I am reading a paper on stochastic optimization. And in this paper, the proofs are based on the Pinelis' 1994 inequality. I read the paper by Pinelis for more information and it is with great frustration that I was not able to find the inequality corresponding to that mentioned in the paper I am reading.


Here's the inequality in the article I'm reading:


$$ \begin{array}l
\text{(The Pinelis' 1994 inequality). Let } X\_1,\dots, X\_T \in \mathbb{R}^d \\ \text{be a random process satisfying } \mathbb{E}[X\_t\mid X\_1,\dots,X\_{t−1} ] = 0 \text{ and} \\ \|X\_t\| ≤ M. \text{ Then } \mathbb{P}[\| X\_1 + \cdots + X\_T\|^2 > 2 \log(2 /\delta)M^2T] \leq \delta.
\end{array}
$$


Nowhere in [Pinelis' 1994 paper](https://projecteuclid.org/download/pdf_1/euclid.aop/1176988477) is this inequality mentioned.


So I ask myself, does this inequality really exist? Do you know it?



Does Pinelis' inequality (1994) exist?
As noted in Carlo Beenakker's comment, your inequality is a direct application of Theorem 3.5 in the linked paper: in that theorem, take $d\_j=X\_j$, $r=\sqrt{2M^2 T\ln(2/\delta)}$, $b\_\*^2=M^2T$, and $D=1$.


