Existence of a limiting sum of random variables

Question

Consider that $X_i$‘s are independent exponentially distributed random variables with mean $1/i$ (and thus variance $1/i^2$).Then the sum of them seems to converge to a “random variable” with finite variance but unbounded mean. What’s the problem here? Why does not such random variable exist? Thanks

Answer 1

Since $X_i$'s are positive $\sum_{i=1}^{n} X_i$ converges (to a possibly infinte ) sum $X$ and Monotone Convergence Theorem gives $EX=\sum_{i=1}^{\infty} \frac 1 i =\infty$. There is no way a random variable with infinite mean can have finite variance. It is not even true that $X <\infty$ almost surely. [This can be shown using Laplace transforms].