Interpret the following sequence $X_n^{(k)} = \underset{1 \leq i_1 < \dots <i_k \leq n }{\sum} \; \xi_{i_1} \dots \xi_{i_k}$

Question

I'm working on a problem in which I have the following set up.

Let $\xi_1, \xi_2, \dots$ be independent random variables with $E[\xi_i] =0$ for all $i$. Then they define the following sequence

$ X_n^{(k)} = \underset{1 \leq i_1 < \dots <i_k \leq n }{\sum} \; \xi_{i_1} \dots \xi_{i_k}$

They gave the following examples:

For k = 1 we get $X_n^{(1)} = S_n = \sum _{i=1}^{n} \xi_i$

For $k=2$ they say that $ 2 X_n^{(2)} = S_n^2 - \sum _{i=1}^{n} \xi^2_i$.

Explain the definition of the sequence $X_n^{(k)}$. For example what is the expression for $k=3$ ? and $k=4$?

The aim afterword is to prove that the sequence for any fixed $k$ is a martingale w.r.t some given filteration.

Answer 1

It looks like they are taking all ordered combinations of k elements and summing them under the condition that $k\leq n$

So for example, with k = 1, we get the regular sum. With $k=2$, we have a sum of terms, with each term a product of two elements from the finte sequence $\xi_1, \dots, \xi_n$.