$X_1,X_2,X_3 ,\ldots$ be independent random variables with distribution $P(X_i=i)=P(X_i=-i)=1/2$ for all $i$. Define $S_n=X_1+X_2+X_3+\cdots+X_n$.

And the question is to show "Does $\{S_n/n^p\}_{n=1}^∞$ converge in distribution? Why?"

I know can use CLT or LLN when $X_i$ be i.i.d random variables, but in this question, $X_i$ has different distribution.

I hope can get some hint. :) And I m confused on how to prove converge in distribution(I know that can use φ,E,CDF)

  • $\begingroup$ What is $p$ here? $\endgroup$ – Kavi Rama Murthy Jun 2 at 5:38
  • $\begingroup$ Oops! p is a number belong to real number. I should use another letter, "p" is confusing. $\endgroup$ – Charlie F Jun 2 at 9:38
  • $\begingroup$ This question can be solved by Using lyapunov central-limit theorem $\endgroup$ – Charlie F Jun 2 at 9:39

can we really use Lyapunov CLT to solve this question? Here the variance of $X_i$ should be $i^2$. Lyapunov central-limit theorem need a finite variance.

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  • $\begingroup$ You are right. I miss this requirement. :( $\endgroup$ – Charlie F Jun 3 at 2:03

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