Convergence in Probability

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 $\dfrac {\xi_1 + \ldots + \xi_n}{n} \xrightarrow {p} 0$?