If I know the limit of Cesaro averages , then can I know limits of uniform Cesaro averages. Details Below

Say that I have a sequence $$\{a_{n}\}_{n\in\mathbb{N}}$$ $$\subset$$ [0,1] such that lim$$_{N\rightarrow\infty}$$ $$\frac{1}{N}$$ $$\sum_{n=1}^{N}$$ a$$_{n}$$ = a.

Can I somehow get the value of lim$$_{(N-M)\rightarrow\infty}$$ $$\frac{1}{N-M}$$ $$\sum_{n=M}^{N-1}$$ a$$_{n}$$ = a.

If that is not the case for general sequences, what specific conditions do the sequence need to satisfy for this? So far, only thing I could come up with was to try to manipulate one of them in hope that somehow the other limit would pop out. I have not been very successful yet.

Put $$K=N-M$$ thus $$N=M+K$$ so $$\frac{1}{N-M}\sum_{n=M}^{N-1}a_n=\frac{1}{K}\sum_{n=M}^{M+K-1}a_n= \frac{1}{K}\sum_{k=1}^{K}a_{k-M+1}$$