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 $\sum_{n}{Var(Y_n)/n^2}<+\infty$.).


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.
  • $\begingroup$ Thanks a lot for your answer. Sorry if I misunderstood something, but it seems to me that this example does not satisfy the assumption $\mathbb{E}Y_n>d>0$, does it? $\endgroup$ – Oliv Mar 24 '17 at 20:01
  • $\begingroup$ @Oliv Sorry, I missed that part of your assumption. See my edited answer. $\endgroup$ – saz Mar 24 '17 at 20:48
  • $\begingroup$ Great answer, many thanks. Since $\sum{\mathbb{P}(Y_i \ne 0)}<\infty $, Borel Cantelli lemma implies that with probability 1 $Y_i \ne 0$ finitely often, and thus $X_n$ is asymptotically constant. Therefore $X_n $ has finite limit a.s.. Hence my conjecture is not true. Is that correct? Thanks again! $\endgroup$ – Oliv Mar 24 '17 at 21:06
  • $\begingroup$ @Oliv Yes, that's correct. $\endgroup$ – saz Mar 24 '17 at 21:34

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