Series Formula, Exponential Moments, and Chernoff Tail Bounds for the Poisson Distribution

Let $(\Omega,\mathcal{F},P)$ be a probability space, let $\mu\ge0$ be a real number, and let $K$ be a random variable on $(\Omega,\mathcal{F},P)$ with the Poisson distribution with parameter $\mu$. Write $\mathbb{N}_0$ for the set consisting of $0$ and the natural numbers, $k!$ for the factorial with $0!=1$, $\mu^{0}=1$, $\exp$ for the exponential function, $\mathbb{E}$ for the expectation (that of a nonnegative random variable being its integral, in $[0,\infty]$), and $\mathcal{B}(\mathbb{R})$ for the Borel $\sigma$-algebra of the real line. Sums of nonnegative terms indexed by $\mathbb{N}_0$ are sums of a nonnegative function over a set, with values in $[0,\infty]$. For $x>0$ put

$$\varpi_\mu(x)=\min\Bigl(\frac{x^{2}}{4\mu},\ \frac{x}{2}\Bigr)\quad\text{if }\mu>0,\qquad \varpi_0(x)=\frac{x}{2}.$$

1. (Series formula.) For every map $g:\mathbb{R}\to[0,\infty)$ which is measurable with respect to $\mathcal{B}(\mathbb{R})$ on both sides, $g(K)=g\circ K$ is a nonnegative random variable and

$$\mathbb{E}\bigl[g(K)\bigr]=\sum_{k\in\mathbb{N}_0}\exp(-\mu)\,\frac{\mu^{k}}{k!}\,g(k)\qquad\text{in }[0,\infty].$$

2. (Exponential moments.) For every real number $\theta$ the random variable $\exp(\theta K)$ is integrable and

$$\mathbb{E}\bigl[\exp(\theta K)\bigr]=\exp\bigl(\mu\,(\exp(\theta)-1)\bigr).$$

3. (Tail bounds.) For every real number $x>0$,

$$P(K\ge\mu+x)\le\exp\bigl(-\varpi_\mu(x)\bigr),\qquad P(K\le\mu-x)\le\exp\bigl(-\varpi_\mu(x)\bigr),$$

and consequently $P(|K-\mu|\ge x)\le2\exp(-\varpi_\mu(x))$. Moreover $\varpi_\mu(x)\ge\varpi_{\bar\mu}(x)$ whenever $0\le\mu\le\bar\mu$, so that all three bounds remain valid with $\varpi_{\bar\mu}$ in place of $\varpi_\mu$ for any real $\bar\mu\ge\mu$.