Weak Convergence

Prove that convergence in probability of a sequence of random variables implies weak convergence of the sequence of corresponding distributions.

Give an example confirming that weak convergence of distributions does not imply, in general, convergence in probability of the sequence of random variables.

Let the sequence of distributions $F_n$ converge weakly to a distribution $F$. On a single probability space, construct a random variable $\xi$ with distribution $F$ and each of the $\xi_n$ with distribution $F_n$ so that the convergence $\xi_n \xrightarrow {p} \xi$ holds.

Let $\xi_n$ converge weakly to some constant $c$. Prove that then $\xi_n \xrightarrow {p} c$.

Let $\{ F_n\}$ be a sequence of degenerate distributions, with $F_n$ concentrated at the point $x_n$. Prove that weak convergence $F_n \Rightarrow F$ is equivalent to the existence of the limit $\lim_{n \rightarrow \infty } x_n = x$ together with $F$ being degenerate at the point $x$.

Construct an example of a sequence of distributions $\{ F_n\}$ and a bounded function $g$ such that $F_n \Rightarrow F$, but the relation

fails to hold.

Construct an example of a sequence of distributions $\{ F_n\}$ and a continuous function $g$ such that $F_n \Rightarrow F$, but the relation

fails to hold.

Let, for every bounded uniformly continuous function $g$, the convergence of integrals

hold. Prove that $F_n \Rightarrow F$.

Let, for every compactly supported infinitely differentiable function $g$, the convergence of integrals

hold. Prove that $F_n \Rightarrow F$.

Let $\{ F_n\}$ be a sequence of distributions on $\mathbb {R}$. Prove that the following three conditions are equivalent:

1. $F_n \Rightarrow F$;

2. $\limsup_{n \rightarrow \infty } F_n(B) \leqslant F(B)$ for every closed set $B$;

3. $\liminf_{n \rightarrow \infty } F_n(B) \geqslant F(B)$ for every open set $B$.

Let $\{ F_n\}$ be a sequence of integer-valued distributions. Prove that $F_n \Rightarrow F$ if and only if $F_n(\{ i\} ) \rightarrow F(\{ i\} )$ for every integer $i$.

Let $F, F_1, F_2, \ldots$ be absolutely continuous distributions with densities $p, p_1, p_2, \ldots$, respectively. Prove that if $p_n(x) \rightarrow p(x)$ as $n \rightarrow \infty$ uniformly on every compact set, then $F_n \Rightarrow F$. Is the converse true?

Consider the sequence of distribution functions

Determine whether this sequence has a weak limit.

Let $F$ be Lebesgue measure on $[0, 1]$, and let the distribution $F_n$ assign probability $1/n$ to points chosen arbitrarily, one per interval $\left[\dfrac {i-1}{n}, \dfrac {i}{n}\right)$, $i = 1, \ldots , n$. Prove that $F_n \Rightarrow F$.

Let the random variable $\xi_n$ be uniformly distributed on the interval $[-\sqrt{n}, n]$. Does weak convergence hold for the sequence of distributions of the random variables:

Let $\xi_n \Rightarrow \xi$ and let $f(x)$ be a continuous function. Prove that $f(\xi_n) \Rightarrow f(\xi )$.

Let $\xi_n \Rightarrow \xi$. Is it true that $\xi_n - \xi \Rightarrow 0$?

Let $\xi_n \Rightarrow \xi$ and $a_n \rightarrow a$, where $\{ a_n\}$ is a numerical sequence, $a \in \mathbb {R}$. Prove that $a_n \xi_n \Rightarrow a \xi$.

Let $\xi_n \Rightarrow \xi$ and $\eta_n \Rightarrow \eta$, where, first, $\xi_n$ and $\eta_n$ are independent for each $n$, and second, $\xi$ and $\eta$ are also independent. Prove that:

Let $\xi_n \Rightarrow \xi$ and $\eta_n \xrightarrow {p} 0$. Prove that:

Let $\xi_n$ be independent identically distributed random variables with finite second moment. Prove that

Let $\xi_1, \xi_2, \ldots$ be independent random variables, each having the exponential distribution with parameter 1. Set $\zeta_n = \max (\xi_1, \ldots , \xi_n)$.

Let $\xi_1, \xi_2, \ldots$ be independent random variables, each uniformly distributed on $[0, 1]$. Set $\zeta_n = \min (\xi_1, \ldots , \xi_n)$. Find the limiting distribution of $n^{\gamma } \zeta_n$ for $\gamma \in \mathbb {R}$.

Let $\xi_1, \xi_2, \ldots$ be independent identically distributed random variables with continuous distribution function $F$. Find the limiting distribution of the random variable $\zeta_n = n(1 - F(M_n))$, where $M_n = \max (\xi_1, \ldots , \xi_n)$.

Prove that the weak limit of a sequence of infinitely divisible distributions is infinitely divisible.