Suppose $X_2,X_3,\ldots$ are independent random variables.
Assume that $X_k$ has the exponential distribution with parameter $\lambda_k=\dbinom{k}{2} $ for all $k$, which means $ E[X_k] = 1/\lambda_k $ and $\mathrm{Var}(X_k) = 1/(\lambda_k)^2$ for all $k$.
Let $ T_n=\sum_{k=2}^{n}kX_k$.
Prove that
$\dfrac{T_n}{2\log (n)}\overset {p}{\rightarrow} 1.$