Probability Theory: A Comprehensive Course

Achim Klenke

Chapter 7 $L^{p}$-Spaces and the Radon-Nikodym Theorem - all with Video Answers

Educators

Section 1

Definitions

02:57

Problem 1

Let $\left(X_{i}\right)_{i \in \mathbb{N}}$ be independent, square integrable random variables with $\mathbf{E}\left[X_{i}\right]=0$ for all $i \in \mathbb{N}$
(i) Show that $\sum_{i=1}^{\infty} \operatorname{Var}\left[X_{i}\right]<\infty$ implies that there exists a real random variable $X$ with $\sum_{i=1}^{n} X_{i} \stackrel{n \rightarrow \infty}{\longrightarrow} X$ almost surely.
(ii) Does the converse implication hold in (i)?

Sagar Singh

Numerade Educator

03:29

Problem 2

Let $f: \Omega \rightarrow \mathbb{R}$ be measurable. Show that the following hold.
(i) If $\int|f|^{p} d \mu<\infty$ for some $p \in(0, \infty)$, then $\|f\|_{p} \stackrel{p \rightarrow \infty}{\longrightarrow}\|f\|_{\infty}$.
(ii) The integrability condition in (i) cannot be waived.

Nick Johnson

Numerade Educator

03:59

Problem 3

Let $p \in(1, \infty), f \in \mathcal{L}^{p}(\lambda)$, where $\lambda$ is the Lebesgue measure on $\mathbb{R}$. Let $T: \mathbb{R} \rightarrow \mathbb{R}, x \mapsto x+1$. Show that
$$
\frac{1}{n} \sum_{k=0}^{n-1} f \circ T^{k} \stackrel{n \rightarrow \infty}{\longrightarrow} 0 \quad \text { in } L^{p}(\lambda).
$$

James Kiss

Numerade Educator