Find two sequences $(X_n)$ for $n\in\mathbb{N}^{+}$, and $(Y_n)$ for $n\in\mathbb{N}^{+}$ of random variables such that all the following conditions are satisfied:

(a) For every $n \in \mathbb{N}^{+}$ the two random variables $X_n$ and $Y_n$ have the same distribution.

(b) The sequence $(X_n)$ converges almost surely to some random variable $X$.

(c) The sequence $(Y_n)$ does not converge almost surely, but it converges in probability to $X$.

(d) It is not true that:

$$ \lim_{n\to\infty} F_{X_n}(t) = F_X(t)$$ for all $t \in\mathbb{R}$.

Once I show that they have the same distribution, how can I show that $(Y_n)$ converges only in probability to $X$ ?


1 Answer 1


First you are to find an example of a sequence of 0-1 random variables converging to 0 in probability, but not a.s. Then these random varables are to be shifted by a small value, say, 1/n.


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