Let $\{X\_n\}$ be an ergodic sequence of random variables, $X\_n:(\Omega,\mathcal{F})\to (S,\mathcal{S})$ where the target set $S$ is a matrix ring. My question is,



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> Can the following limit be found almost surely? $$\frac{\displaystyle\sum\_{N\ge i\_1>i\_2>\cdots>i\_k\ge 1}X\_{i\_1}X\_{i\_2}\cdots X\_{i\_k}}{\displaystyle\binom{N}{k}}$$
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I think this limit could be found, had the target set been a commutative ring, by an application of [Newton's identity](https://en.wikipedia.org/wiki/Newton%27s_identities), but because of the noncommutative nature here, I cannot apply that principle here. Is there any way to tackle this problem?


Please make reference to any material available, as I do not have any proper background on non-commutative ring theory (nor on ergodic theory applied to rings, for that matter), apart from the basics of rings. Thanks in advance.



Ergodicity of elementary symmetric polynomials with noncommutable variables
Let me give an answer for real matrices in the case $k=2$. I believe that larger values of $k$ can be handled by induction. 


Let $A=\mathbb E X\_0$. I will write out three sums for $k=2$:


\begin{align\*}
S\_1&=\Big(X\_0(X\_1+\ldots+X\_{N-1})+X\_1(X\_2+\ldots+X\_{N-1})+
\ldots+X\_{N-2}(X\_{N-1})\Big)\\
S\_2&=\Big((N-1)X\_0A+(N-2)X\_1A+\ldots+1X\_{N-2}A\Big)\\
&=\Big((N-1)X\_0+\ldots+X\_{N-2}\Big)A\\
&=\Big((X\_0+\ldots+X\_{N-2})+(X\_0+\ldots+X\_{N-3})+\ldots+(X\_{0})\Big)A\\
S\_3&=\Big((N-1)A+(N-2)A+\ldots+A\Big)A=\binom{N}2A^2.
\end{align\*}
By the ergodic theorem, each bracketed term in the $S\_1$ differ from the corresponding bracketed term in the $S\_2$ sum by $o(N)$, so that $S\_1-S\_2=o(N^2)$.
Similarly, each bracketed terms in the final version of the $S\_2$ sum differs from the corresponding bracketed terms in the $S\_3$ sum by $o(N)$, so that $S\_3-S\_2=o(N^2)$ also. Hence $S\_1-S\_3=o(N^2)$ and your expression converges to $A^2$. 


