Exchange limit in distribution and pointwise limit

probability-theory

Solution 1:

Let $X_n(c)$ be the deterministic random variable $X_n(c) = \frac{1}{\left(1+\frac{1}{1+|c|}\right)^n}$. Then $\lim_{c\to\infty}X_n(c) = 1 = Y_n$ and $\lim_{n\to\infty}X_n(c) = 0$. But $1$ does not converge to $0$ in distribution.

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