Recently Active 'pr.probability+fa.functional-analysis' Questions - Page 10

2 votes

0 answers

166 views

Must rows of a transition matrix be distinct?

Is it true that for all continuous time Markov processes on a countable state space $S$, we have all rows of the transition matrix $\mathbf{P}_t$ are distinct for all time $t\in[0,\infty)$ ? This ...

Fantastic

165

modified Mar 12, 2016 at 21:11

4 votes

1 answer

300 views

Rank of a sequence of covariance matrices

Let $X_i$ ($i=1, \dots$) be an orthonormal basis for $L^2(\Omega, \mathbb P)$. In particular, it holds that $$\mathbb E[X_iX_j] = \delta_{ij}.$$ Now take $Z\in L^2(\Omega, \mathbb P)$ and define $\...

CommunityBot

1

modified Feb 28, 2016 at 7:54

2 votes

0 answers

168 views

Interchanging integrals and continuous linear forms in RKHS

I am reading Reproducing kernel Hilbert spaces in probability and statistics by A Berlinet, C Thomas-Agnan. In Chapter 5 INTEGRATION OF $\mathcal{H}$-VALUED RANDOM VARIABLES they write One of the ...

Manuel Schmidt

619

modified Feb 23, 2016 at 11:02

5 votes

2 answers

429 views

Does the truncated Hausdorff moment problem admit absolutely continuous solutions?

Let $\mu$ be a (Borel) probability measure on $[0,1]$ and define $m_j(\mu) = \int x^j\,\mu(dx)$. Let $k$ be a positive integer and consider the set $\mathcal C_{\mu,k}$ of probability measures $\nu$ ...

CommunityBot

1

modified Feb 10, 2016 at 19:57

3 votes

1 answer

1k views

If $H$ is a separable Hilbert space, is its dual dense in $L^2(H)$?

Let $H$ be an infinite-dimensional, separable Hilbert space, and let $\gamma$ be a Radon probability measure on $H$ with mean zero and covariance operator the identity $I$. Let $H^*$ denote the space ...

CommunityBot

1

modified Jan 14, 2016 at 15:59

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, ...

Nate Eldredge

29.8k

modified Jan 13, 2016 at 19:38

9 votes

1 answer

385 views

A Generalized Version of Maximal Correlation and Hypercontractivity of Conditional Expectation Operator

Given a pair of random variables $(X,Y)$ over a product space $\mathcal{X}\times \mathcal{Y}$, the maximal correlation coefficient is defined as $$\rho_2(X;Y):=\sup\frac{\mathbb{E}[f(X)g(Y)]}{||f||_2||...

CommunityBot

1

modified Jan 12, 2016 at 6:50

2 votes

0 answers

619 views

Laplace transform of a integral function of CIR/CEV process

The Cox–Ingersoll–Ross model (or CIR model) describes the evolution of interest rates. Constant elasticity of variance model (CEV) is a stochastic volatility model, which attempts to capture ...

KNN

323

modified Dec 27, 2015 at 18:18

0 votes

0 answers

322 views

Comparison of Parameter estimation using maximum likelihood and Maximum entropy

I am not sure if the question is appropriate but I want to try my luck. One can estimate a parameter using maximum likelihood and we know it is optimal. On the other hand there are methods which uses ...

Creator

495

modified Dec 19, 2015 at 22:39

2 votes

0 answers

69 views

How sensitive are the n-th step transition probabilities of the simple random walk to a small perturbation of an infinite graph?

Suppose that $G$ is an infinite, locally finite, connected graph. Fix a vertex $o$ in the graph and for each $n$ and $x$ let $p(n,o,x)$ be the probability that a simple random walk (at each step a ...

ofer zeitouni

7,514

modified Dec 6, 2015 at 7:46

1 vote

2 answers

270 views

Fourier transform localisation (still unanswered, but apparently off-topic?) [closed]

In the context of Pólya's theorem I was reading these notes here on p. 19. In the last paragraph the authors claim (it is the sentence starting like "standard Fourier theory shows...") that the ...

