I am reading a paper on stochastic optimization. And in this paper, the proofs are based on the Pinelis's 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:

$$ \mbox{(The Pinelis's 1994 inequality). Let } X_1,\dots, X_T \in \mathbb{R}^d \mbox{ be a random process satisfying }
\mathbb{E}[X_t|X_1,\dots,X_{t−1} ] = 0 \mbox{ and }  \|X_t\| ≤ M. \mbox{ Then }, \mathbb{P}[\| X_1 + \cdots + X_T\|^2 > 2 \log(2 /\delta)M^2T] \leq \delta. $$

At no time in Pinelis' 1994 paper ([https://projecteuclid.org/download/pdf_1/euclid.aop/1176988477][1]) is this inequality mentioned.

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

  [1]: https://projecteuclid.org/download/pdf_1/euclid.aop/1176988477