# What is a tensor product of random variables?

I am trying to understand the the following paper https://arxiv.org/pdf/1810.10971.pdf, in particular Example 2:

If $$Y \sim N(0,1)$$, the standard normal on $$\mathbb{R}$$, then

\begin{align*} \Big( \mathbb{E} \Big[ \frac{1}{m!} Y^{\otimes m }\Big]\Big)_{m \geq 0 } = \exp\Big(\frac{1}{2} e_1 \otimes e_1 \Big) \in \prod_{m \geq 0 } \mathbb{R}^{\otimes m} \end{align*},

where $$e_1$$ is the unit basis vector of $$\mathbb{R}$$ and $$\exp$$ is defined in the following way:
\begin{align*} \exp &: \prod_{m \geq 0 } V^{\otimes m} \to \prod_{m \geq 0 } V^{\otimes m},\\ exp(s) &:= \sum_{m \geq 0} \frac{s^{\otimes m}}{m!} \end{align*}

What does it mean to take a tensor product of random variables? In particular, what is $$\mathbb{E} \Big[ Y^{ \otimes 3 }\Big]$$, for example?

In this context, I believe the tensor product on random variables is nothing other than the tensor product over the values of the RVs. (In other words, if $$\Omega$$ is a sample space and $$X : \Omega \rightarrow V$$ and $$Y : \Omega \rightarrow W$$ are RVs, then $$X \otimes Y : \Omega \rightarrow V \otimes W$$ is defined by $$(X \otimes Y)(\omega) = X(\omega) \otimes Y(\omega)$$.)

In Example 2, the RV $$Y$$ takes values in $$\mathbb R$$, which makes the tensor powers rather boring since $$\mathbb R^{\otimes n} \simeq \mathbb R$$ for any $$n \geq 0$$; we just have to take apart the notation to read out the usual (normalized) moments for the standard normal.

Thus we can expand (see Appendix A, for example) $$\exp \left ( \frac 12 e_1 \otimes e_1 \right ) = \left (1, 0, \frac 12, 0, \frac 1{2!} \left ( \frac12 \right )^2, 0, \ldots \right),$$

noting that this notation gives the coefficients of $$e_1^{\otimes n}$$ for $$n \geq 0$$.

Thus $$\mathbb E [Y^{\otimes 3}] = 0 \in \mathbb R^{\otimes 3}$$, since the standard normal has zero odd moments.

• Thank you! It all makes much more sense now! Sep 11 at 16:38