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?


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 $\limsup_{n\to +\infty}\mathbb P\{|X_n^n|\gt \delta\}$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.