = 0,

and
1
(1 − ε)c(ln u)q
.
≥1+
u2
t∈[−θ,θ]\∆(u) σu (t + t0 )
inf

In addition, from (14) we have that
lim

sup

u→∞ s,t∈∆(u)
t6=s

1 − r(t0 + t, t0 + s)
− 1 = 0,
a|t − s|α

and
sup
s,t∈[t0 −θ,t0 +θ]


E X(t) − X(s))2 ≤

sup

(2 − 2r(s, t)) ≤ C1 |t − s|α

s,t∈[t0 −θ,t0 +θ]

EXTREMES OF THRESHOLD-DEPENDENT GAUSSIAN PROCESSES

17

hold when θ is small enough. Therefore, by Theorem 2.4

γ
1 R∞

Hα a α wt e−c|t| dt, if α < 2γ,


0
1
2
c|t|γ
Π(u) ∼ u( α − γ )+ Ψ (u)
if α = 2γ,
Pα,a [wt0 , ∞),


 1,
if α > 2γ.

Moreover, since gθ := supt∈[0,T ]\[t0 −θ,t0 +θ] g(t) < 0 we have
)
(
Z
Π1 (u) ≤ P
sup
X(t) > u − gθ ∼ Hα
t∈[0,T ]\[t0 −θ,t0 +θ]

T

0

2
1
dt u α Ψ (u − gθ ) = o(Π(u)), u → ∞,
a(t)

hence the claims follow.
For t0 = 0 and t0 = T , we just need to replace ∆(u) by ∆(u) = [0, δ(u)] and ∆(u) = [−δ(u), 0], respectively.
ii) Applying [10][Theorem 7.1] we obtain
(
)
sup (X(t) + g(t)) > u

P

(

=P

t∈[A,B]

sup X(t) > u
t∈[A,B]

)

∼

Z

P

≥

sup (X(t) + g(t)) > u

t∈[0,T ]

(

sup (X(t) + g(t)) > u

t∈[0,T ]

)

)

sup (X(t) + g(t)) > u ,

P

t∈[A,B]

≤

P





sup (X(t) + g(t)) > u + P

t∈∆ε

Since g is a continuous function and gε := supt∈[0,T ]\∆ε g(t) < 0
(
)
(
sup

P

sup (X(t) + g(t)) > u

t∈∆ε



sup

(X(t) + g(t)) > u .

)

≤



C2 u2/α Ψ(u − gε ) = o u2/α Ψ(u) ,

≤

P

∼

Z

Hence the claims follow.

sup

t∈[0,T ]\∆ε



X(t) > u − gε

 Z
sup X(t) > u ∼

t∈∆ε

B

)

t∈[0,T ]\∆ε

P

Further, we have


(

≤

(X(t) + g(t)) > u

t∈[0,T ]\∆ε

P

2

(a(t))1/α dtHα u α Ψ (u) .

A

Set ∆ε = [A − ε, B + ε] ∩ [0, T ] for some ε > 0, then we have
(
)
(
P

B

B+ε

u → ∞, ε → 0.

2

1

(a(t)) α dtHα u α Ψ(u)

A−ε

1

2

(a(t)) α dtHα u α Ψ(u),

A

u → ∞, ε → 0.


Proof of Proposition 3.3: We give the proof only for t0 = 0. In this case, x ∈ (0, ∞). By definition
n
o
n
o P supt∈[0,u−1/γ x] (X(t) + g(t)) > u
n
o .
P u1/γ (τu − t0 ) ≤ x τu ≤ T =
P supt∈[0,T ] (X(t) + g(t)) > u

Set ∆(u) = [0, u−1/γ x]. For all u large
(
P

sup (X(t) + g(t)) > u

t∈∆(u)

u
and σu (t) =
Denote Xu (t) = X(t) u−g(t)

)

(

)
u
=P
sup X(t)
>u .
u − g(t)
t∈∆(u)

u
u−g(t) .

As in the proof


(
)


2
(α
− γ1 )+
P
sup (X(t) + g(t)) > u ∼ u
Ψ (u)

t∈∆(u)



of Proposition 3.1 i), by Theorem 2.2 we obtain
1

a α Hα
c|t|

γ

Rx
0

Pα,a [0, x],
1,

γ

e−c|t| dt, if α < 2γ,
if α = 2γ,
if α > 2γ.

Consequently, by Proposition 3.1 statement i), the results follow.
Proof of Proposition 3.6: Clearly, for any u > 0
(
)
P

sup (X(t) + g(t)) > u

t∈[0,T ]

(

mu (t)
u − g(tu )
>
= P sup X(t)
mu (tu )
σ(tu )
t∈[0,T ]


)

,

18

LONG BAI, KRZYSZTOF DȨBICKI, ENKELEJD HASHORVA, AND LANPENG JI

and A1’ is satisfied. By the continuity of σ(t), limu→∞ tu = t0 and σ(t0 ) = 1, we have that for u large enough
u − g(tu )
∼ u, u → ∞.
σ(tu )

σ(tu ) > 0, and
Set next
Xu (t) =X(t)
which has standard deviation function σu (t) =

mu (t)
,
mu (tu )

mu (tu +t)
mu (tu )

t ∈ [0, T ],

and correlation function ru (s, t) = r(s, t) satisfying assump-

tions A2–A4. Further, X u (t) = X(t) implies A5. Hence the claims follow from Theorem 2.4.



