# Prove convergence in distribution of sum of non-i.i.d random variables.

$$X_1,X_2,X_3 ,\ldots$$ 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)

• What is $p$ here? – Kavi Rama Murthy Jun 2 '20 at 5:38
• Oops! p is a number belong to real number. I should use another letter, "p" is confusing. – Charlie F Jun 2 '20 at 9:38
• This question can be solved by Using lyapunov central-limit theorem – Charlie F Jun 2 '20 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.