# What is an example of a sequence of random variables $|Y_n|\leq 1$ converging in probability to $0$ but the variance is either constant or blows up?

I am trying to come up with an example of a sequence of random variables $$|Y_n|\leq 1$$ converging in probability to $$0$$, but where the variance $$Var(Y_n)$$ does not converge to $$0$$ in limit. Is there a way I can use a transformation of a variable $$Y_n ~ Bern\left(\frac{1}{n}\right)$$?

• Doesn't such example contradict dominated convergence theorem for convergence in probability? – Shashi Oct 17 at 20:57
• Which part does it contradict? thanks! – user321627 Oct 17 at 21:25
• I mean the variance should go to zero by DCT... So you cannot find an example – Shashi Oct 17 at 21:31
• How could I prove the variance goes to zero by the DCT? – user321627 Oct 17 at 22:11

Such example cannot exist.

Why? Well, let $$a_n:=\operatorname{Var}(Y_n)$$ Take a subsequence $$a_{n_k}$$, note that $$Y_{n_k}$$ still converges in probability to zero. It has a further subsequence $$Y_{n_{k_j}}$$ converging almost surely to zero. Since $$|Y_{n_{k_j}}|\leq 1$$ a.s., we have $$|Y_{n_{k_j}}^2|\leq 1$$ a.s.. Hence by DCT $$a_{n_{k_j}} = \mathbb E(Y_{n_{k_j}}^2)-\mathbb E(Y_{n_{k_j}}) ^2\to 0$$ as $$j\to\infty$$.

So every subsequence of $$a_n$$ has a further subsequence converging to zero. So the sequence $$a_n$$ converges to zero by this fact.

Reversed binomial distribution described bellow worked.

EDIT: The $$Y_m$$ I constructed may not satisfy the condition "converging in probability to $$0$$". It may be modified but I couldn't find precisely what "converging in probability to $$0$$" means.

EDIT: I found the definition of "converging in probability to $$0$$" in the Wikipedia and that my example does not work.

(My trial was: Let $$P( X_m=j/m ) = (1/4)^m * (2m)!/(m-j)!(m+j)!$$ for $$-m\leq j\leq m$$,
for $$-m< j<0$$, $$P( Y_m=-1-j/m ) = P( X_m=j/m )$$
for $$0
for $$j=-m,0,m, P( Y_m=j/m ) = P( X_m=j/m )$$)