Proof of Proposition 3.8: For all u large

E [X(tu + t) − X(tu + s)]2 − [σ(tu + t) − σ(tu + s)]2
1 − r(tu + t, tu + s) =
(61)
.
2σ(tu + t)σ(tu + s)
Using that

E [X(tu + t) − X(tu + s)]2

[σ(tu + t) − σ(tu + s)]

we have, as u → ∞

= σ ′2 (tu + t)(t − s)2 + o((t − s)2 ),


E X ′2 (tu + t) − σ ′2 (tu + t)
(t − s)2 + o((t − s)2 ).
1 − r(tu + t, tu + s) =
2σ(tu + t)σ(tu + s)

E{X ′2 (t)}−σ′2 (t)
2σ(s)σ(t)

Since D(s, t) :=

2


= E X ′2 (tu + s) (t − s)2 + o((t − s)2 ),

is continuous at (t0 , t0 ), then setting D = D(t0 , t0 ) we obtain
lim

sup

u→∞ t∈∆(u),s∈∆(u)
t6=s

1 − r(tu + t, tu + s)
− 1 = 0,
D|t − s|2

which implies that A3 is satisfied. Next we suppose that σ(t) >
σ(t) ≤

1
2 },

by Borell-TIS inequality
P



sup (X(t) + g(t)) > u

t∈E1





2

E (X(t) − X(s))

for any t ∈ [0, T ], since if we set E1 = {t ∈ [0, T ] :

≤ exp −2 u − sup g(t) − C1
t∈[0,T ]

n
o
as u → ∞, where C1 = E supt∈[0,T ] X(t) < 0. Further by (61)


1
2

≤ 2 − 2r(t, s) ≤ 4

!2 

 

 = o Ψ u − g(tu )
σ(tu )


sup E X ′2 (θ) (t − s)2 − inf σ ′2 (θ)(t − s)2

θ∈[0,T ]

θ∈[0,T ]

!

,

then A5 is satisfied. Consequently, the conditions of Proposition 3.6 are satisfied and hence the claim follows.



Proof of Proposition 3.9: Without loss of generality we assume that g(t) satisfies (4) with g(t0 ) = 0.
First we present the proof for t0 ∈ (0, T ). Clearly, mu attains its maximum at the unique point t0 . Further, we have
1
g(t0 + t)
mu (t0 )
−1=
(1 − σ(t0 + t)) −
.
mu (t0 + t)
σ(t0 + t)
uσ(t0 + t)
Consequently, by (2) and (4)

mu (t0 )
c γ
β
= 1 + b |t| + |t| (1 + o(1)), t → 0
mu (t0 + t)
u
2/β ∗

q
holds for all u large. Further, set ∆(u) = [−δ(u), δ(u)], where δ(u) = (lnuu)
for some constant q > 1 with
(62)

β ∗ = min(β, 2γ), and let f (t) = b|t|β I{β=β ∗ } + c|t|γ I{2γ=β ∗ } . We have


mu (t0 )
2
2/β ∗
t)
mu (t0 +t) − 1 u − f (u
(63)
= 0.
lim
sup
u→∞ t∈∆(u),t6=0
f (u2/β ∗ t) + I{β6=2γ}
By (2)
(64)





E (X(t) − X(s))2 = E (X(t))2 + E (X(s))2 − 2E X(t)X(s) = 2 − 2r(s, t) ≤ C1 |t − s|α

EXTREMES OF THRESHOLD-DEPENDENT GAUSSIAN PROCESSES

19

holds for s, t ∈ [t0 − θ, t0 + θ], with θ > 0 sufficiently small. By (62), for any ε > 0
(ln u)q
mu (t0 )
≥ 1 + C2 (1 − ε)
mu (t0 + t)
u

(65)
holds for all t ∈ [−θ, θ] \ ∆(u). Further
(
Π(u) := P

sup

(X(t) + g(t)) > u

t∈[t0 −θ,t0 +θ]

with
Π1 (u) := P

(

)

≤P

(

sup (X(t) + g(t)) > u

t∈[0,T ]

sup

)

≤ Π(u) + Π1 (u),

)

(X(t) + g(t)) > u .

t∈([0,T ]\[t0 −θ,t0 +θ])

By(63), (2), (65), (64) which imply A2–A5 and Proposition 3.6, we have

R∞

Hα a1/α wt e−f (t) dt,


0
2
2
f
Π(u) ∼ u( α − β∗ )+ Ψ (u)
(66)
Pα,a
[wt0 , ∞),


 1,

if α < β ∗ ,
if α = β ∗ ,
if α > β ∗ .

