Tensor Product Notation for Probability Distributions

Question

I have a quick question about notation in probability theory. I have come across a couple of examples in which a random vector $X$ with $n \in \mathbb{N}$ iid entries is said to have the distribution $$ X \sim \mathcal{N}^{\otimes n}(\mu, \sigma^2).$$ My question is, what does this mean exactly? Is every entry normally distributed with parameters $\mu, \sigma^2$? Or is this supposed to denote a joint distribution?

Answer 1

It denotes a specific joint distribution.

For two probability measures $\mu$ and $\nu$, you can find the definition of their product measure $\mu\otimes\nu$ for instance here.

In terms of random variables, you can show that a random vector $X=(X_1,X_2)$ is such that $X\sim\mu\otimes\nu$ iff $X_1\sim\mu$, $X_2\sim\nu$ and $X_1$ is independent of $X_2$.

This definition has a straightforward extension to the case of $n$ probability measures. Then you write that $(X_1,\cdots,X_n)\sim\mu_1\otimes\cdots\otimes\mu_n$ iff $X_1,\cdots,X_n$ are independent and $X_i\sim\mu_i$ for all $i$.

If for all $i$, $\mu_i=\mu$, you can write $\mu^{\otimes n}=\mu_1\otimes\cdots\otimes\mu_n$.

So back to your case, $$ X \sim \mathcal{N}^{\otimes n}(\mu, \sigma^2) $$ means that $X$ is a random vector with independent components, each normally distributed with mean $\mu$ and variance $\sigma^2$.