# Poincare inequality for Poisson random variables

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.

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!

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)!