Quadratic exponentials of Gaussian random vectors

2024

The goal of this note is to gather some elementary facts on integrals of the form, say, $\mathbb{E}[e^{\lambda X^2}]$ where $X$ is a Gaussian random variable. These kind of integrals arise in many situations: for instance, in the Laplace transform of chi-square distributions. I'll cover the one-dimensional case first, then the multi-dimensional case.

The one-dimensional case

We recall the fundamental Gaussian integral, valid for $\lambda >0$ :

\int_{-\infty}^{\infty} e^{-\lambda t^2/2}dt = \sqrt{\frac{2\pi}{\lambda}}.

Let

X \sim N(0,1)

. Then, for any real numbers

a,b,c

\mathbb{E}[e^{\frac{aX^2+bX+c}{2}}] = +\infty

a\geq 1

, and otherwise

\mathbb{E}[e^{\frac{aX^2+bX+c}{2}}] = \frac{\exp\left(\frac{c}{2} + \frac{b^2}{8(1-a)}\right)}{\sqrt{1-a}}.

Proof. Set $q(x) = ax^2+bx+c$ . Then, $\mathbb{E}[e^{q(X)/2}]$ is equal to $\frac{1}{\sqrt{2\pi}}\int e^{-\frac{t^2}{2} + \frac{at^2+bt+c}{2}}dt,$ which obviously diverges iff $a\geq 1$ . Otherwise, simple algebraic manipulations show that

\frac{(a-1)t^2}{2}+\frac{bt}{2}+\frac{c}{2} = -\frac{\gamma}{2}\left(t-\frac{b}{2\gamma}\right)^2 +\frac c 2 + \frac{b^2}{8\gamma}

where $\gamma = 1-a>0$ . Plugging this into (3) yields

e^{c/2 + b^2/8\gamma}\int e^{-\frac{1}{2}\gamma\left(t-\frac{b}{2\gamma}\right)^2}dt

which, after changing variables $t-b/2\gamma$ into $s$ , then using (1) and replacing $\gamma$ by its value $1-a$ , gives exactly (2).

We now list a few consequences of this formula.

Laplace transform of a chi-square distribution:

\mathbb{E}[e^{-\lambda X^2}]=\frac{1}{\sqrt{1 + 2\lambda}}.

Analytic continuation. Take $b=c=0$ for simplicity. The function $1/\sqrt{1-z}$ is defined for any $z$ not in $[1,+\infty)$ if we use the principal branch of the Logarithm. But outside of this set, the formula remains valid, and thus we can compute $\mathbb{E}[e^{zX^2/2}] = (1 - z)^{-1/2}$ for any $z \in \mathbb{C}\setminus [1,+\infty)$ . Of course, proving that $\mathbb{E}[e^{zX^2/2}]$ is holomorphic on this domain is a bit more delicate.

Fresnel integral. Taking $b=c=0$ and $a=1+2i$ (which is not in $[1,+\infty[$ as requested) we see that

\mathbb{E}[e^{aX^2 / 2}] = \frac{1}{\sqrt{2\pi}}\int e^{it^2}dt.

The value of the LHS was proven to be $1 / \sqrt{-2i} = (1+i)/2$ . We thus get $\int e^{it^2}dt = (1+i)\sqrt{\pi /2}$ . In particular, we recover the famous Fresnel integral

\int_0^\infty \sin(t^2)dt = \int_0^\infty \cos(t^2)dt = \frac{\sqrt{\pi/2}}{2} = \sqrt{\frac{\pi}{8}}.

The multi-dimensional case

For simplicity, in this case we'll only consider standard Gaussians. The general formula can nonetheless be derived using the same proof, see right after the proof.

Let $X \sim N(0,I_d)$ and let $A$ be a real symmetric matrix. Then, $\mathbb{E}[e^{\frac{1}{2}\langle X, AX\rangle}]$ diverges if $\lambda_{\max}(A)\geq 1$ , and otherwise

\mathbb{E}[e^{\frac{1}{2}\langle X, AX\rangle}] = \sqrt{\det(I_d-A)^{-1/2}}.

The proof is trivial once we have the one-dimensional case : since

A = UDU^*

and

UX

has the same distribution as

X

, we can write

\langle X, AX\rangle

as a sum of

\lambda_i \xi_i^2

where the

\lambda_i

are the eigenvalues of

A

(all of which are smaller than

1

by assumption) and the

\xi_i

are i.i.d.

General formula. Now if $\mu$ was to be nonzero, one could write $X = \mu+Y$ with $Y \sim N(0,I_d)$ , and then develop:

\langle X, AX\rangle = \langle \mu, A\mu\rangle + 2\langle A\mu, Y\rangle + \langle Y, AY\rangle.

Set $A = U^* DU$ and $Z = UY$ . Then, $Z \sim N(0,I_d)$ and

\mathbb{E}[e^{\frac{1}{2}\langle X, AX\rangle}] = e^{\frac{1}{2}\langle \mu, A\mu\rangle}\mathbb{E}[e^{\frac{1}{2}(\langle Z, DZ\rangle+ 2\langle U^* A \mu, Z \rangle)}] = e^{\frac{1}{2}\langle \mu, A\mu\rangle}\prod_{i=1}^d \mathbb{E}[e^{\frac{\lambda_i}{2}Z_i^2 + 2b_i Z_i}]

where we introduced $b = U A \mu$ . Using the first result, we get

\mathbb{E}[e^{\frac{1}{2}\langle X, AX\rangle}] = e^{\frac{1}{2}\langle \mu, A\mu\rangle}\prod_{i=1}^d \frac{e^{\frac{b_i^2}{8(1-\lambda_i)}}}{\sqrt{1-\lambda_i}}.

Chi-square, again. If $X \sim N(0,I_d)$ , then $Y:=|X|^2 \sim \chi^2(d)$ , and we recover the Laplace transform of a chi-square distribution $Y$ with $d$ degrees of freedom:

\mathbb{E}[e^{-\lambda Y}] = \left(\frac{1}{1+2\lambda}\right)^{\frac{d}{2}}.

Simon Coste

Quadratic exponentials of Gaussian random vectors

The one-dimensional case

The multi-dimensional case