Almost sure convergence to infinity of a sum of independent variables

Question

Let $Y_n$ be independent real-valued random variables such that $\mathbb{E}{Y_n^2}<+\infty$ for all $n$, and the sequence $\mathbb{E}Y_n=d_n$ is bounded below by some $d>0$.

Let $X_n=Y_1+\cdots+Y_n$. Is it true that $X_n \rightarrow +\infty$ almost surely?

I am learning some probability theory, so I would prefer a hint rather than a complete answer if possible. Many thanks in advance.

Note: my approach so far has been to look for an appropriate version of the strong law of large numbers for sums of independent but not iid variables, but it seems to me than they all require additional assumptions (for instance Kolmogorov's strong law of large numbers requires $\sum_{n}{Var(Y_n)/n^2}<+\infty$.).

Answer 1

Hint: Consider a sequence of independent random variables $(Y_i)_{i \in \mathbb{N}}$ such that $\mathbb{P}(Y_i = 2^i) = 2^{-i}$ and $\mathbb{P}(Y_i =0) = 1- 2^{-i}$.

1. Using the Borel Cantelli lemma show that $\mathbb{P}(\limsup_{i \to \infty} \{Y_i \neq 0\})=0$.
2. Conclude that for almost all $\omega$ there exists $N \in \mathbb{N}$ such that $Y_n(\omega)=0$ for all $n \geq N$.
3. Conclude.