Random Variable with Values in $\mathcal{L}(\mathbb{R}^n)$

Question

Let $(\Omega,\Sigma,\mathbb{P})$ a probability space and $\mathcal{L}(\mathbb{R}^n)$ be the vector space of all linear transformation $T:\mathbb{R}^n\to \mathbb{R}^n$.

Suppose that $X:\Omega \to \mathcal{L}(\mathbb{R}^n)$ is a random variable.

My question is: how can I define a notion of expected value such that $\mathbb{E}[\lambda (X)]=\lambda (\mathbb{E}[X])$ for all linear functionals $\lambda :\mathcal{L}(\mathbb{R}^n)\to \mathbb{R}$?

The author of the book "Mathematical Statistics" (written by Wiebe R. Pestman) uses this notion in the chapter 8 without explaining it.

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Thank you for your attention!

Answer 1

$\mathcal L(\mathbb R^n)$ can be represented as $n \times n$ matrices, and the expected value taken coordinate-wise:

$\mathbb E[X]$ is the $n \times n$ matrix with entries $(\mathbb E[X])_{ij} = \mathbb E[X_{ij}]$.

Answer 2

After choosing a basis for $\mathbb{R}^n$ we have $\mathcal{L}(\mathbb{R}^n)\cong\mathbb{R}^{n\times n}$ so we may assume that $X$ is a random matrix and each entry $X_{ij}$ of $X$ is a random variable on $\mathbb{R}$. In this case we may define the expectation as a matrix with coordinates $$\mathbb{E}[X]_{ij}:= \mathbb{E}[X_{ij}].$$

The linearity of $\mathbb{E}[X]$ then follows from the linearity of $\mathbb{E}[X_{ij}]$.