Convergence in distribution $\frac{X_n}{n}$, $X_n \sim$ geo$ (\frac{1}{n+1})$

Asked Feb 10 '20 at 20:38

Active Feb 10 '20 at 20:56

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Let $(X_n)_{n \geq 1}$ be a sequence of random variables, and $X_n \sim$ geo$ (\frac{1}{n+1})$. Find a limit in distribution of $(\frac{X_n}{n})_{n \geq 1}$.

Is there any general rule one can use to find whether a series converges in distribution or not? I know that convergence in probability implies convergence in distribution but I guess it might not always be the appropriate way to tackle this kind of exercise.

edited Feb 10 '20 at 20:56

asked Feb 10 '20 at 20:38

alladinsane

You could try to find the limit of the moment generating function or the characteristic function – Henry Feb 10 '20 at 20:44
Are you interested in the convergence of $X_n$ or $\sum_n X_n$ ? The name "series" is confusing – Gabriel Romon Feb 10 '20 at 20:53
$X_n$, sequence. Sorry. @Gabriel Romon – alladinsane Feb 10 '20 at 20:56
Let $\phi_{X_n}$ denote the characteristic function of $X_n$ (its closed form is known). You need only find the limit of $\phi_{X_n}(\frac tn)$, which is standard calculus. – Gabriel Romon Feb 10 '20 at 21:02

Convergence in distribution $\frac{X_n}{n}$, $X_n \sim$ geo$ (\frac{1}{n+1})$

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