# Convergence a.s. of a series of positive random variables.

Suppose $$\{X_{n}\}$$ is a sequence of non-negative but not necessarily independent random variables, $$S_n$$ is the sum of the first $$n$$ random variables, and $$S_{n}$$ converges to $$S$$ in probability. I would like to show in fact that $$S_{n}$$ converges to $$S$$ almost surely. I tried mimicking the argument in how to show convergence in probability imply convergence a.s. in this case? but I believe since we don't have independence here this won't work. I tried using the fact that $$X_{n}$$ is Cauchy in probability to try and show that set where $$S_{n}$$ goes to infinity has probability 0 but couldn't figure out the details. Appreciate if someone could help out.

Independence is not necessary. Since $$X_n \geq 0$$ it follows that $$(S_n)$$ is increasing. So $$T=\lim S_n$$ exists (but may be $$\infty$$). But $$S_n \to S$$ in probability and this implies there is a subsequence $$(S_{n_k})$$ which converges a.s to $$S$$. [Theorem 4.2.3 of K L Chung's "A Course in Probability Theory']. Now $$(S_{n_k})$$ converges to $$T$$ as well as $$S$$ so $$S=T$$ a.s.. Hence $$S_n \to T=S$$ with probability $$1$$.