Almost Sure Convergence

Let $\xi_n \rightarrow \xi$ and $\xi_n \rightarrow \eta$ a.s. Prove that then $\mathbb {P}\left(\xi = \eta \right) = 1$.

Let a sequence of independent random variables $\xi_n$ converge almost surely. Prove that the variance of the limit is zero.

Prove that $\xi_n \rightarrow \xi$ a.s. if and only if for every $\varepsilon > 0$

Prove that a sequence of random variables converges almost surely to some random variable if and only if it is fundamental with probability 1.

Prove that $\xi_n \rightarrow \xi$ a.s. if and only if for every $\varepsilon > 0$

Prove that a sequence of random variables $\{ \xi_n\}$ is fundamental with probability 1 if and only if for every $\varepsilon > 0$

Let independent random variables $\xi_n$ take the values 1 and 0 with probabilities $p_n$ and $1 - p_n$, respectively. Under what conditions on $p_n$ does the sequence of random variables $\{ \xi_n\}$ converge a.s.:

Let $\Omega = [0, 1]$, let $\mathscr {F}$ be the $\sigma$-algebra of Borel sets, and let $\mathbb {P}\left(\right)$ be Lebesgue measure. Prove that the sequence of random variables

converges a.s. and find the limit.

Let $\xi_n \rightarrow \xi$ a.s. Prove that:

Let $\xi_n \rightarrow 1$ and $\eta_n \rightarrow 1$ a.s. Prove that:

Let $(\xi_n - \xi )^2 \rightarrow 0$ a.s. Prove that $\xi_n^2 \rightarrow \xi^2$ a.s.

Let $\xi_n \rightarrow \xi$ a.s. and let $g(x)$ be a continuous function. Prove that $g(\xi_n) \rightarrow g(\xi )$ a.s.

Let $\xi_n \rightarrow \xi$, $\eta_n \rightarrow \eta$ a.s. and let $g(x, y)$ be a continuous function. Prove that $g(\xi_n, \eta_n) \rightarrow g(\xi , \eta )$ a.s.

Let $\{ \xi_n\}$ be a sequence of random variables such that for every $\varepsilon > 0$

Prove that $\xi_n \rightarrow \xi$ a.s.

Let $\{ \xi_n\}$ and $\{ \eta_n\}$ be two sequences of random variables such that for every $\varepsilon > 0$

Prove that if $\eta_n \rightarrow a$ a.s., then $\xi_n \rightarrow a$ a.s.

Let the random variables $\xi_1, \xi_2, \ldots$ be independent and let $\xi_n \rightarrow 0$ a.s. Prove that the series $\sum_{n=1}^{\infty } \mathbb {P}\left(\left|\xi_n\right| > 1\right)$ converges.

Let $\{ \xi_n\}$ be a sequence of random variables. Prove that if for some summable sequence of positive numbers $\{ \varepsilon_n\}$

then the sequence $\{ \xi_n\}$ converges with probability 1 to some random variable.

Let $\{ \xi_n\}$ be a sequence of random variables. Prove that if the series $\sum_{n=1}^{\infty } \mathbb {E}\left[\xi_n^2\right]$ converges, then $\xi_n \rightarrow 0$ a.s.

Let $\xi , \xi_1, \xi_2, \ldots$ be random variables such that the series $\sum_{n=1}^{\infty } \mathbb {E}\left[\left|\xi_n - \xi \right|^a\right]$ converges for some $a > 0$. Prove that $\xi_n \rightarrow \xi$ a.s.

Let $\xi_1, \xi_2, \ldots$ be a sequence of independent random variables such that $\mathbb {E}\left[\xi_n^2\right] < \sigma^2$ and $\mathbb {E}\left[\xi_n\right] = 0$. Prove that the series $\sum_{n=1}^{\infty } \dfrac {\xi_n}{2^n}$ converges a.s.

Prove that the radius of convergence $R$ of the random power series $\sum_{i=0}^{\infty } \xi_i z^i$, whose coefficients $\xi_i$ are independent, is a constant (almost surely).

Prove that the radius of convergence $R$ of the random power series $\sum_{i=0}^{\infty } \xi_i z^i$, whose coefficients $\xi_i$ are independent and identically distributed with a nondegenerate distribution, equals 1 or 0 according to whether the moment $\mathbb {E}\left[\ln (1 + \left|\xi_0\right|)\right]$ is finite or infinite.

Let $\xi_1, \xi_2, \ldots$ be independent random variables with the Cauchy distribution. Prove that

while for $\alpha > 0$

Let the random variables $\xi_1, \xi_2, \ldots$ be independent and uniformly distributed on $[0, 1]$. Prove that the series $\sum_{n=1}^{\infty } \prod_{i=1}^{n} \xi_i$ converges a.s.

Let independent identically distributed random variables $\xi_1, \xi_2, \ldots$ take values in the set $\{ 1, \ldots , J\}$, with $p(k) = \mathbb {P}\left(\xi_1 = k\right) > 0$ for every $k \in \{ 1, \ldots , J\}$. Denote by $N_n(k) = \sum_{i=1}^{n} \; \mathbb {1}_{\xi_i = k}$ the number of elements among $\xi_1, \ldots , \xi_n$ equal to $k$; set $\eta_n = \prod_{k=1}^{J} (p(k))^{N_n(k)}$. Prove that the sequence $\dfrac {\ln \eta_n}{n}$ converges a.s., and find its limit.