All Questions
Tagged with fa.functional-analysis pr.probability
596
questions
4
votes
1
answer
976
views
weak convergence in infinite dimensional spaces
Weak convergence can be tricky when dealing with infinite dimensional spaces. For example, the usual Levy's continuity theorem does not extend readily to separable Banach spaces.
Consider a (...
- 2,133
65
votes
9
answers
11k
views
Polish spaces in probability
Probabilists often work with Polish spaces, though it is not always very clear where this assumption is needed.
Question: What can go wrong when doing probability on non-Polish spaces?
- 651
4
votes
2
answers
2k
views
Convergence of Gaussian measures
Let $X$ be a separable Banach space with its Borel $\sigma$-algebra $\mathcal F$. Let $x_n \to x$ in $X$. Fix a Gaussian covariance operator $K$, and let $\mathbb P_n$ and $\mathbb P$ be Gaussian ...
- 8,422
14
votes
3
answers
2k
views
Disintegrations are measurable measures - when are they continuous?
This is a sequel to another question I have asked.
The notion of disintegration is a refinement of conditional probability to spaces which have more structure than abstract probability spaces; ...
- 8,422
4
votes
2
answers
1k
views
Can we extract information about how fast a function decay from its Laplace transform?
My question is whether we can extract information about how fast an integrable function converges to zero by looking at the asymptotics of its Laplace transform.
More concrete case, let $f:\mathbb{R} ...
- 1,839
6
votes
2
answers
3k
views
Dense inclusions of Banach spaces and their duals
This seems like a really simple question, but I'm struggling with it. Let $X$ be a separable Banach space, $H$ be a separable Hilbert space, and suppose $i : H \hookrightarrow X$ is a dense, ...
- 8,422
7
votes
4
answers
928
views
On operator ranges in Hilbert & Banach spaces
Lemma 1 from Anderson & Trapp's Shorted Operators, II isLet $A$ and $B$ be bounded operators on the Hilbert space $\mathcal H$. The following statements are equivalent:
(1) ran($A$) $\subset$ ...
- 8,422
5
votes
0
answers
533
views
Conditional probabilities in Banach spaces
This is the infinite-dimensional sequel to my question, Conditional probabilities are measurable functions - when are they continuous?.
Let $\Omega = \Omega_1 \times \Omega_2$ be a probability space ...
- 8,422
17
votes
5
answers
3k
views
Conditional probabilities are measurable functions - when are they continuous?
Let $\Omega$ be a Banach space; for the sake of this post, we will take $\Omega = {\mathbb R}^2$, but I am more interested in the infinite dimensional setting. Take $\mathcal F$ to be the Borel $\...
- 8,422
3
votes
3
answers
2k
views
Conditional expectation of convolution product equals..
Let $X, Y$ be two $L^1$ random variables on the probablity space $(\Omega, \mathcal{F}, P)$. Let $\mathcal{G} \subset \mathcal{F}$ be a sub-$\sigma$-algebra.
Consider the conditional expectation ...
- 47
2
votes
2
answers
1k
views
Decoupling lemma for the Lambda(p) problem
I'm attempting to work through Bourgain's paper "Bounded orthogonal systems and the $\Lambda(p)$-set problem". There is a step in the proof of the decoupling lemma that I am stuck on, and thought ...
- 12k