Exercise: CF-Based Derivation of Moments and Cumulants

Prerequisites: Characteristic Functions, Expectation and Variance

Problem

For a random variable $X$ with finite moments up to order $k$, the CF is $k$-times differentiable at $0$ and

1. For $X \sim \mathcal{N}(\mu, \sigma^2)$, compute $\kappa_1, \kappa_2, \kappa_3, \kappa_4$ from $\ln\varphi_X(t) = i\mu t - \tfrac{1}{2}\sigma^2 t^2$. What is the cumulant structure of a Gaussian?

2. For $X \sim \text{Poisson}(\lambda)$, compute $\kappa_1, \kappa_2, \kappa_3, \kappa_4$ from $\ln\varphi_X(t) = \lambda(e^{it} - 1)$.

3. Excess kurtosis is defined as $\kappa_4/\kappa_2^2$ (zero for a Gaussian). Compute the excess kurtosis of $\text{Poisson}(\lambda)$ and interpret: does the Poisson have heavier or lighter tails than a Gaussian with matched variance?
4. Additivity property. Show that if $X, Y$ are independent, $\kappa_k(X + Y) = \kappa_k(X) + \kappa_k(Y)$ for every $k$. Which CF property drives this additivity (which moments do not have)?

Hint

For parts 1–2, compute $\ln\varphi_X(t)$ directly, Taylor-expand in $t$ near $0$, and read off the coefficients: $\ln\varphi_X(t) = \sum_{k=1}^\infty \kappa_k (it)^k / k!$.