let {x_i} be an infinite sequence
x_i = a for i < N,
x_i = 0 for i >= N.
Average calculated over 1 < i < n, where n < N is a. The intuitively "correct" average of the sequence is (approaching) 0.
i.e. for a = 1, N = 10 the sequence is
> 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 0, 0, 0, ...
then the average of samples
>5
>5, 5
>5, 5, 5
>...
> 5, 5, 5, 5, 5, 5, 5, 5, 5, 5
is 5. But the average of samples of size > N
> 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 0
> 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 0, 0
> 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 0, 0, 0
> ...
> 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 0, 0, 0, ...
starts approaching 0
