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For every $n\in\Bbb{N}$, let $X_n$ be a random variable which gets the values $\{-1, -\frac{n-1}n,...,-\frac 1 n, 0, \frac 1 n,...,\frac{n-1}n, 1\}$ with equal probability.

  1. Find a random variable $X$ such that $X_n$ converges in distribution to $X$.

I got $X\sim U[-1,1]$.

  1. Find a random variable $Y$ such that $X_1, X_2^2, ..., X_n^n,...$ converges in distribution to $Y$.

Any ideas on this one?

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Hint: We have for any $\delta\gt 0$, $$\tag{*}\mathbb P\{|X_n^n|\gt \delta\}=\mathbb P\{|X_n|\gt \delta^{1/n}\}.$$ Let $\delta$ be fixed and consider $\varepsilon\gt 0$. There is some $n_0=n_0(\varepsilon,\delta)$ such that if $n\geqslant n_0$, then $\delta^{1/n}\gt 1-\varepsilon$. For such $n$'s, in view of (*), we have $$\mathbb P\{|X_n^n|\gt \delta\}\leqslant \mathbb P\{|X_n|\gt 1-\varepsilon\}.$$ Now use the first part to find an upper bound for $\limsup_{n\to +\infty}\mathbb P\{|X_n^n|\gt \delta\}$.

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