Convergence of Random Variables and Distributions

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.

Let $\xi_n \xrightarrow {p} \xi$ and $\xi_n \xrightarrow {p} \eta$. Prove that $\mathbb {P}\left(\xi = \eta \right) = 1$.

Let $\xi_n \xrightarrow {p} \xi$, $\eta_n \xrightarrow {p} \eta$, and $\mathbb {P}\left(\xi = \eta \right) = 1$. Prove that $(\xi_n - \eta_n) \xrightarrow {p} 0$ as $n \rightarrow \infty$.

Prove that almost sure convergence implies convergence in probability.

Prove that convergence in probability does not imply, in general, almost sure convergence.

Let $\Omega$ be the unit circle, let $\mathscr {F}$ be the $\sigma$-algebra of Borel sets, and let $\mathbb {P}\left(\right)$ be the uniform distribution on the circle. Let $\xi (\omega ) \equiv 1$, and

where $r(n) = \sum_{k=1}^{n} \dfrac {1}{k}$. Determine whether the sequence $\{ \xi_n\}$ converges to the random variable $\xi$:

Prove that if the sequence $\xi_n$ converges in probability to $\xi$, then there exists a subsequence of indices $n_k$ such that $\xi_{n_k} \rightarrow \xi$ a.s. as $k \rightarrow \infty$.

Prove that if $\xi_1 \leqslant \xi_2 \leqslant \ldots$ and $\xi_n$ converge in probability to $\xi$, then $\xi_n$ converge to $\xi$ a.s.

Let the random variables $\xi_1, \xi_2, \ldots$ be independent. Prove that if the series $\sum_{n=1}^{\infty } \xi_n$ converges in probability, then it also converges a.s.

Let $\{ \xi_n\}$ be a sequence of random variables, where $\xi_n$ takes the values $e^{-\alpha n}$ and $e^{\alpha n}$ with probabilities $1 - e^{-\beta n}$ and $e^{-\beta n}$, respectively. For which values of $\alpha$ and $\beta$ does the convergence $\xi_n \xrightarrow {p} 0$ hold?

Let the random variables $\xi_1, \xi_2, \ldots$ be independent, where $\xi_n$ has:

Prove that $\xi_n$ converge in probability to $\xi$ and $\eta_n$ converge in probability to $\eta$ if and only if the sequence of random vectors $(\xi_n, \eta_n)$ converges in probability to $(\xi , \eta )$.

Let $\xi_n \xrightarrow {p} \xi$. Prove that:

Let $\xi_n \xrightarrow {p} \xi$ and let $g(x)$ be a uniformly continuous function. Prove that $g(\xi_n) \xrightarrow {p} g(\xi )$.

Let $\xi_n \xrightarrow {p} \xi$ and let $g(x)$ be a continuous function. Prove that $g(\xi_n) \xrightarrow {p} g(\xi )$.

Let $\xi_n \xrightarrow {p} \xi$, with $\mathbb {P}\left(\xi = 0\right) = 0$. Prove that $\dfrac {1}{\xi_n} \xrightarrow {p} \dfrac {1}{\xi }$.

Let $\xi_n \xrightarrow {p} \xi$ and let $A$ be a set of points on the line such that $\mathbb {P}\left(\xi \in A\right) = 0$. Consider a function $g(x)$ whose set of discontinuity points is contained in $A$. Prove that $g(\xi_n) \xrightarrow {p} g(\xi )$.

Let $\xi_n \xrightarrow {p} 1$ and $\eta_n \xrightarrow {p} 1$. Prove that:

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

Let $\xi_n \xrightarrow {p} \xi$, $\eta_n \xrightarrow {p} \eta$, and let $g(x, y)$ be a uniformly continuous bounded function. Prove that $g(\xi_n, \eta_n) \xrightarrow {p} g(\xi , \eta )$.

Let $\xi_n \xrightarrow {p} \xi$, $\eta_n \xrightarrow {p} \eta$, and let $g(x, y)$ be a continuous function. Prove that $g(\xi_n, \eta_n) \xrightarrow {p} g(\xi , \eta )$.

Let $(\xi_n - \xi )^2 \xrightarrow {p} 0$. Prove that $\xi_n^2 \xrightarrow {p} \xi^2$. Is the converse true?

Prove that if $\xi_n \xrightarrow {p} \xi$ as $n \rightarrow \infty$ and $g(x)$ is a continuous bounded function, then $\mathbb {E}\left[g(\xi_n)\right] \rightarrow \mathbb {E}\left[g(\xi )\right]$.

Give an example confirming that, under the conditions of Problem 19.22, the requirement that the function $f$ be continuous and bounded cannot, in general, be dropped.

Let $g$ be a continuous, monotonically increasing, and bounded function on the half-line $[0, \infty )$ with $g(0) = 0$. Prove that the condition $\lim_{n \rightarrow \infty } \mathbb {E}\left[g(\left|\xi_n\right|)\right] = 0$ is necessary and sufficient for $\xi_n \xrightarrow {p} 0$.

Prove that $\xi_n$ converge in probability to $\xi$ if and only if

Let $\{ \xi_n\}$ be a sequence of random variables with zero expectations. Does convergence $\xi_n \xrightarrow {p} 0$ as $n \rightarrow \infty$ imply convergence $\dfrac {\xi_1 + \ldots + \xi_n}{n} \xrightarrow {p} 0$?

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.

Prove that convergence in the mean of order $\alpha > 0$ implies convergence in the mean of order $\beta$ for every $\beta \in (0, \alpha )$.

Prove that convergence in the mean of some order implies convergence in probability.

Give an example confirming that convergence in the mean of order $\alpha$ does not imply, in general, almost sure convergence.

