Expectation of the Ratio of the Sum of Squares to the Square of the Sum: Exact and Asymptotic Results

Let $X_i$, $i=1,\dots,n$, be a sequence of positive independent identically distributed random variables. Define
$$ R_n:=\mathbf{E}\frac{X_1^2+X_2^2+\dots+X_n^2}{(X_1+X_2+\dots+X_n)^2}. $$
Let $\varphi(s)=\mathbf{E}e^{-sX}$. We give an explicit representation of $R_n $ in terms of $\varphi$, and with the help of the Karamata theory of functions of regular variation, we study the asymptotic behavior of $R_n$ for large $n$.