In order to complete the proof it suffices to show that

Π1 (u) = o(Π(u)).
Since σθ := maxt∈([0,T ]\[t0 −θ,t0 +θ]) σ(t) < 1 , by the Borell-TIS inequality we have
(
)


(u − C3 )2
= o(Π(u)),
Π1 (u) ≤ P
sup
X(t) > u ≤ exp −
2σθ2
t∈([0,T ]\[t0 −θ,t0 +θ])
n
o
where C3 = E supt∈[0,T ] X(t) < ∞.

For the cases t0 = 0 and t0 = T , we just need to replace ∆(u) by [0, δ(u)] and [−δ(u), 0], respectively. Hence the proof
is complete.



Proof of Proposition 3.10: i) We shall present the proof only for the case t0 ∈ (0, T ). In this case, [x1 , x2 ] = R. By

definition, for any x ∈ R

n
o
P
sup
(X(t)
+
g(t))
>
u
−λ

t∈[0,tu +u x]
n
o .
P uλ (τu − tu ) ≤ x τu ≤ T =
P supt∈[0,T ] (X(t) + g(t)) > u

For u > 0 define

Xu (t) = X(tu + t)
As in the proof of Proposition 3.6, we obtain
(
P

sup

mu (tu + t)
,
mu (tu )

(X(t) + g(t)) > u

t∈[0,tu +u−λ x]

)

σu (t) =

mu (tu + t)
.
mu (tu )

(

u − g(tu )
=P
sup
Xu (t) >
σ(tu )
t∈[0,tu +u−λ x]

)

,

and A1’, A2–A5 are satisfied with ∆(u) = [−δu , u−λ x]. Clearly, for any u > 0
(
)
u − g(tu )
π(u) ≤ P
sup
Xu (t) >
≤ π(u) + π1 (u),
σ(tu )
t∈[0,tu +u−λ x]
where

(

u − g(tu )
π(u) = P
sup
Xu (t) >
σ(tu )
−λ
t∈[tu −δ(u),tu +u x]

)

,

(

u − g(tu )
π1 (u) = P
sup
Xu (t) >
σ(tu )
t∈[0,tu −δ(u)]

Applying Theorem 2.2 we have

(67)

π(u) ∼ Ψ



u − g(tu )
σ(tu )









R x −f (t)
Hα
e
dt,
−
uλ ←
ρ (u−1 ) −∞
f
Pα,η (−∞, x],
supt∈(−∞,x] e−f (t) ,

if η = ∞,

if η ∈ (0, ∞),

if η = 0.

)

.

20

LONG BAI, KRZYSZTOF DȨBICKI, ENKELEJD HASHORVA, AND LANPENG JI

In view of (57)

 
u − g(tu )
,
π1 (u) = o Ψ
σ(tu )
hence
P

(

sup

(X(t) + g(t)) > u

t∈[0,tu +u−λ x]

)

u → ∞,

∼ π(u),

u→∞

and thus the claim follows by (67) and Proposition 3.6.
ii) We give the proof of t0 = T . In this case x ∈ (−∞, 0) implying
n
o
∗
n
o
P
sup
(X(t)
+
g(t))
>
u
−2/β
t∈[0,T +u
x]
∗
n
o
.
P u2/β (τu − T ) ≤ x τu ≤ T =
P supt∈[0,T ] (X(t) + g(t)) > u
Set δu =



(ln u)q
u

2/β ∗

for some q > 1 and let
∗

∆(u) = [−δu , u−2/β x],

σu (t) =

mu (t)
,
mu (T )

with
mu (t) =
For all u large, we have
(
π(u) ≤ P

sup

σ(t)
,
1 − g(t)/u

(X(t) + g(t)) > u

t∈[0,T +u−2/β ∗ x]

where
π(u) := P

(

Xu (t) = X(t)

)

≤ π(u) + P

sup (X(T + t) + g(T + t)) > u
t∈∆(u)

)

(

=P

mu (t)
.
mu (T )

sup

)

(X(t) + g(t)) > u ,

t∈[0,T −δu ]

(

)

sup Xu (T + t) > u .
t∈∆(u)

∗

As in the proof of Proposition 3.9 it follows that the Assumptions A2–A5 hold with ∆(u) = [−δu , u−2/β x]. Hence
an application of Theorem 2.2 yields


R∞

a1/α Hα −x e−f (t) dt, if α < β ∗ ,


2
2
f
π(u) ∼ u( α − β∗ )+ Ψ (u)
Pα,a
[−x, ∞),
if α = β ∗ ,


 e−f (x) ,
if α > β ∗ .

