# Distribution and convergence of two random variables

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

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.