probability and almost surely convergence

Question

I need help with these proofs: $\{X_n\}$ independent random variables with $E[|X_n|] = \frac{1}{2^n}$. Prove:

A) $S_n= \sum X_n$ converges in probability to a random variable $Y$ using Chebyshev's inequality.

B) $S_n = \sum X_n$ converges almost surely (using martingale theory, I guess Doob's theorem)

Any ideas?

Answer 1

Consider the series with nonnegative terms $T=\sum\limits_n|X_n|$, then $E[T]$ is finite hence $T$ is almost surely finite. In particular, the series $S=\sum\limits_nX_n$ converges (absolutely) almost surely. Since almost sure convergence always implies convergence in probability, $S$ also converges in probability.

(Nota: Independence, Chebyshev and Doob are irrelevant.)