(68)

In view of (57)
P

(

sup

(X(t) + g(t)) > u

t∈[0,T −δu ]

implying
P

(

sup

)

t∈[0,T +u−2/β ∗ x]

=P

(

sup

Xu (t) > u

t∈[0,T −δu ]

(X(t) + g(t)) > u

)

)

∼ π(u),

= o (Ψ (u)) ,

u→∞

u → ∞.

Consequently, the proof follows by (68) and Proposition 3.9.
Rt
Proof of Proposition 3.12: Set next A(t) = 0 e−δv dB(v) and define
Z t
e (t) = u + c
U
e−δv dv − σA(t), t ≥ 0.



0

Since


1
sup E [A(t)]2 =
2δ
t∈[0,∞)

e (∞) :=
implying supt∈[0,∞) E {|A(t)|} < ∞, then by the martingale convergence theorem in [57] we have that U
e (t) exists and is finite almost surely. Clearly, for any u > 0
limt→∞ U


e
p(u) = P
inf U (t) < 0
t∈[0,∞)

EXTREMES OF THRESHOLD-DEPENDENT GAUSSIAN PROCESSES

= P

(

(

21

)


Z t
−δv
σA(t) − c
e dv > u
sup

t∈[0,∞]

0

)


1
c
1
= P sup σA(− ln t) − (1 − t 2 ) > u .
2δ
δ
t∈[0,1]
The proof will follow by applying Proposition 3.6, hence we check next the assumptions therein for this specific model.
1
ln t) with variance function given by
Below, we set Z(t) = σA(− 2δ
!
Z − 2δ1 ln t
σ2
2
−δv
VZ (t) = V ar σ
(1 − t),
e dB(v) =
2δ
0

t ∈ [0, 1].

We show next that for u sufficiently large, the function
√
√σ
1−t
uVZ (t)
2δ
Mu (t) :=
,
=
c
Gu (t)
1 + δu (1 − t1/2 )
1

0 ≤ t ≤ 1,

with Gu (t) := u + δc (1 − t 2 ) attains its maximum at the unique point tu =
dMu (t)
[Mu (t)]t :=
dt

=



c
δu+c

2

. In fact, we have

"
#
1
u
dVZ2 (t)
dVZ (t)
u
VZ (t)  cu − 1 
ct− 2
2
− t 2 =
·
−
Gu (t) + VZ (t)
dt
Gu (t) G2u (t)
2δ
2G2u (t)Vz (t)
dt
δ

uσ 2 t−1/2 h c 
c 1i
t2 .
− u+
2
4δGu (t)VZ (t) δ
δ

2
c
Letting [Mu (t)]t = 0, we get tu = δu+c
. By (69), [Mu (t)]t > 0 for t ∈ (0, tu ) and [Mu (t)]t < 0 for t ∈ (tu , 1], so tu
(69)

=

is the unique maximum point of Mu (t) over [0, 1]. Further

σ
σu
= √ (1 + o(1)), u → ∞.
Mu := Mu (tu ) = √
2
2δu + 4cu
2δ
We set δ(u) =



(ln u)q
u

2

for some q > 1, and ∆(u) = [−tu , δ(u)]. Next we check the assumption A2. It follows that

[Gu (tu + t)VZ (tu )]2 − [Gu (tu )VZ (tu + t)]2
Mu
−1=
.
Mu (tu + t)
VZ (tu + t)Gu (tu )[Gu (tu + t)VZ (tu ) + VZ (tu + t)Gu (tu )]

We further write
[Gu (tu + t)VZ (tu )]2 − [Gu (tu )VZ (tu + t)]2
h
h
i2 σ 2
c c√
c  c √ i2 σ 2
= u+
−
−
(1 − tu ) − u +
(1 − tu − t)
tu + t
tu
δ
δ
2δ
δ
δ
2δ


√
c 2 σ 2
c2 σ 2
c  cσ 2 √
= u+
( tu + t − tu )(1 − tu ) −
t
t−2 u+
2
δ
2δ
δ 2δ
2δ 3


√ √
√
c 2 σ 2
c 2 σ 2
t(1 − tu ) − 2 u +
(1 − tu ) tu ( tu + t − tu )
= u+
δ
2δ
δ
2δ


√ 2
c 2  c 2 √
σ2 
u+
( t + tu − tu )
−
=
2δ
δ
δ


√
√
σ2
2c
2
=
u + u ( t + tu − tu )2 .
2δ
δ
Since for any t ∈ ∆(u)
r
r
c
cp
σ2
σ2
c
(1 − tu − δ(u)) ≤ VZ (tu + t) ≤
, u+ −
tu + δ(u) ≤ Gu (tu + t) ≤ u + ,
2δ
2δ
δ δ
δ
we have for all large u