CommunityBot

1

modified Dec 1, 2015 at 0:10

18 votes

4 answers

1k views

Reference for a strong intermediate value theorem for measures

Let $\mu$ be a finite nonatomic measure on a measurable space $(X,\Sigma)$, and for simplicity assume that $\mu(X) = 1$. There is a well-known "intermediate value theorem" of Sierpiński that states ...

Salvo Tringali

10.5k

answered Nov 10, 2015 at 21:23

1 vote

0 answers

147 views

Bounding Rayleigh quotient for stochastic matrix

Suppose you have an irreducible, stochastic matrix $A$ with left Perron-Frobenius eigenvector $v$ (corresponding to the eigenvalue $1$), and suppose the next largest eigenvalue for $A$ is $\lambda$. ...

Stefan Kohl♦

19.6k

modified Oct 16, 2015 at 13:06

8 votes

1 answer

360 views

Can we recover a topological space from the collection of Borel probability measures living on it?

Let $(X, \tau)$ be a topological space, and $\mathcal{P}(X, \tau)$ be the Borel probability measures living on $X$. Can we recover $(X, \tau)$ from $\mathcal{P}(X, \tau)$?

Liviu Nicolaescu

34.7k

modified Oct 5, 2015 at 23:45

10 votes

1 answer

253 views

Approximation via finite rank Cameron-Martin projections

Let $(W, \|\cdot\|_W)$ be a real separable Banach space equipped with a non-degenerate Gaussian Borel measure $\mu$. Let $H \subset W$ be the corresponding Cameron-Martin Hilbert space (also known as ...

Bill Johnson

31.5k

answered Sep 27, 2015 at 21:26

5 votes

0 answers

244 views

Tensorization of Orlicz norm?

Associated with a convex function $\phi:[0,\infty)\mapsto[0,\infty)$ satisfying $\lim_{x\to 0} \frac{\phi(x)}{x} = 0, \lim_{x\to\infty}\frac{\phi(x)}{x} = \infty,$ the Orlicz norm of a random variable ...

Hedonist

1,269

modified Aug 27, 2015 at 14:52

5 votes

2 answers

332 views

General additive function of probability

Let $H$ be a function of finite sequences of probabilities (non-negative numbers summing up to 1) into real numbers, such that: $H$ is continuous, $H$ is symmetric w.r.t. the order of its arguments, $...

CommunityBot

1

modified Aug 17, 2015 at 16:43

2 votes

1 answer

164 views

Is there any parameter space of Cramér–Rao_bound

It is known that Cramér–Rao_bound is the lower bound of variance of a parameter. A useful link is https://en.wikipedia.org/wiki/Cram%C3%A9r%E2%80%93Rao_bound There is also a term called '...

CommunityBot

1

modified Aug 12, 2015 at 10:10

3 votes

1 answer

2k views

Are most random variables trivially sub-gaussian? [closed]

I'm trying to understand sub-gaussian RVs to see if they could be relevant to my work. The common definition of a sub-gaussian RV is the following. X is $\sigma$ sub-gaussian if its laplace transform /...

CommunityBot

1

modified Jul 30, 2015 at 18:19

3 votes

0 answers

69 views

Dilation of positive operators into martingales

In Rota's paper (An Alternierende Verfahren for General Positive Operators), Theorem 2 says that: Let $P$ be a doubly stochastic operator which is selfadjoint in $L^2 (S, \Sigma, \mu)$. Then there is ...

CommunityBot

1

modified Jun 17, 2015 at 5:11

1 vote

1 answer

218 views

Row-stochasticity of the Jacobian matrix of a stationary distribution

Let $P_{\mathbf{p}}$ be a $n \times n$ row-stochastic matrix whose entries are a function of a probability vector $\mathbf{p} \in \mathbf{R}_{> 0}^n$, $\sum_i p_i = 1$ and define the following ...

