Let $X\_1, X\_2, \dots, X\_n$ be a martingale difference sequence such that
$$
X\_i \leq y \quad \text{and} \quad \sum\_{i=1}^{n} \operatorname{Var}(X\_i) \leq B^2.
$$
Question 1: Does the following hold?
$$
\mathbb{P}\left[ \sum\_{i=1}^{n}X\_i \geq x \right] \leq \exp{\left(\frac{-x^2}{2B^2 + \frac{2}{3}xy}\right)}.
$$


A similar bound (albeit for independent random variables) is given in Corollary 2 in [Pinelis–Utev (1990)](https://doi.org/10.1137/1134032) (DOI link). I have seen that exponential inequalities for sums of independent random variables can be extended to martingales generally.


Question 2: If the bound given in question 1 doesn't hold, does any other similar exponential inequality exist for the LHS?
I have came across Freedman's inequality (Theorem 1.6 in [Freedman (1975)](https://projecteuclid.org/euclid.aop/1176996452)) which deals with similar quantities but it contains $\operatorname{Var}(X\_i | \mathcal{F}\_{i-1})$. As seen from the above, I would rather have the bound in terms of $\operatorname{Var}(X\_i)$.


Thank you for your time and consideration.



Exponential inequality for the sum of martingale differences $X_1, \dots, X_n$ when $\sum_{i=1}^{n} \operatorname{Var}(X_i) \leq B^2$
**Theorem 1:** In the known exponential bounds for martingales, the conditional variances cannot be replaced by the unconditional ones.


*Proof:* Otherwise, we would most likely have such bounds. $\Box$ :-)


This "proof" of "Theorem 1" is not so non-serious as it may look.




---


Perhaps more seriously, we have


**Theorem 2:** The following statement is false:



> 
> There is a real constant $c>0$ such that for all natural $n$, all real $y>0$, all real $B>0$, and all martingale difference sequences $(X\_1,\dots,X\_n)$ such that
> \begin{equation\*}
> X\_i\le y \ \forall i\quad\text{and}\quad\sum\_{i=1}^n Var\,X\_i\le B^2\label{0}\tag{0}
> \end{equation\*}
> we have
> \begin{equation\*}
>  P\Big(\sum\_{i=1}^n X\_i\ge x\Big)\le\exp\frac{-cx^2}{B^2+xy}\label{1}\tag{1}
> \end{equation\*}
> for all real $x>0$.
> 
> 
> 


*Proof:* This proof would be a bit simpler if, instead of using Corollary 2 in the Pinelis--Utev paper, you used the better bound in Theorem 3 in that paper. Indeed, one can show that, at least in the case when the $X\_i$'s are conditionally symmetric (given $\mathcal F\_{i-1}$), that theorem implies the Rosenthal-type inequality
\begin{equation\*}
 ES\_n^4\ll B^4+A^{(4)}\_n,
\end{equation\*}
where $$S\_n:=\sum\_{i=1}^n X\_i,$$ $a\ll b$ means $a\le Cb$ for some real $C$ depending only on $c$, and
\begin{equation\*}
 A^{(p)}\_n:=\sum\_{i=1}^n E|X\_i|^p. 
\end{equation\*}


Because the bound in \eqref{1} is suboptimal, it only implies an ugly version of the Rosenthal-type inequality:


**Lemma 1:** If the highlighted statement is true, then for conditionally symmetric martingale difference sequences $(X\_1,\dots,X\_n)$ such that $\sum\_{i=1}^n Var\,X\_i\le B^2$ we have
\begin{equation\*}
 ES\_n^4\ll B^4+A^{(6)}\_n/B^2. \label{2}\tag{2}
\end{equation\*}


This lemma will be proved at the end of this answer.


