Probability Theory: A Comprehensive Course

Achim Klenke

Chapter 6 Convergence Theorems - all with Video Answers

Educators

Section 1

Almost Sure and Measure Convergence

01:58

Problem 1

Let $\Omega$ be countable. Show that convergence in probability implies almost everywhere convergence.

Nick Johnson

Numerade Educator

01:24

Problem 2

Give an example of a sequence that
(i) converges in $L^{1}$ but not almost everywhere,
(ii) converges almost everywhere but not in $L^{1}$.

Lauren Shelton

Numerade Educator

03:52

(Egorov's theorem (1911), [41]) Let $(\Omega, \mathcal{A}, \mu)$ be a finite measure space and let $f_{1}, f_{2}, \ldots$ be measurable functions that converge to some $f$ almost everywhere. Show that, for every $\varepsilon>0$, there is a set $A \in \mathcal{A}$ with $\mu(\Omega \backslash A)<\varepsilon$ and $\sup _{\omega \in A}\left|f_{n}(\omega)-f(\omega)\right| \stackrel{n \rightarrow \infty}{\longrightarrow} 0$

Donald Albin

Numerade Educator

04:19

Problem 4

Let $X_{1}, X_{2}, \ldots$ be independent, square integrable, centered random variables with $\sum_{i=1}^{\infty} \operatorname{Var}\left[X_{i}\right]<\infty .$ Show that there exists a square integrable $X$ with $X=\lim _{n \rightarrow \infty} \sum_{i=1}^{n} X_{i}$ almost surely.

Linda Hand

Numerade Educator

Problem 1

Problem 2

Problem 3

Problem 4