I have a written statement of the form "Let $X_1$ and $X_2$ be independent order-$n$ matrices whose elements are i.i.d. normally-distributed random variables".
Because I frequently use such statements in my current paper, I require a compact notation for declaring random variables such as $X_1$ and $X_2$. What seems natural to me is to define a symbol $\mathcal{N}(n)$ as "the set of all order-$n$ matrix random variables with i.i.d. normally-distributed elements" and then write $X_1,X_2 \in \mathcal{N}(n)$. Although I'm certain the readership could deduce what's meant by this, the definition of $\mathcal{N}(n)$ is erroneous since there's really only one order-$n$ matrix RV with i.i.d. normally-distributed elements, not a set of them. That is, the sample space is already encapsulated by the variable.
If I amend the definition of $\mathcal{N}(n)$ to "the sample space of an order-$n$ matrix random variable with i.i.d. normally-distributed elements", this is also erroneous since it would mean $X_1$ and $X_2$ are deterministic rather than random variables.
I could also define $\mathcal{N}(n)$ as "an i.i.d. process returning order-$n$ matrices whose elements are i.i.d. normal random variables", in which case $X_1,X_2 \in \mathcal{N}(n)$ would imply $X_1$ and $X_2$ were realizations of this process, but it seems to me that the first question in the reader's mind will be "Where is this 'process' coming from? What's generating these variables?", for which there's no good answer since 'process' here is just a linguistic contrivance.
I'm certain the literature must have standard notation for (what I'm denoting as) $\mathcal{N}(n)$ and $X_1,X_2 \in \mathcal{N}(n)$, as well as a standard definition for $\mathcal{N}(n)$. Can some kind soul please enlighten me as to what these are?
Best Answer
Why not use the $\sim$ symbol?
If $X\sim N(\mu,\sigma^2)$ is used to define a normal random variable with mean $\mu$ and variance $\sigma^2$, you can use $X\sim N_n(\mu,\sigma^2)$ (or $N_{n\times n}$) for an $n\times n$ matrix where all entries are iid $N(\mu,\sigma^2)$.