Hoeffding's Lemma for bounded complex random variables?

Question

If we have a real random variable $X$ such that $a\leq X\leq b$ almost surely, we can establish the following inequality: \begin{align} \mathbb{E}\left[\exp\Big(t(X-\mathbb{E}[X])\Big)\right]\leq\exp\left(\frac{1}{8}t^2(b-a)^2\right). \end{align} Is there a similar bound for a complex random variable $Y$ such that it satisfies the condition of $r_1\leq |Y|\leq r_2$ with almost certain probability?

Answer 1

Restricting $Y$ to an annulus doesn't seem useful as any bounds are likely to be satisfied also inside the annulus.

A bound with $Y$ restricted to a disk, or more generally to a region with bounded diameter, appears in Mikhail Isaev and Brendan McKay. "On a bound of Hoeffding in the complex case." Electron. Commun. Probab. 21 1-7, 2016. https://doi.org/10.1214/16-ECP4372

Free copy here.

The proof is different from the real case because the exponential function is not concave in the complex plane.

The complex condition most similar to the real condition $a\le X\le b$ is bounded diameter: $$\mathrm{diam}(Y) = \inf\{c\in\mathbb{R}\,:\, P(|Y_1-Y_2|>c)=0\},$$ where $Y_1,Y_2$ are independent copies of $Y$. We show that $$|\mathbb{E}e^{Y-\mathbb{E}Y}-1| \le e^{\mathrm{diam}(Y)^2/8}-1,$$ where $\frac 18$ is the best possible constant. For diameter less than 3 we give the optimum bound.