1
vote
1answer
64 views

Weak convergence in $L^2$ and CDF

Assume that for sequence $X_n \in L^2(\Omega,F,P)$ which converges in distribution to CDF $F_X$ ($F_n(t)\rightarrow F_X(t)$ for every point of continuity of $F_X$), we have also that $X_n$ converges ...
2
votes
1answer
181 views

Weak convergence in Hilbert space L2 implies convergence in distribution?

Does weak convergence in $L^2$ (for $X_n, X \in L^2$ we say that $X_n$ converges weakly to $X$ ($X_n \rightarrow^w X$) if for every $Y\in L^2$ we have $\mathbb{E}X_nY \rightarrow \mathbb{E}XY$) ...
6
votes
1answer
190 views

Measure on a separable Hilbert space

Let $H$ be a real separable Hilbert space. Is it true that there exist a probability space $(\Omega, \mu)$ and a measurable function $\pi\colon \Omega \to H$ such that for any $h \in H$ we have $$ ...
6
votes
0answers
277 views

Is this question solvable? $2$ non-linear equations and the proof that the solution is unique (with asymmetric bounty option)

As mentioned in the title I want to show the uniqueness of the solution to $2$ non-linear equations. However, it seems that I can not solve this question with my current mathematical knowledge. More ...
2
votes
1answer
146 views

Prove or disprove that the given expression is “always” positive

I have previously asked a question and I tried to solve it by my own and it led to the question below: Prove or disprove that ...
4
votes
1answer
113 views

functional analytic interpretation of the (co)variation and the doob decompostion

I have a question concerning the covariation of two time-discrete stochastic processes. Let $(\mathcal{F}_i)_{i\in T}$ be a filtration. We call a time-discrete, real-valued, adapted process $X$ ...
7
votes
1answer
208 views

Reproducing Kernel Hilbert Space is dense?

Let $E=C[0,1]$, space of all real-valued continuous functions on $[0,1]$, $\mathcal{E}$ be its Borel $\sigma$-algebra and $\mu$ a Gaussian measure on $E$. Let $E^*$ be a space of all continuous ...
3
votes
1answer
593 views

Dual of $C[0,1]$, Hilbert space and Riesz representation.

Let $E=C[0,1]$, space of all real-valued continuous functions on $[0,1]$, $\mathcal{E}$ be its Borel $\sigma$-algebra and $\mu$ a Gaussian measure on $E$. I need help proving the following claim: ...
25
votes
4answers
1k views

Given two basis sets for a finite Hilbert space, does an unbiased vector exist?

Let $\{A_n\}$ and $\{B_n\}$ be two bases for an $N$-dimensional Hilbert space. Does there exist a unit vector $V$ such that: $$(V\cdot A_j)\;(A_j\cdot V) = (V\cdot B_j)\;(B_j\cdot V) = 1/N\;\;\; \ ...