VZ (tu + t)Gu (tu )[Gu (tu + t)VZ (tu ) + VZ (tu + t)Gu (tu )] ≤
and
VZ (tu + t)Gu (tu )[Gu (tu + t)VZ (tu ) + VZ (tu + t)Gu (tu )] ≥

c 2
σ2 
u+
δ
δ


2
σ2
c
cp
(1 − tu − δ(u)) u + −
tu + δ(u)
δ
δ
δ

22

LONG BAI, KRZYSZTOF DȨBICKI, ENKELEJD HASHORVA, AND LANPENG JI

≥
Thus as u → ∞
(70)

inf

t∈∆(u),t6=0

Mu /Mu (tu + t) − 1
−1≥
q
2
c2
1
c
2
−2
u
u t + δ2 − δ
2

2
2c
1u +δ u √
2 (u+ δc )2 ( t

1
2

q
t+

where we used the fact that for t ∈ ∆(u)

√
√
( t + tu − tu )2 ≥



σ2 
c 2
u+
−u .
δ
δ

+ tu −

c2
(δu)2

−

√ 2
tu )
u2 + 2c
δ u
2 − 1 ≥
c 2 − 1 → 0,
(u + δ )
c

δu

s

c
c2
−
t+
(δu)2
δu

!2

.

Furthermore, since
0

≤

=
we have as u → ∞
sup
t∈∆(u),t6=0

q
q
√
c
c2
c2
√
√
t + (δu)
t + (δu)
t + tu
2 + δu
2 −
t + tu − tu
q
√
√
√
√
−
1
=
−
1
≤
c
c2
t + tu + tu
t + tu + tu
t + (δu)
2 − δu
q
r
c2
c2
−
t
2
u
(δu)2 − tu
c 2
(δu)
√
q
− 1,
1
+
≤
=
√
√
√
δu
c2
tu
( t + tu + tu )( t + (δu)
t + tu )
2 +

Mu /Mu (tu + t) − 1
−1 ≤
q
2
c
c2
1
−2
2
u t + δ2 − δ u
2
≤

(71)

2
2c
1 u +δ u √
2 (u+ δc )2 −u ( t

1
2

q
t+

+ tu −

c2
(δu)2

u2 + 2c
δ u
(u + δc )2 − u

−

1+

√

c
δu

tu )2

2

r

−1

!2
c 2
− 1 − 1 → 0.
1+
δu

Consequently, (70) and (71) imply

(72)

lim

sup

u→∞ t∈∆(u),t6=0

Mu /Mu (tu + t) − 1
− 1 = 0.
q
2
c2
1
c
2
−2
u t + δ2 − δ u
2

Since for 0 ≤ t′ ≤ t < 1, the correlation function of Z(t) equals
n R 1
o
R − 1 ln t′ −δv
−
ln t −δv
√
E (σ 0 2δ
e dB(v))
e dB(v))(σ 0 2δ
1−t
t − t′
q
√
q
√
r(t, t′ ) =
=1− √
,
= √
′
′
σ2
σ2
1−t
1 − t ( 1 − t′ + 1 − t)
′)
(1
−
t)
(1
−
t
2δ
2δ

we have

sup
t,t′ ∈∆(u),t′ 6=t

1 − r(tu + t, tu + t′ )
−1
1
′
2 |t − t |

=
≤

(73)

1
1−

c
( c+δu
)2

Further, for some small θ ∈ (0, 1), we obtain (set below Z(t) =
(74)
for t, t′ ∈ [0, θ]. For all u large
Π(u) := P
where
e
Π(u)
:= P

2
√
√
√
−1
′
1 − t − tu ( 1 − t − tu + 1 − t − tu )

sup
t,t′ ∈∆(u),t′ 6=t

q

− ( (lnuu) )2

− 1 → 0, u → ∞.

Z(t)
VZ (t) )

2
E Z(t) − Z(t′ ) = 2 − 2r(t, t′ ) ≤ C1 |t − t′ |

(
(

sup
t∈[0,θ]

sup
t∈[θ,1]





1
c
Z(t) − (1 − t 2 ) > u
δ


1
c
Z(t) − (1 − t 2 ) > u
δ

)

)

e
≤ p(u) ≤ Π(u) + Π(u),
≤P

(

)

sup Z(t) > u .
t∈[θ,1]

EXTREMES OF THRESHOLD-DEPENDENT GAUSSIAN PROCESSES

23

Moreover, for all u large
1
1
−
Mu (t) Mu

≥
≥

(75)

√
√ 2
σ2
2c
2
tu )
[Gu (t)VZ (tu )]2 − [Gu (tu )VZ (t)]2
2δ (u + δ u)( t −
=
√
2
3
c
σ
3/2
2uVZ (tu )Gu (tu )
2u[ 2δ (1 − tu )] [u + δ (1 − tu )]
√
√
C2 δ 2 (u)
(ln u)2q
C2 ( t − tu )2 ≥ p
≥
C

3
√ 2
u2
δ(u) + tu + tu

holds for any t ∈ [tu + δ(u), θ], therefore

(ln u)q
Mu
.
≥ 1 + C3
u2
t∈[tu +δ(u),θ] Mu (t)
inf

The above inequality combined with (72), (73), (74) and Proposition 3.6 yields

 2
  p
c
1
h
2δu2 + 4cu , u → ∞.
− 2,∞ Ψ
Π(u) ∼ P1,δ/σ
2
δ
σ
Finally, since

sup
t∈[θ,1]

by Borell-TIS inequality
e
Π(u)
≤P

(

VZ2 (t)