Iosif Pinelis

128k

modified Jun 10, 2015 at 20:07

6 votes

1 answer

444 views

When does a matrix define a convolution operator on a hypergroup?

Let $H$ be a discrete hypergroup. Suppose I have a matrix $A=(A_{x,y})$ indexed over $H$ with nonnegative entries which defines a bounded operator on $\ell^2(H)$. When does there exist $f\in\ell^1(H)$ ...

CommunityBot

1

modified Jun 5, 2015 at 5:40

1 vote

1 answer

720 views

Question about uniform continuity under Skorokhod Metric

Let $D=D([0,1], \mathbb{R})$ be the space of cadlag functions $x$ with $x(0)=0$ and $x$ is continuous on $1$. If we endow $D$ with Skorokhod Metric, see: http://en.wikipedia.org/wiki/C%C3%A0dl%C3%A0g ...

Mostafa Mirabi

1,872

modified Jun 3, 2015 at 17:46

16 votes

3 answers

708 views

An inequality for two independent identically distributed random vectors in a normed space

Suppose that $X$ and $Y$ are independent identically distributed random vectors in a separable Banach space $B$. Does it always follow that $E\|X-Y\|\le E\|X+Y\|$? Some background information on ...

Bill Johnson

31.5k

modified Jun 3, 2015 at 11:16

2 votes

1 answer

453 views

Weak convergence of probability measures on weak versus strong dual

The space of temperate distributions $S'(\mathbb{R}^d)$ is often equipped with the weak-$\ast$ or with the strong topology. When defining the notion of a probability measure on $S'(\mathbb{R}^d)$, ...

CommunityBot

1

modified May 8, 2015 at 20:41

-1 votes

1 answer

148 views

Continuity of function mapping $\mathcal{P}(\mathcal{P}(X))$ to $\mathcal{P}(X)$ [closed]

Given a topological space $Y$, let $\mathcal{P}(Y)$ be the set of all probability measures on $Y$, endowed with the weak* topology. Let $X$ be a topological space (for convenience, it might be Polish ...

Rahman. M

2,381

modified May 4, 2015 at 20:41

3 votes

0 answers

134 views

Poincare inequality for the measure of Brownian path

I am wondering if the Poincare inequality holds for the Brownian path space. As the simplest example, let $\{w_t, t \in [0, 1]\}$ be a 1-d standard BM: has independent increments and continuous ...

gregarki khayal

199

asked May 1, 2015 at 6:24

21 votes

2 answers

3k views

A measure on the space of probability measures

This question was originaly posted in the stackexchange https://math.stackexchange.com/questions/1226701/a-measure-on-the-space-of-probability-measures but since it only got a comment I decided to ...

Otis Chodosh

7,197

modified Apr 21, 2015 at 20:01

7 votes

4 answers

1k views

Is the space of tempered distribution second countable?

Let $\mathcal S '(\mathbb R^d)$ be the space of Schwartz tempered distributions equipped with the weak-* topology. I need to know if this space is second countable, i.e. if this topology has a ...

Abdelmalek Abdesselam

23k

answered Apr 5, 2015 at 21:24

1 vote

1 answer

125 views

A differential inequality and a special value

Let $G \colon [0,1] \to [0,1]$ be a monotonically decreasing function with $G(0) = 1$ and $G(1) = 0$. Suppose that $G$ is differentiable infinitely many times, and that: $$G(x)G''(X) \leq 2{G'(x)}^2.$$...

Hachino

902

answered Mar 26, 2015 at 13:49

3 votes

1 answer

340 views

Relatively compact sets in Ky Fan metric space

Let $(\Omega,P,\mathcal{F})$ be a probability space. $X$, $Y$ are two random variables. The Ky Fan metric defined as: $d_F(X,Y)=\inf\{\epsilon: P(|X-Y|> \epsilon)<\epsilon\}$ (or $d'_F(X,Y)=E \...

