Let $X_n$ be any sequence of random variables. Find a suequence of non-zero numbers $a_n$ such that $a_nX_n$ converges to $0$ in distribution.

Can someone give me a hint? - I stuck on it.


closed as off-topic by Did, quid, Davide Giraudo, Michael Albanese, user147263 Nov 24 '15 at 23:21

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Let $F_n$ be the distribution function of $|X_n|$, i.e., $F_n(x) = \mathbb P(|X_n| \le x) $.

Define $F_n^{-1}(y) = \inf\{x:F_n(x) \ge y\}$. Let $a_n$ be such that $$\frac 1 {a_n n} = F_n^{-1}(1-1/n).$$ Then $$ \mathbb P \left\{|X_n| \ge \frac 1 {a_n n}\right\} \le \frac 1 n. $$ In other words, $a_n X_n$ converges in probability to $0$, which implies that it also converges in distribution to $0$.

  • 2
    $\begingroup$ +1. Exercise: Adapt this exact idea to ensure almost sure convergence. $\endgroup$ – Did Nov 24 '15 at 15:17

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