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Let $X_1, X_2, \dots$ be independent, with

$$P(X_n=1) = P(X_n = -1) = \frac{1}{2} \left( 1 - \frac{1}{n^2} \right),$$

and

$$P(X_n = n) = P(X_n = -n) = \frac{1}{2n^2}.$$

Let $S_n = X_1 + \cdots + X_n$ and $S_n^* = S_n/\sqrt{n}$. The first part of the problem asked to show that $\text{Var}(S_n^*) \rightarrow 2$, which was a pretty straightforward calculation.

The second part asks us to show that $S_n^* \stackrel{d}{\underset{n\to\infty}{\longrightarrow}} Z$, where $Z$ is a normal random variable. I tried to do this by using characteristic functions. We have that

$$\varphi_{X_n}(t) = \cos t + \frac{1}{n^2}\left( \cos (nt) - \cos t \right).$$

However, writing the characteristic function of $S_n^*$ in terms of $\varphi_{X_n}(t)$ made it really messy and not workable much. Is there a more efficient way of approaching this problem? Would Lindeberg-Feller CLT work here?

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The lindeberg CLT can be applied for this sequence of random variables.

Hint: figure out a convinient lower bound for the cumulative variance

$s_n^2=\sum_{i=1}^n \sigma_k^2$

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  • $\begingroup$ I don't think Lindeberg-Feller CLT will give the result here. However, I did get the result by using characteristic functions. $\endgroup$ – symmetricuser Jan 27 '15 at 20:55

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