I would like to show that the sequence of random variables $\frac{S_n}{n}$ defined below converges to zero. The proof requires the application of Cesaro lemma and I don't understand how.
First, let the sequence $M_n$ be an $L^2$-bounded martingale. By the martingale convergence theorem, $M_n$ converges almost surely.
Second, let $S_n$ be a sequence of random variables such that
$$ S_n = \sum_{k=1}^n k(M_k-M_{k-1}) \\ =\sum_{k=1}^n kM_k - \sum_{k=1}^n kM_{k-1} \\ = \sum_{k=0}^n kM_k- \sum_{k=0}^{n-1} (k+1)M_k \\ = nM_n - \sum_{k=0}^{n-1} M_k$$ where $M_0=0$.
Claim: by Cesaro lemma, $\frac{S_n}{n}$ converges to zero.
Question: Could you help me understand how I can apply Cesaro lemma to show the claim? I found this statement of the Cesaro theorem on wiki. Based on it, I would set $b_n\equiv n$ and $a_n\equiv S_n$. However, I don't see how $\lim_{n\rightarrow\infty} \frac{S_{n+1}-S_n}{n+1-n}=\ell$, which is required by the theorem, where $$\lim_{n\rightarrow\infty} \frac{S_{n+1}-S_n}{n+1-n}=\lim_{n\rightarrow\infty} S_{n+1}-S_n = (n+1)M_{n+1}-\sum_{k=0}^n M_k - nM_n+\sum_{k=0}^{n-1} M_k=(n+1)M_{n+1}-(n+1)M_n=(n+1)(M_{n+1}-M_n). $$ I have also found a statement of Cesaro lemma on this forum but, again, I am unable to map $b_n$ and $\nu_n$ to my setting.
You just need the particular case of arithmetic mean of Cesàro's lemma, which yields that $\frac{1}{n}\sum_{k=0}^{n-1}M_k$ converges almost surely as $n\to+\infty$ to the same limit as that of $M_n$. Hence $$ \frac{S_n}{n}=M_n-\frac{1}{n}\sum_{k=0}^{n-1}M_k $$ vanishes almost surely as $n$ goes to infinity.