Tagged Questions
0
votes
1answer
42 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 ...
1
vote
1answer
29 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 $X\in L_p(P)$, ...
1
vote
1answer
20 views
If a function is $L^p$ small, is its expectation with respect to a $\sigma$-algebra $L^p$ small?
This came up in my homework, but isn't strictly my homework. I've just gotten very curious, and I keep going in circles trying to prove it.
Consider a probability measure space $(X,\Sigma,\mu)$ and ...
2
votes
1answer
75 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 ...
0
votes
1answer
57 views
Proving the following moment distribution function.
I am trying to prove the following relation ,
If $u \in L^p(\Omega)$ $\Omega \subset R^n $and $0 < p <\infty$ , the the following relation is valid ,
$$\|u\|_{L^p(\Omega)}^p = p\int_0^\infty ...
1
vote
2answers
93 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 ...
