Cesaro sum convergence in probability

Question

I saw in this question that for random variables $\{X_n\}$ converging to 0 in probability it is not necessarily true that their Cesaro sum $\frac{X_1+X_2+\ldots X_n}{n}$ converge in probability to 0. I am interested in sort of the converse of this. That is, if the Cesaro sum converge in probability to zero, then does the sequence $\{ (\frac{X_n}{n})_{n \geq 1} \}$ converge in probability to 0?

Answer 1

If you let $S_n = X_1 + \dots + X_n$ then $$\frac{X_n}{n} = \frac{S_n}{n} - \frac{n-1}{n}\cdot\frac{S_{n-1}}{n-1}$$ Since $\frac{n-1}{n} \to 1$ you get that $\frac{X_n}{n} \to 0$ in probability.