All Questions
8 questions
12
votes
2
answers
3k
views
Does there exist an event independent of a given sigma-algebra?
The following question came up in a discussion with my advisor:
Let $(\Omega, \mathcal F, \mathbb P)$ be a non-trivial probability space, and suppose that $\mathcal G$ is a proper sub-$\sigma$-...
- 8,512
8
votes
2
answers
640
views
Does a random sequence of vectors span a Hilbert space?
Let $\mathcal{H}$ be a separable Hilbert space. Let $v$ be a random variable taking values in $\mathcal{H}$ such that $P(v \perp h) < 1$ for all $h \in \mathcal{H}.$ Suppose we sample an infinite ...
- 1,429
5
votes
1
answer
512
views
Concentration inequality for Hilbert space valued random variables
I have read in a paper about the following result:
Let $V$ be a separable Hilbert space and $(\Omega,A_{\Omega},P)$ a probability space. Suppose that $Y_1,Y_2,...$ is a sequence of independent $V$-...
- 115
5
votes
1
answer
341
views
Reference request for Deterministic $\subset$ Random $\subset$ Quantum
I hope this post is on topic as a reference request.
I have seen somewhere the idea of (and saw it written just like this):
$$\text{Deterministic }\subset\text{ Random }\subset\text{ Quantum }.$$
I am ...
- 1,037
4
votes
1
answer
863
views
Hoeffding's inequality for Hilbert space valued random elements
Suppose that $\mathbb H$ is a separable Hilbert space and $X_1,\ldots,X_n$ are independent zero mean $\mathbb H$-valued random elements such that $\|X_i\|\le s$ for each $1\le i\le n$, where $\|\cdot\|...
- 423
3
votes
1
answer
159
views
Tight L2 bound on moments approximation and reference
Consider $f\in L^2(I)$, where $I$ is the unit interval and $L^2$ is w.r.t. Lebesgue measure, and consider an approximation of $f$ denoted by $\tilde{f}\in L^2$.
The error in approximated the moments ...
- 3,574
3
votes
0
answers
198
views
Karhunen-Loeve expansion convergence rate for Gaussian Proccess
Consider A Gaussian Procces $X(t):\mathbb{R}\times \Omega \to \mathbb{R}$ with $\Omega$ a probability space and $\mathbb{E} \left[ X_t \right] = 0$ for all $t\in \mathbb{R}$.
Consider also its KL ...
- 3,574
1
vote
0
answers
83
views
Embedding random variables in infinite-dimensional spaces
Let $H$ be a reproducing kernel Hilbert space of functions $f:E\to F$ with kernel $k$. A point in $E$ may be embedded into $H$ via the canonical embedding $x\mapsto k(x,\cdot)$. Similarly, a random ...
- 710