All Questions
Tagged with lp-spaces probability-theory
198
questions
7
votes
1
answer
167
views
Sandwiching the Lp norm sequence of random variable
Let X be a random variable with $\Vert X\Vert_p = E[\vert X\vert^p]^{1/p}<\infty$ for all $p\geq 1$.
Assume for some fixed $r$, and for all $q,s$ satisfying $1\leq q < r < s$,
$$\lim_{p\to\...
- 292
1
vote
1
answer
70
views
Showing $L^2$ - convergence of $\phi_n(X_n)$ to $\phi(X)$ when $\phi_n \to \phi$ and $X_n\to X$
Let $(X_n)$ be a sequence of square integrable random variables converging to $X$ in $L^2$ and $(\phi_n)$ a sequence of smooth functions converging to $\phi$ uniformly on compact sets where $\phi$ is $...
- 376
1
vote
0
answers
40
views
Adjoint operator and random variable
Let $(\Omega,\mathcal{A},\mathbb{P})$ be a probability space and $\mathbb{D}$ a dense subset of $L^2(\Omega,\mathcal{A},\mathbb{P})$.
I consider a linear map $D$ from $\mathbb{D}\subset L^2(\Omega)$ ...
- 1,381
1
vote
0
answers
83
views
Var$[X]$ exists if and only if $X\in L^2$
In Baldi's textbook Probability, he writes that a random variable $X$'s variance exists if and only if $X\in L^2(\mu)$.
I went searching through the text and cannot find where he defines what it means ...
- 5,696
1
vote
1
answer
60
views
Why almost sure convergence cannot imply convergence in $L^p$? [closed]
Almost sure convergence cannot imply convergence in $L^p$. However I can't find a rigorous counterexample?
Can anybody help me? Any helpful ideas would be greatly appreciated!
- 23
1
vote
1
answer
88
views
How to prove: ${X_n}$ Uniform absolute continuous and $X_n$ Convergence in probability to $X$ equivalent to $X$ in $L_p$?
Help me my friends! Our teacher left a remark on the notes without providing a proof, which I am very curious about. It says that
If $\{|X_n|^p,n\geq1\}$ is uniform absolute continuous, and $X_n\...
- 23
1
vote
0
answers
111
views
A question about probability theory and a little functional analysis
Consider the sequences of mean zero random variables $(X_n)_{n \in \mathbb N}$ with finite variances, i.e., $E[X_n]=0$ and $E\left[X_n^2\right] =: \sigma_n< \infty$, for each $n$. Moreover, ...
- 75
1
vote
1
answer
61
views
When are sub-$\sigma$-algebras 'orthogonally complementable' in $L^2$?
For a probability space $(\Omega, \mathscr{F}, \mathbb{P})$ with sub-$\sigma$-algebra $\mathscr{G}\subseteq \mathscr{F}$, we have the orthogonal $L^2$-decomposition
\begin{equation}
L^2(\Omega, \...
- 1,064
1
vote
0
answers
99
views
Analogue of Skorohod representation theorem for Wasserstein metric and convergence in $L^p$?
Background. The Skorohod representation theorem says that if $X_n \to_{\mathrm{d}} X$ (i.e. $X_n$ converges to $X$ in distribution as $n \to \infty$), then we can construct a new probability space ...
- 346
2
votes
0
answers
79
views
Does $\mathbb{E}[g(X)Y]=0$ for all functions $g$ imply $\mathbb{E}[Y|X]=0$?
Is there any result saying something like
$$\mathbb{E}[g(X)Y]=0 \,\,\forall g\in\mathcal{G}\,\,\,\implies\mathbb{E}[Y\mid X]=0$$
where $\mathcal{G}$ is some function class? Have been searching online ...
- 65
0
votes
1
answer
40
views
Element in Lp Vs Lp-Bounded
I'm studying uniform integrability of random variables and am confused by the difference between some definitions regarding $L^p$ spaces. Please let me know why the following is incorrect:
A random ...
- 461
2
votes
0
answers
124
views
An $L^2 $ martingale $M_n$ is bounded in $L^2$ if and only if $\sum_{k=1}^{\infty} E[M_k-M_{k-1}]^2 < \infty$
In the following proof
These notes assume that bounded in $L^p$ means $\sup_n E[|X|^p]< \infty$
I see clearly the if part:
If $\sum_{k=1}^{\infty} E[M_k-M_{k-1}]^2 < \infty$, then $E[M_n^2]<\...
- 3,162
2
votes
1
answer
122
views
Existence proof of conditional expectation
I am self-learning introductory stochastic calculus from the text A first course in Stochastic Calculus, by Louis Pierre Arguin. I'm struggling to understand a particular step in the proof, and I ...
- 5,450
1
vote
1
answer
83
views
Weak-$\star$ Convergence in $L_1$
Let
$(X,d)$ be a complete separable metric space;
$\mu$ be a Borel probability measure on $X$;
$(f_n),(g_n)\subset L_1(X,\mu)$ be sequences of non-negative uniformly bounded sequences, bounded by $1$,...
- 310
0
votes
1
answer
83
views
On the charateristic function of random variables
For all random variables that admit a probability density function (PDF), their characteristic function provides an alternative way to completely define its probability distribution.
Why is that? The ...
- 141