σ2
(1 − θ), and E
≤
2δ

sup Z(t) > u

t∈[θ,1]

)

(

)

sup Z(t)
t∈[θ,1]

≤ C4 < ∞,



δ(u − C4 )2
≤ exp − 2
= o(Π(u)),
σ (1 − θ)

u → ∞,

which establishes the proof. Next, we consider that
o
n
(
)

2 !
e (t) < 0
P inf t∈[− 2δ1 ln(tu +u−2 x),∞) U
c
n
o
P u2 e−2δτu −
≤ x τu < ∞ =
δu + c
e (t) < 0
P inf
U
t∈[0,∞)


o
n

1
1
ln t) − δc (1 − t 2 ) > u
P supt∈[0,tu +u−2 x] σA(− 2δ
n


o
=
1
1
ln t) − δc (1 − t 2 ) > u
P supt∈[0,1] σA(− 2δ

= P u2 (τu∗ − tu ) ≤ x τu∗ < 1 ,

where

τu∗ = {t ∈ [0, 1] : σA(−

c
1
1
ln t) − (1 − t 2 ) > u}.
2δ
δ

The proof follows by Proposition 3.10 i).


5. Appendix

Proof of (11): Let ξ(t), t ∈ R be a centered stationary Gaussian process with unit variance and correlation function

r satisfying

1 − r(t) ∼ a|t|α , t → 0, a > 0, α ∈ (0, 2].
In view of by Theorem 2.2, for −∞ < x1 < x2 < ∞ and f ∈ C0∗ ([x1 , x2 ]) we have
(
)
ξ(t)
f
P
sup
> u ∼ Ψ(u)Pα,a
[x1 , x2 ],
−2 f (u2/α t)
1
+
u
−2/α
−2/α
t∈[u
x1 ,u
x2 ]
and for any y ∈ R

u→∞

)
ξ(t)
>u
P
sup
−2 f (u2/α t)
t∈[u−2/α x1 ,u−2/α x2 ] 1 + u
)
(
ξ(t + yu−2/α )(1 + u−2 f (y))
−2
> u(1 + u f (y))
=P
sup
1 + u−2 f (y + u2/α t)
t∈[u−2/α (x1 −y),u−2/α (x2 −y)]
(

fy (t)−f (y)
∼ Ψ(u(1 + u−2 f (y)))Pα,a
[x1 − y, x2 − y]
fy (t)
∼ Ψ(u)Pα,a
[x1 − y, x2 − y].

24

LONG BAI, KRZYSZTOF DȨBICKI, ENKELEJD HASHORVA, AND LANPENG JI

Let
Zu (t) =

ξ(t + yu−2/α )(1 + u−2 f (y))
,
1 + u−2 f (y + u2/α t)

t ∈ [u−2/α (x1 − y), u−2/α (x2 − y)]

2
and denote its variance function by σZ
(t). Then
u




f (y + u2/α t) − f (y)
1 + u−2 f (y + u2/α t)
1
2
− 1 u2 =
−
1
u
=
,
σZu (t)
1 + u−2 f (y)
1 + u−2 f (y)

i.e.,
lim

sup

u→∞ t∈[u−2/α (x −y),u−2/α (x −y)]
1
2



1
σZu (t)


− 1 u2

f (y + u2/α t) − f (y)

− 1 = 0.

Consequently, we have
f
fy
[x1 − y, x2 − y].
Pα,a
[x1 , x2 ] = Pα,a
f

y
f
[x1 − y, ∞). This completes the proof.
Further, letting x2 → ∞ yields Pα,a
[x1 , ∞) = Pα,a



Proof of Example 3.4: We have t0 = 0, γ = 1, gm = 0. Then by Proposition 3.1 statement i)
(


c−1 a1/α u2/α−1 Hα , α ∈ (0, 2),
P max (X(t) − ct) > u ∼ Ψ(u)
ct
t∈[0,T ]
Pα,a
[0, ∞),
α = 2.
Since for all u large
n
o
n
o P supt∈[0,u−1 x] (X(t) − g(t)) > u
n
o ,
P uτu ≤ x τu ≤ T =
P supt∈[0,T ] (X(t) − g(t)) > u

then using Proposition 3.3, we obtain for x ∈ (0, ∞)


n
o 
P uτu ≤ x τu ≤ T ∼


Bα (t)
V ar(Bα (t))

Proof of Example 3.5: We have that X(t) = √
α

rX (t, t + h) =
2πt
T

α



α/2

α ∈ (0, 2),
α = 2.

is locally stationary with correlation function

α

|t| + |t + h| − |h|
2 |t(t + h)|

R x −ct
e
dt
R 0∞
,
e−ct dt
0
ct
Pα,a
[0,x]
ct [0,∞) ,
Pα,a

=1−

1
|h|α + o(|h|α ),
2tα

, t ∈ [T, (n + 1)T ] attains its maximum at tj =
 π 2
g(t) = c − 2c
|t − tj |2 (1 + o(1)), t → tj , j ≤ n
T

for any t > 0. Since g(t) = c sin

h→0
(4j+1)T
4

, j ≤ n and

the claim follows by applying Remarks 3.2 statement i).