Give an example confirming that convergence in probability does not imply, in general, convergence in the mean of order $\alpha$.

Give an example confirming that almost sure convergence does not imply, in general, convergence in the mean of order $\alpha$.

Prove that a sequence of random variables $\xi_n$ converges in the mean of order $\alpha$ if and only if the sequence $\left|\xi_n\right|^{\alpha }$ is uniformly integrable and $\xi_n \xrightarrow {p} \xi$.

Let $\sum_{n=1}^{\infty } \mathbb {E}\left[\left|\xi_n - \xi \right|^{\alpha }\right] < \infty$. Prove that $\xi_n \rightarrow \xi$ a.s.

Let $\left|\xi_n\right| \leqslant c < \infty$ a.s. for all $n$, and let $\xi_n \rightarrow \xi$ in probability as $n \rightarrow \infty$. Prove that $\mathbb {E}\left[\left|\xi_n - \xi \right|^{\alpha }\right] \rightarrow 0$ for every $\alpha > 0$.

Let $\xi , \xi_1, \xi_2, \ldots$ be nonnegative random variables such that $\xi_n \rightarrow \xi$ in probability and $\mathbb {E}\left[\xi_n\right] \rightarrow \mathbb {E}\left[\xi \right] < \infty$ as $n \rightarrow \infty$. Prove that $\mathbb {E}\left[\left|\xi_n - \xi \right|\right] \rightarrow 0$.

Let $\alpha > 0$ and $\mathbb {E}\left[\left|\xi_n\right|^{\alpha }\right] < \infty$ for all $n$. Prove that the following statements are equivalent:

1. $\xi_n \rightarrow \xi$ in probability and $\mathbb {E}\left[\left|\xi_n\right|^{\alpha }\right] \rightarrow \mathbb {E}\left[\left|\xi \right|^{\alpha }\right] < \infty$ as $n \rightarrow \infty$;

2. $\mathbb {E}\left[\left|\xi_n - \xi \right|^{\alpha }\right] \rightarrow 0$ as $n \rightarrow \infty$.

Let the family $\{ \xi_n\}$ of random variables be uniformly integrable. Prove that $\sup_n \mathbb {E}\left[\left|\xi_n\right|\right] < \infty$.

Let the sequence $\{ \xi_n\}$ of random variables be such that $\sup_n \mathbb {E}\left[\left|\xi_n\right|^{1+\delta }\right] < \infty$ for some $\delta > 0$. Prove that this sequence is uniformly integrable.

Let $\{ \xi_n\}$ be a sequence of random variables. Prove that the following conditions are equivalent:

1. the family $\{ \xi_n\}$ is uniformly integrable;

2. there exists a nonnegative function $g(x)$ such that $g(x) \rightarrow \infty$ as $x \rightarrow \infty$ and $\sup_n \mathbb {E}\left[\left|\xi_n\right| g(\left|\xi_n\right|)\right] < \infty$.

Let the sequence $\{ \xi_n\}$ of random variables be such that $\mathbb {P}\left(\left|\xi_n\right| \geqslant x\right) \leqslant \mathbb {P}\left(\left|\eta \right| \geqslant x\right)$ for every $x > 0$, with $\mathbb {E}\left[\left|\eta \right|\right] < \infty$. Prove that $\{ \xi_n\}$ is uniformly integrable.

Let $\{ \xi_n\}$ be a sequence of nonnegative random variables with finite means such that the distribution of $\xi_n$ converges weakly to the distribution of a random variable $\xi$ and $\mathbb {E}\left[\xi_n\right] \rightarrow \mathbb {E}\left[\xi \right] < \infty$ as $n \rightarrow \infty$. Prove that this sequence is uniformly integrable.

Let $\xi_n \Rightarrow \xi$ as $n \rightarrow \infty$. Prove that the following statements are equivalent:

1. the sequence $\{ \xi_n\}$ is uniformly integrable;

2. $\mathbb {E}\left[\left|\xi_n\right|\right] < \infty$, $\mathbb {E}\left[\left|\xi \right|\right] < \infty$, and as $n \rightarrow \infty$

   $$\mathbb {E}\left[\max (0, \xi _n)\right] \rightarrow \mathbb {E}\left[\max (0, \xi )\right], \qquad \mathbb {E}\left[\min (0, \xi _n)\right] \rightarrow \mathbb {E}\left[\min (0, \xi )\right].$$

Let the sequences $\xi_n$ and $\eta_n$ be uniformly integrable. Prove that the following sequences are uniformly integrable:

Let the independent random variables $\xi_1, \xi_2, \ldots$ be identically distributed with finite mean; set $S_n = \xi_1 + \ldots + \xi_n$. Prove that the sequence of random variables $\{ S_n / n, n \geqslant 1\}$ is uniformly integrable.

Let $x \in \mathbb {R}$ and let $\{ \xi_n\}$ be a sequence of random variables with finite means such that $\mathbb {E}\left[\left|\xi_n\right|\right] \rightarrow \mathbb {E}\left[\left|\xi \right|\right]$ as $n \rightarrow \infty$. Determine whether this implies the convergence $\mathbb {E}\left[\left|\xi_n + x\right|\right] \rightarrow \mathbb {E}\left[\left|\xi + x\right|\right]$.

Let $\xi , \xi_1, \xi_2, \ldots$ be nonnegative random variables such that $\mathbb {E}\left[\xi_n\right] \rightarrow \mathbb {E}\left[\xi \right]$ and $\mathbb {P}\left(\xi > \xi_n + \varepsilon \right) \rightarrow 0$ as $n \rightarrow \infty$ for every $\varepsilon > 0$. Prove that $\mathbb {E}\left[\left|\xi_n - \xi \right|\right] \rightarrow 0$.