Suppose we have a triangular array of positive random variables $\{\{X_{1i},\ldots,X_{ii}\},i=1,\ldots,n\}$ such that the random variables in the array are independent, all random variables have mean $\mu$ and the random variables on row $i$ are identical with variance $\operatorname{Var}[X_{1i}]=i^c-1$ where $c$ satisfies $0<c<1$. The random variables in the triangular array are unimodal, but are not symmetric. Their tails behave as follows $P(X_{1i}>x)\sim \frac{\exp\left[-\frac{1}{2}(g_i(x))^2\right]}{g_i(x)}$ where $g_i(x)\approx \frac{\log x}{a\sqrt{\log i}}+a\sqrt{\log i}$ with $a>0$ a constant.

I am interested in the asymptotic behavior of the sequence of arithmetic averages of the rows $S_1,\ldots,S_n$, where $S_i=\frac{1}{i}\sum_{j=1}^iX_{ji}$. Specifically, I am wondering if, as $n\rightarrow\infty$, $S_n$ converges in distribution to a random variable that is symmetric around its mean.

I am not even sure where to look for an answer to this question. Since the random variables $X_{1i}$ are positive, the condition for the left tail of the generalized CLT is not met (and I don't think the right tail meets the condition either, as it may be too light, but, obviously not light enough for the standard CLT to apply). However, all I need to know is whether the arithmetic averaging across the rows of the triangular array yields a symmetric distribution. Is there any way to learn this?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.