2

Proof of Example 3.11: First
√ note that the variance function of X(t) is given by σ (t) = t(1 − t) and correlation
s(1−t)
function is given by r(t, s) = √
, 0 ≤ s < t ≤ 1.
t(1−s)
√
t(1−t)
u
∈
Case 1) The proof of (22): Clearly, mu (t) := 1+ct/u attains its maximum over [0, 1] at the unique point tu = c+2u
1
1
∗
(0, 1) which converges to t0 = 2 as u → ∞, and mu := mu (tu ) = √
. Furthermore, we have
2

m∗u
−1 =
mu (t)

1+c/u

p
p
p
(u + ct) tu (1 − tu ) − (u + ctu ) t(1 − t)
tu (1 − tu )
u + ct
p
p
−1=
t(1 − t) u + ctu
t(1 − t)(u + ctu )

(u + ct)2 tu (1 − tu ) − (u + ctu )2 t(1 − t)
p
p
p
.
t(1 − t)(u + ctu )[(u + ct) tu (1 − tu ) + (u + ctu ) t(1 − t)]
i
h
q
q
Setting ∆(u) = − (lnuu) , (lnuu) , and (tu + ∆(u)) ⊂ [0, 21 ] for all u large, we have
(76)

=

(u + ct)2 tu (1 − tu ) − (u + ctu )2 t(1 − t)

(77)

= u2 [(tu − t2u ) − (t − t2 )] + 2cuttu (t − tu ) + c2 ttu (t − tu )
= (t − tu )2 u(u + c)

EXTREMES OF THRESHOLD-DEPENDENT GAUSSIAN PROCESSES

25

and
u4
2 u+
for all t ∈ (tu + ∆(u)). Then
lim

(78)

sup

u→∞ t∈∆(u),t6=0

−1/2
≤ 2(u + ct)2 [t(1 − t)] ≤
 −u
c 2

2

c 2
1
u+
2
2

m∗u /mu (tu + t) − 1
m∗u /mu (tu + t) − 1
−
1
=
lim
sup
− 1 = 0.
u→∞ t∈∆(u),t6=0
2t2
2(ut)2 u−2

Furthermore, since

p
p
p
s(1 − t)
s(1 − t) − t(1 − s)
t−s
p
p
p
=1+
=1− p
,
r(t, s) = p
t(1 − s)
t(1 − s)
t(1 − s)( s(1 − t) + t(1 − s))

and

p
p
1 1 p
1 1
− ≤ t(1 − s)( s(1 − t) + t(1 − s)) ≤ +
2 u
2 u

for all s < t, s, t ∈ (tu + ∆(u)), we have
lim

sup

u→∞ t,s∈∆(u)
t6=s

1 − r(tu + t, tu + s)
− 1 = 0.
2|t − s|

Next for some small θ ∈ (0, 12 ), we have


|t − s|
E (X(t) − X(s))2 = 2(1 − r(t, s)) ≤ 1
( 2 − θ)2

holds for all s, t ∈ [ 21 − θ, 12 + θ]. Moreover, by (76), (77) and
2 
2


1
1
+θ
+θ
2(u + ct)2 [t(1 − t)] ≤ 2 u + c
2
2
for all t ∈ [ 12 − θ, 21 + θ], we have that for any t ∈ [ 21 − θ, 12 + θ] \ (tu + ∆(u))
m∗u
(ln u)2q
−1≥
,
mu (t)
2[u + c( 12 + θ)]2 ( 12 + θ)2
and further
(ln u)q
m∗u
,
≥ 1 + C1
mu (t)
u2

(79)

Consequently, by Proposition 3.6
(
sup

P

(X(t) − ct) > u

t∈[t0 −θ,t0 +θ]

)

1
1
t ∈ [ − θ, + θ] \ (tu + ∆(u)).
2
2

∼ 8H1 u

Z

∞

−∞

 p

2
2
e−8t dtΨ 2 cu + u2 ∼ e−2(u +cu) .

In addition, since σθ := maxt∈[0,1]/[t0 −θ,t0 +θ] σ(t) < σ(t0 ) = 12 , by Borell-TIS inequality
 
n
o2 
(
)
(
)
u
−
E
sup
X(t)
t∈[0,1]


P
sup
(X(t) − ct) > u
≤ P
sup
X(t) > u ≤ exp −

2
2σ
t∈[0,1]\[t0 −θ,t0 +θ]
t∈[0,1]\[t0 −θ,t0 +θ]
θ
2

= o(e−2(u

(80)

Thus, by the fact that
P

(

sup (X(t) − ct) > u

t∈[0,1]

and
P

(

sup (X(t) − ct) > u

t∈[0,1]

)

+cu)

≤P

(

sup
t∈[t0 −θ,t0 +θ]

we conclude that
P

(

)

).

