Convergence in probability and convergence of Cesaro means

Question

Consider the random variable $X_n$, not necessarily iid. If $X_n\rightarrow 0$ almost surely, then the Cesaro means $\frac 1n\sum_{k=1}^nX_n$ converge almost surely to 0. This cannot be weakened to convergence in probability, as has been discussed here before by considering an independent sequence $\{X_n\}$, such that $X_n=2^n,$ with probability $1/n$,and $X_n=0$ with probability $1-1/n$.

My question is, if $|X_n|\leq 1$ surely, does convergence in probability of $X_n$ to zero imply the almost sure convergence of $\frac 1n\sum_{k=1}^nX_k$ to zero?

Answer 1

Consider a sequence of random variables $(Y_i)_{i\geqslant 1}$ such that $0\leqslant Y_i\leqslant 1$ and $Y_i\to 0$ in probability but not almost surely (for example, $Y_i=\mathbf 1(A_i)$, where $(A_i)_{i\geqslant 1}$ is a sequence of independent events). Now consider an increasing sequence of positive integers $(n_j)_{j\geqslant 0}$ such that $$\forall j\geqslant 1, \frac{n_j-n_{j-1} } {n_j} \geqslant \frac 12, n_0=1$$ and define $$X_j=Y_i\mbox{ if }n_{i-1} \leqslant j\lt n_{i}.$$
Then $X_j\to 0$ in probability, $|X_j|\leqslant 1$ and $$\frac 1{n_\ell}\sum_{j=1}^{n_\ell}X_j=\frac 1{n_\ell}\sum_{i=1}^\ell(n_i-n_{i-1})Y_i\geqslant \frac{n_\ell-n_{\ell-1}}{n_\ell}Y_\ell\geqslant \frac{Y_\ell}2,$$ hence the sequence $1/n\sum_{j=1}^n X_j$ cannot converge to $0$ almost surely.