The independent binary random variables $X_k$ take values $\pm 1$ with the probabilities $(1 \pm k^{\frac{-1}{2}})/2, k = 1,2,...$. Find $\lim_{n \rightarrow \infty} \mathbb{P}(X_1 + ... + X_n \leq 0)$.
Here are my thoughts so far:
I'd like to use the Central Limit Theorem here -- if $X_1,...,X_n$ are a sequence of i.i.d. random variables, each with mean $\mu$ and variance $\sigma^2$, then the distribution of $\frac{X_1 + ... + X_n - n\mu}{\sigma \sqrt{n}}$ tends to the standard normal as $n \rightarrow \infty$.
However, here, although $X_1,..,X_n$ are independent, they are not identically distributed.
I first let $Y_k = \frac{X_k + 1}{2}$, so that $P(Y _k = 0) = \frac{1 - k^{\frac{-1}{2}}}{2}$ and $P(Y _k = 1) = \frac{1 + k^{\frac{-1}{2}}}{2}$. This gives that $Y_k$ is a Bernoulli random variable with probability of success $\frac{1 + k^{\frac{-1}{2}}}{2}$. This makes our calculations easier to handle, since we're now working with random variables we're familiar with -- but still, $Y_1,...,Y_n$ are independent but not identically distributed.
Where can I continue from here ? Another idea is to compute the moment generating function of $X_1 + ... + X_n$, and map this back to the p.d.f. of a familiar random variable by the uniqueness theorem -- but, the resulting moment generating function doesn't map to such a familiar p.d.f., as the sum of Bernoulli random variables with different success probabilities does not give a nice, familiar distribution. Thus, I'm sticking with my instinct that the Central Limit Theorem is the way to go here.
Any help would be appreciated. Thanks!