Nate Eldredge

29.8k

answered Mar 20, 2015 at 3:43

4 votes

0 answers

107 views

Is Wiener's Tauberian theorem true in Wiener space?

Let $\gamma$ be the standard product Gaussian measure in $\mathbb{R}^\infty$, and let $\mu$ be a finite variation measure, not necessarily positive, such that $\mu \ll \gamma$. Is the following true? ...

CommunityBot

1

modified Mar 20, 2015 at 0:36

1 vote

0 answers

383 views

Topological properties of space of Radon measures

Let $M$ denote the space of signed unbounded Radon measures on $\mathbb{R}$ as is defined by Bourbaki, i.e. $M$ is the dual of $C_c$ where $C_c$ is the space of continuous functions on $\mathbb{R}$ ...

yada

1,773

asked Mar 19, 2015 at 11:03

4 votes

1 answer

356 views

Monotonicity of a ratio of conditional expectation operator

Let a pair of random variables $(X, Y)$ over a finite product space $\mathcal{X}\times \mathcal{Y}$ be given. The conditional expectation operator is defined as $$(T_Yf) (y):=\mathbb{E}[f(X)|Y=y],$$ ...

Anthony Quas

23.2k

answered Mar 17, 2015 at 5:09

5 votes

2 answers

356 views

$L^\infty$ properties of an infinite-dimensional Gaussian semigroup

Let $W$ be a separable Banach space and $\mu$ a Gaussian Borel measure on $W$ which is centered and non-degenerate. For $F : W \to \mathbb{R}$ bounded Borel and $t \ge 0$, let $$P_t F(x) = \int_W F(x+...

Nate Eldredge

29.8k

modified Mar 6, 2015 at 5:48

2 votes

1 answer

186 views

Does $\int \Phi \left( \frac{u}{\xi} \right) f_t(\xi) \mathrm{d} \xi \to \Phi(u)$ imply that $f_t \to \delta_1$?

I'm looking at a family $(f_t)$ of densities of some continuous random variables and know that $$\int_{-\infty}^{\infty} \Phi \left( \frac{u}{\xi} \right) f_t(\xi) \mathrm{d} \xi \xrightarrow{t \to \...

CommunityBot

1

modified Feb 18, 2015 at 12:38

2 votes

0 answers

188 views

Equivalence of two non-degenerate Gaussian measures on Banach space

The motivation of this question is to show that two probabilities on $C_{0}^{n}(0,1)$ (the space of continuous $\mathbb R^{n}$ valued process on $[0,1]$ starting from zero) induced by two non-...

kenneth

1,399

modified Feb 14, 2015 at 23:19

5 votes

1 answer

664 views

Is the space of Radon measures a Prohorov space?

Consider the spaces $C_c(\mathbb{R})$ of compactly supported continuous functions equipped with the inductive limit topology and the Banach space $C_0(\mathbb{R}) = \overline{C_c(\mathbb{R})}^{\, _{||....

yada

1,773

modified Feb 13, 2015 at 13:21

1 vote

0 answers

260 views

Generating the sigma algebras on the set of probability measures

I was wondering if somebody could help me see/provide a reference to the following fact: Let $X$ be a metrizable set, $\mathcal{F}$ the corresponding Borel sigma-algebra on $X$, and $\triangle\left(X,\...

Mark

11

modified Jan 28, 2015 at 3:10

6 votes

1 answer

713 views

Equivalence of Gaussian measures

Let $H$ be a separable Hilbert space and $N(0, C)$ and $N(0, D)$ be Gaussian measures on it. Further, for each $v \in H$, define $R_v = \frac{\left\langle v,Cv \right\rangle}{\left\langle v,Dv \right\...

