Expected Bilinear Forms: Trace Formula and Mean-Square Continuity

Let $(\Omega,\mathcal{F},P)$ be a probability space let $\mathbb{R}$ be the real numbers, and let $p,q\ge1$ be natural numbers. A real-valued function defined on a closed interval $[a,b]\subseteq\mathbb{R}$ is called continuous on $[a,b]$ when it is continuous relative to $[a,b]$, both $[a,b]$ and the codomain $\mathbb{R}$ carrying the metric of the real line. Dot products are the dot product applied componentwise to tuples of random variables, matrix actions are the matrix-vector product, and $\operatorname{tr}$ is the trace.

1. (Trace formula) Let $Y=(Y^{1},\dots,Y^{p})$ be a tuple of square-integrable random variables and let $M$ be a real $p\times p$ matrix. Then the random variable $Y\cdot(MY)=\sum_{i,j}M_{ij}Y^{i}Y^{j}$ is integrable, and with the expectation, the covariance, $\mu:=(\mathbb{E}[Y^{1}],\dots,\mathbb{E}[Y^{p}])$, and $C:=\bigl(\operatorname{Cov}(Y^{i},Y^{j})\bigr)_{1\le i,j\le p}$,

moreover $\operatorname{tr}(M^{\top}C)=\operatorname{tr}(MC)$, the matrix $C$ being symmetric.

2. (Mean-square continuity of expected bilinear forms) Let $a<b$ be real numbers, let $(Y_t)_{t\in[a,b]}$ and $(Z_t)_{t\in[a,b]}$ be families of tuples $Y_t=(Y^{1}_t,\dots,Y^{p}_t)$ and $Z_t=(Z^{1}_t,\dots,Z^{q}_t)$ of square-integrable random variables whose component families are mean-square continuous on $[a,b]$, and let $M$ assign to each $t\in[a,b]$ a real $p\times q$ matrix $M(t)$ whose entries are continuous functions of $t$. Then for every $t$ the random variable $Y_t\cdot(M(t)Z_t)$ is integrable, and the function

is defined, finite, and continuous on $[a,b]$.