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?