# Matrix-valued cumulant generating function for Wishart matrices

Suppose we have an axis-aligned Gaussian vector $$v \sim \mathcal{N}(\mu, \sigma^2 I_{d \times d})$$, and consider the Wishart matrix $$W = vv^\top$$.

Is there a simple closed form/"Lowener order upper bound" for the matrix-valued cumulant generating function $$\log \mathbb{E}(e^{tW})$$ for real values $$t$$? Even the special case of $$\mu = 0$$ is interesting for me. I attempted to Google, but the closest thing I found are real-valued versions of moment/cumulant generating functions, e.g. $$t$$ is a matrix and the exponent is $$\mathrm{tr}(tW)$$.

Thanks!

• Do you have a response to the answer below? Commented Dec 12, 2021 at 16:16

$$\newcommand\R{\mathbb R}$$You want to find $$\ln M_W(t)$$, where $$M_W$$ is the moment generating function (mgf) of $$W$$, given by $$\begin{equation*} M_W(t)=E e^{tW}=\sum_{n=0}^\infty\frac{t^n EW^n}{n!} \end{equation*}$$ for real $$t$$. By obvious rescaling, without loss of generality $$\sigma=1$$.
For natural $$n$$, $$\begin{equation*} W^n=|v|^{2(n-1)} vv^\top, \end{equation*}$$ where $$|v|$$ is the Euclidean norm of $$v$$, and $$W^0=I$$, the identity matrix.
So, for $$\mu=(\mu_1,\dots,\mu_d)\in \R^d$$ and independent standard normal $$Z_,\dots,Z_d$$, the $$(i,j)$$-entry of the matrix $$EW^n$$ is \begin{align*} (EW^n)_{i,j}&=E(\mu_i+Z_i)(\mu_j+Z_j)\Big(\sum_{k=1}^d(\mu_k+Z_k)^2\Big)^{n-1} \tag{1} \\ &=\sum\nolimits'\frac{(n-1)!}{j_1!\cdots j_d!}\, E(\mu_i+Z_i)(\mu_j+Z_j)\prod_{k=1}^d(\mu_k+Z_k)^{2j_k}, \notag \end{align*} where $$\sum'$$ denotes the sum over all $$d$$-tuples $$(j_1,\dots, j_d)$$ of nonnegative integers such that $$j_1+\cdots+j_d=n-1$$. Expanding $$(\mu_i+Z_i)(\mu_j+Z_j)\prod_{k=1}^d(\mu_k+Z_k)^{2j_k}$$ to get an explicit polynomial in $$Z_1,\dots,Z_d$$ and using the known expression for the moments of the standard normal distribution, we get an explicit expression for $$EW^n$$ and hence for $$M_W(t)=E e^{tW}$$. The resulting expression may be unwieldy.
However, in the special case $$\mu=0$$ (also of interest to you), the result is simple. Indeed, then by (1) and symmetry we get $$\begin{equation*} (EW^n)_{i,j}=1(i=j)EZ_i^2\Big(\sum_{k=1}^d Z_k^2\Big)^{n-1} =\frac{1(i=j)}d\, EX^n, \end{equation*}$$ where $$X:=\sum_{k=1}^d Z_k^2$$, which has the gamma distribution with parameters $$d/2,2$$. So, \begin{align*} M_W(t)&=\Big(1+\frac1d \sum_{n=1}^\infty\frac{t^n EX^n}{n!}\Big)I \\ &=\Big(1+\frac1d \,[Ee^{tX}-1]\Big)I \\ &=\Big(1+\frac1d \,[(1-2t)^{-d/2}-1]\Big)I \end{align*} for real $$t<1/2$$, and $$M_W(t)=\infty I$$ for real $$t\ge1/2$$. So, $$\begin{equation*} \ln M_W(t)=\ln\Big(1+\frac1d \,[(1-2t)^{-d/2}-1]\Big)\,I \end{equation*}$$ for real $$t<1/2$$.