Poincare inequality for Poisson random variables

Question

My question comes from exercise 3.21 in this monograph. Specifically, let $f$ be a real-valued function defined on the set of non-negative integers and denote its "discrete derivative" by $Df(x) = f(x+1)-f(x)$. Let $X$ be a Poisson random variable with parameter $\mathbb E[X] = \mu$. Prove that $$\mathrm{Var}(f(X)) \leq \mu\,\mathbb E\left[(Df(X))^2\right].$$ The hint is to use the Efron-Stein inequality and the infinite divisibility of the Poisson distribution.

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My attempt: the infinite divisibility of the Poisson distribution means that we can write $X := \sum_{i=1}^n X_i$, where $X_i$'s are i.i.d. with Poisson distribution $\mathrm{Poisson}(\mu/n)$. However, I do not see how the Efron-Stein inequality (Theorem 3.1) $$\mathrm{Var}(f(X)) \leq \sum_{i=1}^n \mathbb{E}\left[\left(f(X) - \mathbb{E}^{(i)}f(X)\right)^2\right] $$ will lead us to the advertised inequality. Here the conditional expectation operator $\mathbb{E}^{(i)}$ means that we are taking the expectation of $f(X)$ with respect to $X_i$ (while keeping $X_1,\ldots,X_{i-1},X_{i+1},\ldots,X_n$ fixed). Any help will be greatly appreciated!

Answer 1

OK I think I figured it out. The way to use the infinite divisibility of the Poisson distribution is actually the key towards the proof. We let $(X_i)_{i=1}^n$ to be i.i.d. Bernoulli distributed random variables with parameter $\mu/n$. Then we have $S_n := \sum_{i=1}^n X_i \to X$ as $n \to \infty$, where the convergence is in the sense of distribution. A routine computation gives us $$\mathrm{Var}^{(i)}\left(f(S_n)\right) = \frac{\mu}{n}\left(1-\frac{\mu}{n}\right)\left[f(S_n-X_i+1) - f(S_n-X_i)\right]^2.$$ By the i.i.d. property of $(X_i)_{i=1}^n$ and the Efron-Stein inequality, we end up with $$\mathrm{Var}\left(f(S_n)\right) \leq \left(1-\frac{\mu}{n}\right)\mu \mathbb{E}\left[\left(f(S_{n-1}+1) - f(S_{n-1})\right)^2\right] = \left(1-\frac{\mu}{n}\right)\mu \mathbb{E}\left[\left(Df(S_n)\right)^2\right].$$ The advertised Poisson Poincare inequality follows by sending $n \to \infty$. Overall, the strategy of the proof is pretty similar to the one used in the proof of Theorem 3.20 in the aforementioned monograph, where a Gaussian Poincare inequality is demonstrated. I welcome any other approaches as well (either functional-analytic approach or geometric approach)!