Solution: CF-Based Derivation of Moments and Cumulants

Exercise: CF-Based Derivation of Moments and Cumulants

Part 1 — Gaussian

\ln\varphi_X(t) = i\mu t - \tfrac{1}{2}\sigma^2 t^2

is a polynomial of degree

2

t

, so all higher derivatives vanish:

$\kappa_1 = \mu$ , $\kappa_2 = \sigma^2$ , $\kappa_3 = 0$ , $\kappa_4 = 0$ , and $\kappa_k = 0$ for all $k \ge 3$ .

Result: a Gaussian is entirely determined by its first two cumulants. Every higher cumulant is zero — this is the defining property of the normal distribution among continuous laws.

Part 2 — Poisson

$\ln\varphi_X(t) = \lambda(e^{it} - 1) = \lambda\sum_{k=1}^\infty (it)^k/k!$ . Matching coefficients with $\sum \kappa_k(it)^k/k!$ :

$\kappa_k = \lambda$ for every $k \ge 1$ .

Result: every cumulant of a Poisson distribution equals its rate parameter. In particular:

\mathbb{E}[X] = \lambda

\operatorname{Var}(X) = \lambda

, skewness

\kappa_3/\kappa_2^{3/2} = 1/\sqrt\lambda

, and...

Part 3 — Excess kurtosis

\text{excess kurtosis of Poisson} = \frac{\kappa_4}{\kappa_2^2} = \frac{\lambda}{\lambda^2} = \frac{1}{\lambda}.

For small $\lambda$ (say $\lambda = 1$ ) the Poisson has excess kurtosis $1$ — meaningfully heavier tails than Gaussian. For large $\lambda$ (say $\lambda = 100$ ) the excess kurtosis is $0.01$ — essentially Gaussian, consistent with the CLT ( $\text{Poisson}(\lambda)$ is a sum of $\lambda$ independent Poisson(1)s, so it converges to Gaussian).

Interpretation. Poisson has heavier tails than a matched-variance Gaussian — at low intensity, extreme counts (many defaults at once) are more likely than Gaussian would predict. This matters in credit-risk where default rates are low and the normal approximation undersells tail risk.

Part 4 — Additivity

For $X, Y$ independent, $\varphi_{X+Y}(t) = \varphi_X(t)\varphi_Y(t)$ , so $\ln\varphi_{X+Y}(t) = \ln\varphi_X(t) + \ln\varphi_Y(t)$ . Matching Taylor coefficients:

\kappa_k(X + Y) = \kappa_k(X) + \kappa_k(Y) \text{ for every } k.

Which moments fail. Moments do not add; only

\mathbb{E}[X+Y] = \mathbb{E}[X] + \mathbb{E}[Y]

is clean, because it is actually

\kappa_1

. For

k = 2

\mathbb{E}[(X+Y)^2] = \mathbb{E}[X^2] + 2\mathbb{E}[XY] + \mathbb{E}[Y^2]

, which has a cross term. Cumulants are the clean additive "centred" moment surrogates —

\kappa_2 = \operatorname{Var}

\kappa_3 =

third central moment, etc.

Multiplicativity of CFs drives the additivity of cumulants; this is the structural advantage of cumulants over ordinary moments in anything involving sums.

Takeaways

Cumulants are the additive version of moments. For sums of independents, cumulants simply add; moments don't.
Gaussians have only two non-zero cumulants. This is the formal sense in which the Gaussian is "minimally structured" among distributions with a given mean and variance.
Excess kurtosis $= \kappa_4/\kappa_2^2$ is the standard heavy-tail diagnostic. $> 0$ = heavier tails than Gaussian; $< 0$ = lighter.
The Poisson sits between discrete and continuous. Each cumulant equals $\lambda$ ; as $\lambda \to \infty$ , the distribution Gaussianises. This is the CLT for counting processes seen in cumulant form.