What is an example where we have convergence in distribution to a constant, but that doesn't imply convergence almost surely?

Question

I have been trying to disprove that if I have a sequence of random variables $X_n$, that $X_n \to a$, where $a$ is a constant, in distribution doesn't imply $X_n \to a$ almost surely. One example I came up with was where $X_n \sim N(0, n)$. $X_n$ converges in distribution to $1/2$, but I am not sure how to show almost convergence fails here. Does anyone have a hint or know if this is a valid example? Thanks.

http://math.stackexchange.com/questions/149775/convergence-of-random-variables-in-probability-but-not-almost-surely — , Feb 08 '16 at 06:59

score 4 · Accepted Answer · edited Dec 28 '16 at 02:28

4

For exemple , take the sequence where $X_n$ are independent bernoulli variables such that: $$P(X_n = 1 ) = 1/n$$ So , Borel-Cantelli tells you that the event $(X_n =1 )$ will happen infintely many times but $P(X_n=0)=1-1/n$ converges to 1 , so the sequence converges in distribution to 0

edited Dec 28 '16 at 02:28

BCLC

14,197

answered Dec 15 '16 at 23:36

Ariel Serranoni

504

What is an example where we have convergence in distribution to a constant, but that doesn't imply convergence almost surely?

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