What is the proper notation for the "average of the last N elements of a sequence"?

Question

I use sliding window average in my calculations: average for the last N elements and average for the N elements before that. What would be the correct mathematical notation to describe this?

Answer 1

If $x_n$ is your sequence I would just define another sequence $y_n$

$$y_n=\frac{1}{N}\sum_{k=n-N+1}^n x_k$$

and from then on use $y_n$ to denote your running average over the last $N$ elements.