Nate Eldredge

29.8k

modified Jan 21, 2015 at 21:29

15 votes

2 answers

2k views

Intuitive explanation of Dvoretzky's theorem

I am wondering if anyone has an enlightening explanation of why Dvoretzky's theorem (which says that a high-dimensional convex body has an almost round central section) is true -- there are a number ...

CommunityBot

1

modified Jan 7, 2015 at 11:30

7 votes

1 answer

444 views

Is an infinite-dimensional "Lebesgue measure" uniquely determined by a set of positive finite measure?

Let $\mu$ be a probability measure on a subset $C \subset \mathbb{R}^\infty$ of the space of sequences, and assume, for simplicity, that $C$ is closed and convex. We say that $\mu$ admits shifts if ...

Alexander Shamov

4,010

answered Jan 5, 2015 at 15:46

1 vote

0 answers

417 views

Defining density of a random function using Radon-Nikodym Theorem

Let $(\Omega,\mathbb{F},P)$ be a probability space and $E$ be an infinite dimensional Banach space and $\mathbb{B}$ be the $\sigma$-algebra of Borel subset of $E$. Let $X$ be random function defined ...

Janak

213

modified Nov 28, 2014 at 8:50

0 votes

1 answer

375 views

About an integral equation

I would like to obtain $g$ by solving the following integral equation $$ \int_s^T R(u) dg(u) + f(s,T)\int_s^T g(u)du =0$$ where $f,R:\mathbb R _+ ^*\rightarrow \mathbb R _+ $and $g: \mathbb R _+ ...

CommunityBot

1

modified Nov 20, 2014 at 17:21

1 vote

0 answers

80 views

Rate of convergence in narrow convergence

Does anyone help me in the following question? I have a sequence of probability measures $\mu_n$ and know that $\mu_n$ converges narrowly to a probability measure $\mu$. Is there any way to estimate ...

Hongdmhut Hong

31

asked Oct 22, 2014 at 15:24

2 votes

0 answers

238 views

Examples for Markov generators with pure point spectrum

I'm looking at symmetric diffusion Markov generators $L$ with pure point spectrum, i.e. infinitesimal generators of symmetric diffusion Markov semigroups, which are defined on $L^2(\mu)$ where $\mu$ ...

herrsimon

235

modified Sep 30, 2014 at 10:56

0 votes

1 answer

195 views

Existence of bounded $n-$th derivative of the solution of differential equation

This question is the copy from mat.stackexchange.com here. I requestioned here due to the very limited responses there. Let $\phi:\mathbb{R}\mapsto\mathbb{R}$ be the standard normal density, $$\phi(x)...

CommunityBot

1

modified Sep 23, 2014 at 16:24

3 votes

1 answer

99 views

Regularity of finite variation kernels in the (intersection) of the semimartingale spaces $H^p$

Suppose you have a continuous semimartingale $S_t=M_t + A_t$ where $A_t$ is the continuous finite variation part which has the form $A_t = \int_0^t b_s \, \mathrm{d} s$, where $\int_0^{\infty} |b_s| \,...

CommunityBot

1

modified Sep 23, 2014 at 2:40

5 votes

0 answers

161 views

$L^p$ estimates for Ornstein-Uhlenbeck: what is known beyond hypercontractivity?

Consider an infinite-dimensional Gaussian random vector $X$, and a positive random variable $f(X) \in L^p, p > 1$. Let $f(X) \sim \sum_n f_n(X)$ be its (formal) chaos expansion. Let $(U_\rho, \rho \...

Alexander Shamov

4,010

modified Sep 20, 2014 at 23:27

0 votes

1 answer

222 views

Behavior of the integral of products of probability densities

Assume $z \in \mathbb{R}^m$ and $x \in \mathbb{R}^n$. Assume we have proper density function $P(z)$ and proper conditional density function $P(x|z)$. We give the definition $$ T(x_1,\ldots,x_n) := \...

Colm O'C

21

answered Aug 30, 2014 at 16:47

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