Now consider the following construction of a conditionally symmetric martingale difference sequence $(X\_1,\dots,X\_n)$: Let $V\_1:=R\_1$, where $R\_1$ is a Rademacher random variable, so that $P(R\_1=\pm1)=1/2$. For natural $k\ge2$, let
\begin{equation\*}
 V\_k:=a\_k R\_k,\quad a\_k:=\frac1{\sqrt{k\ln k}},
\end{equation\*}
where $R\_2,R\_3,\dots$ are independent copies of $R\_1$. Let then $X\_1:=V\_1$, and for natural $k\ge2$ let
\begin{equation\*}
 X\_k:=S\_{k-1}V\_k,
\end{equation\*}
where $S\_j:=\sum\_{i=1}^j X\_i$, as before. So, for natural $k\ge2$,
\begin{equation\*}
 S\_k=S\_{k-1}(1+V\_k). 
\end{equation\*}
So, for any even natural $p$ and any natural $k\ge2$, we have $M\_k^{(p)}:=ES\_k^p=M\_{k-1}^{(p)} E(1+V\_k)^p$ and hence
\begin{equation\*}
 M\_k^{(p)}=\prod\_{j=2}^k E(1+V\_j)^p. 
\end{equation\*}
In particular,
\begin{equation\*}
 M\_k^{(2)}=\prod\_{j=2}^k (1+a\_k^2)=\prod\_{j=2}^k \Big(1+\frac1{k\ln k}\Big)
 =\exp\Big\{(1+o(1))\int\_2^k\frac{dx}{x\ln x}\Big\}
 =(\ln k)^{1+o(1)} 
\end{equation\*}
(as $k\to\infty$). Similarly,
\begin{equation\*}
 M\_k^{(4)}=\prod\_{j=2}^k (1+6a\_k^2+a\_k^4)=(\ln k)^{6+o(1)}, 
\end{equation\*}
\begin{equation\*}
 M\_k^{(6)}=\prod\_{j=2}^k (1+15a\_k^2+15a\_k^4+a\_k^6)=(\ln k)^{15+o(1)}. 
\end{equation\*}
Hence,
\begin{equation\*}
 A^{(6)}\_n=1+\sum\_{k=2}^n M\_{k-1}^{(6)}a\_k^6\ll1+\sum\_{k=2}^n (\ln k)^{15+o(1)}\frac1{k^3\ln^3k}\ll1. 
\end{equation\*}
Also, we may take
\begin{equation\*}
 B^2=\sum\_{i=1}^n Var\,X\_i=ES\_n^2=M\_n^{(2)}=(\ln n)^{1+o(1)}.
\end{equation\*}
So, for $n\to\infty$ the right-hand side of \eqref{2} is
\begin{equation\*}
 B^4+A^{(6)}\_n/B^2=(\ln n)^{2+o(1)}+O(1)/(\ln n)^{1+o(1)}=(\ln n)^{2+o(1)},
\end{equation\*}
whereas the left-hand side of \eqref{2} is
\begin{equation\*}
ES\_n^4=M\_n^{(4)}=(\ln n)^{6+o(1)}. 
\end{equation\*}
Thus, \eqref{2} fails to hold for large enough $n$.


It remains to give


*Proof of Lemma 1:* Suppose the highlighted statement is true. Take any conditionally symmetric martingale difference sequence $(X\_1,\dots,X\_n)$ such that $\sum\_{i=1}^n Var\,X\_i\le B^2$. Take any real $y>0$. Let $X\_{i,y}:=X\_i\,1(|X\_i|\le y)$ for all $i$. Then $(X\_{1,y},\dots,X\_{n,y})$ is a martingale difference sequence with $|X\_{i,y}|\le y$ and $Var\,X\_{i,y}\le Var\,X\_i$ for all $i$. So,
\begin{align\*}
 P(|S\_n|\ge x)&\le\sum\_{i=1}^n P(|X\_i|>y)+P\Big(\Big|\sum\_{i=1}^nX\_{i,y}\Big|\ge x\Big) \\ 
 &\le \sum\_{i=1}^n P(|X\_i|>y)+2\exp\frac{-cx^2}{B^2+xy}
\end{align\*}
by the highlighted statement, for all real $x>0$.
Using this inequality with $y=B(x/B)^{2/3}$, integrating in $x>0$, and using the substitutions $z=B(x/B)^{2/3}$ and $x/B=t$, we have
\begin{align\*}
 ES\_n^4&=\int\_0^\infty dx\,4x^3P(|S\_n|\ge x) \\ 
 &\le\sum\_{i=1}^n \int\_0^\infty dx\,4x^3 P(|X\_i|>B(x/B)^{2/3}) \\ 
& +\int\_0^\infty dx\,4x^3 2\exp\frac{-cx^2}{B^2+xB(x/B)^{2/3}} \\
&\ll A^{(6)}\_n/B^2+B^4. 
\end{align\*}
This completes the proof of Lemma 1 and thus the proof of Theorem 2.
$\Box$




---


Whereas, as has just been shown, the highlighted statement is false even for conditionally symmetric martingale difference sequences $(X\_1,\dots,X\_n)$, note [Theorem 3.6](https://www.jstor.org/stable/2244912?seq=1), which implies that for any conditionally symmetric martingale difference sequences $(X\_1,\dots,X\_n)$ such that $\sum\_{i=1}^n X\_i^2\le B^2$ for some real $B>0$, we have
\begin{equation\*}
 P\Big(\Big|\sum\_{i=1}^n X\_i\Big|\ge x\Big)\le2\exp\frac{-x^2}{2B^2}
\end{equation\*}
for all real $x>0$.