≥P

(

(X(t) − ct) > u

sup (X(t) − ct) > u

t∈[0,1]

sup
t∈[t0 −θ,t0 +θ]

)

)

+P

(X(t) − ct) > u

(

sup

t∈[0,1]\[t0 −θ,t0 +θ]

2

∼ e−2(u

+cu)

.

)
)

(X(t) − ct) > u ,

26

LONG BAI, KRZYSZTOF DȨBICKI, ENKELEJD HASHORVA, AND LANPENG JI

For any u > 0
 
P u τu −

u
c + 2u



≤ x τu ≤ 1



n
o
P supt∈[0,tu +u−1 x] (X(t) − ct) > u
n
o
=
P supt∈[0,1] (X(t) − ct) > u

and by Theorem 2.2


Z x



 p
2
P
sup
(X(t) − ct) > u ∼ 8H1 u
e−8t dtΨ 2 cu + u2 .
t∈ t − (ln u)q ,t +u−1 x

−∞
[u
]
u
u

The above combined with (79) and (80) implies that as u → ∞


(
)
Z x



 p
2
P
sup
(X(t) − ct) > u ∼ P
sup
(X(t) − ct) > u ∼ 8H1 u
e−8t dtΨ 2 cu + u2 .
t∈ t − (ln u)q ,t +u−1 x

t∈[0,tu +u−1 x]
−∞
[u
]
u
u

Consequently,

 
P u τu −

u
c + 2u



≤ x τu ≤ 1

Case 2) The proof of (23): We have tu =

u
c+2u



∼

Rx

−8t2
dt
−∞ e
R∞
2
−8t
dt
−∞ e

= Φ(4x), x ∈ (−∞, ∞).

∈ (0, 21 ) which converge to t0 =
1
c
− tu ∼
,
2
4u

1
2

as u → ∞. Since

u → ∞,

by Proposition 3.6
P

(

sup (X(t) − ct) > u

t∈[0,1/2]

)

∼ 8H1 u

Z

c/4

−∞


 p
2
2
e−8t dtΨ 2 cu + u2 ∼ Φ(c)e−2(u +cu) .

As for the proof of Case 1) we obtain further
 
P u τu −

u
c + 2u



1
≤ x τu ≤
2



∼

Rx

2
e−8t dt
−∞
R c/4
e−8t2 dt
−∞

∼ Φ(4x)/Φ(c), x ∈ (−∞, c/4].

Case 3) The proof of (24): We have that σ(t) attains its maximum over [0, 1] at the unique point t0 = 12 , which is also
the unique maximum point of

c
2

−c t−

1
2

, t ∈ [0, 1]. Furthermore,

σ(t) =
and

2

p
1
1
1
t(1 − t) ∼ − t −
, t→
2
2
2

r(t, s) ∼ 1 − 2|t − s|,

s, t →

1
.
2

By Proposition 3.9 as u → ∞
(
)


Z ∞
2
2
c
1
P sup X(t) + − c t −
e−(8|t| +4c|t|) dtΨ (2u − c) ∼ 2Ψ(c)e−2(u −cu)
> u ∼ 8H1 u
2
2
t∈[0,1]
−∞
and in view of Proposition 3.10 ii)

 R x −(8|t|2 +4c|t|)
 
e
dt
1
≤ x τu ≤ 1 ∼ R−∞
P u τu −
, u → ∞.
∞
2 +4c|t|)
−(8|t|
2
dt
−∞ e



Acknowledgments
Thanks to Swiss National Science Foundation Grant no. 200021-166274. KD acknowledges partial support by NCN
Grant No 2015/17/B/ST1/01102 (2016-2019).

EXTREMES OF THRESHOLD-DEPENDENT GAUSSIAN PROCESSES

27

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Long Bai, Department of Actuarial Science, University of Lausanne UNIL-Dorigny, 1015 Lausanne, Switzerland
E-mail address: Long.Bai@unil.ch
Krzysztof Dȩbicki, Mathematical Institute, University of Wroclaw, pl. Grunwaldzki 2/4, 50-384 Wroclaw, Poland
E-mail address: Krzysztof.Debicki@math.uni.wroc.pl
Enkelejd Hashorva, Department of Actuarial Science, University of Lausanne,, UNIL-Dorigny, 1015 Lausanne, Switzerland
E-mail address: Enkelejd.Hashorva@unil.ch
Lanpeng Ji, Department of Actuarial Science, University of Lausanne, UNIL-Dorigny, 1015 Lausanne, Switzerland
E-mail address: jilanpeng@126.com

