2
votes
2answers
59 views

Properties of a sequence of iid rv's

I cannot do part a), and Im fairly sure that b) and c) will follow from it. If possible could I please have a solution to part a) and hints if you feel necessary to parts b) and c).
3
votes
1answer
35 views

Find a sequence of r.v's satisfying the following conditions

I think part a) can be solved by using $X_n=\frac{1}{n}\chi_{[0,n^2]}$ Not sure about part b).
2
votes
1answer
26 views

$L_p$ spaces and tail estimates

I can prove the main identity in this question. Not sure how the "and deduce" bit works. I think $O(\lambda^{-q})$ is some kind of tail estimate.
4
votes
2answers
71 views

If the sum of two independent random variables is in $L^2$, is it true that both of them are in $L^1$?

Let $X$ and $Y$ be two independent random variables. If $\mathbb E(X+Y)^2 < \infty$, do we have $\mathbb E |X| < \infty$ and $\mathbb E |Y| < \infty$? What I actually want is that $X$ and ...
4
votes
1answer
83 views

Conditional expectation: Is $X/E[X \mid \mathcal{G}] \in L^p$?

Let $(\Omega, \mathcal{F}, P)$ be a probability space, and let $X$ be a strictly positive random variable with finite moments of all orders (i.e. $E[X^q] < \infty$ for all $1 \le q < \infty$). ...
2
votes
1answer
94 views

A question on Orlicz norms

I was reading Empirical Processes from "Weak Convergence and Empirical processes" by Van Der Waart and Jon Wellner. There I was studying Orlicz norms of random variables which are defined as follows: ...
2
votes
1answer
78 views

If $X_n \to X$ in $L^1$ does $E[X_n|\mathcal{G}] \to E[X| \mathcal{G}]$?

We work on a probability space $(\Omega,\mathcal{F},P)$, and $\mathcal{G}$ is a sub-$\sigma$-algebra of $\mathcal{F}$. Suppose that $X_n\to X$ in $L^1$, i.e. that $E[|X_n-X|] \to 0$. When does this ...
0
votes
1answer
65 views

uniform integrability characterization

How to show the following: When a family of random variables $ \{X_n\}_{n \geq 1}$ is $L^p$ bounded for some $p > 1$ then $ \{X_n\}_{n \geq 1}$ is uniformly integrable. Also why does the above ...
8
votes
1answer
209 views

$L^{p}$ functions from Rudin Exercises 3.5

I am attempting a question from Rudin's "Real and Complex Analysis" Chapter 3 question 5. I shall summarise the question as below: Suppose that $f$ is a complex measurable function on $X$, $\mu$ a ...
1
vote
1answer
59 views

Continuity of conditional expectation in $L_p$

I'm looking at a probability space $(\Omega,\mathcal{F},P)$. Let $1\leq p<\infty$, and let $\mathcal{G}$ be a sub-$\sigma$-algebra of $\mathcal{F}$. I'm then asked to show that, for $X\in L_p(P)$, ...
3
votes
1answer
174 views

Convergence of $L^p$ norm as $p \downarrow 0$ [duplicate]

Consider a measurable space $(\Omega, \mathscr{F}, P)$ with $P(\Omega) = 1$. Define for measurable functions $X$ the following $\| X \|_p := \left(\int |X|^p dP\right)^{1/p}$. We know that for $p \in ...
2
votes
1answer
107 views

How compute $\lim_{p\rightarrow 0} \|f\|_p$ in a probability space?

I not solve the follow limit $$\lim_{p\rightarrow 0} \bigg[\int_{\Omega} |f|^p d\mu \bigg]^{1/p} = \exp\bigg[ \int_{\Omega} \log|f|d\mu \bigg],$$ where $(\Omega, \mathcal{F}, \mu)$ is a probability ...
1
vote
2answers
105 views

Examples that are not Lebesgue integrable for any $p$

I've been trying to think up different examples of functions such that $EZ^p = \infty$ (with $Z>0$) for all $p$, but each time it becomes rather messy. Can anyone suggest some interesting but ...
3
votes
2answers
289 views

$L^p$ norm of multivariate standard normal random variable

Given $X_i\sim \mathcal{N}(0,1)$ what is the behaviour of $$ ||X||_{l^p}=(\sum_{i=1}^n|X_i|^p )^{1/p}$$ as $n\rightarrow \infty$? For $p=2$ results about $\chi$-distribution tell us that ...