In a number of different contexts, I have wanted to estimate hitting times for a monotonic process $(T\_n)$ taking values in the reals (or sometimes a process $(T\_n,X\_n)$ taking values in $\mathbb R^2$ where the first component is monotonic). I'm assuming the step size is small, and am interested in the first time, $\tau\_1$, that $T\_n$ exceeds some threshold $a$ (and in the 2d case, possibly the distribution of $X\_n$ at this hitting time). 


For some concreteness, assume $a=1$, $T\_0=0$ and $(T\_n)$ is Markov, where the jump size, $T\_{n+1}-T\_n$ is much smaller than 1. If necessary, we can assume that the jump sizes are i.i.d., although I'd ultimately prefer to have something more flexible where the distribution of $T\_{n+1}-T\_n$ is in some sense continuously dependent on $T\_n$. I would like to obtain reasonably precise information on the distribution of $\tau\_1$. 



> 
> Is there an established machinery that can address questions of this type?
> 


Here is a specific simple (made up) instance that I would be very interested to see a clean answer to (especially, as indicated above, an answer that might be generalizable away from the i.i.d. jump case): 
Let $T\_{n+1}-T\_n\sim \exp(\text{Unif}[-2k,-k])$ and $T\_0=0$. What can be said about the distribution of $\tau\_1$? How does the distribution of $\tau\_1$ depend on $k$?



Hitting time estimates
$\newcommand{\de}{\delta}
\newcommand{\be}{\beta}
\newcommand{\si}{\sigma}$
Consider the iid case, when $T\_n=X\_1+\dots+X\_n$, where $X\_1,X\_2,\dots$ are positive iid random variables with, say, $EX\_1=\de\to0$, $Var\,X\_1=\de^2\si^2$, $E|X\_1|^3=\de^3\be<\infty$ (scaling with $\de$) such that 
\begin{equation}
\be\sqrt\de/\si^3\to0,\quad \de\si^2\to0. \tag{0} 
\end{equation}


The simple key observation is that 
\begin{equation}
 P(\tau\_1\le n)=P(T\_n>1). \tag{1}
\end{equation}
Note that $ET\_n=n\de$ and $Var\,T\_n=n\de^2\si^2$. 


So, by Chebyshev's inequality, for any fixed $t\in(0,1)$ and any $n\le(1-t)/\de$,
\begin{equation}
 P(\tau\_1\le n)=P(T\_n>1)\le\frac{n\de^2\si^2}{(1-n\de)^2}
 \le\frac{(1-t)\de\si^2}{t^2}\to0; 
\end{equation}
similarly, $P(\tau\_1\ge n)\to0$ if $n\sim(1+t)/\de$ and hence if $n\gtrsim(1+t)/\de$. 
So, 
\begin{equation}
 \de\tau\_1\to1
\end{equation}
in probability and hence 
without loss of generality we need consider only 
\begin{equation}
 n\sim1/\de 
\end{equation}
in (1). 


Next, by the Berry--Esseen inequality, 
\begin{equation}
 P(\tau\_1\le n)=P(T\_n>1)=P\Big(Z>\frac{1-n\de}{\de\si\sqrt n}\Big)+R
 =P\Big(Z\le\frac{n-1/\de}{\si\sqrt n}\Big)+R,
\end{equation}
where $Z\sim N(0,1)$,
\begin{equation}
 |R|\le C\frac\be{\si^3\sqrt n}\sim C\frac{\be\sqrt\de}{\si^3}\to0 
\end{equation}
by (0), 
and $C$ is a universal constant. 
Also, assuming $|\frac{n-1/\de}{\si\sqrt n}|=O(1)$, we have 
\begin{equation}
 P\Big(Z\le\frac{n-1/\de}{\si\sqrt n}\Big)
 =P\Big(Z\le\frac{n-1/\de}{\si/\sqrt\de}\Big)+o(1). 
\end{equation}


So, $\tau\_1$ is asymptotically normal with asymptotic mean $1/\de$ and asymptotic variance $\si^2/\de$. 


