0
votes
0answers
10 views

Hypercontractivity of Markov Operator

I have been reading a paper by Ahlswede and Gacs on hypercontractivity of Markov operator (see here 1) and its application to information theory. To be honest, I could not fully understand the ...
0
votes
0answers
24 views

Intuition about generating functions

I am trying to gain some intuition about moment generating functions. In particular, for a random variable $X$, we have $$ \newcommand{\E}[1]{\mathbf{E}\!\left[#1\right]} M_X(t) = \E{e^{Xt}} = ...
1
vote
0answers
45 views

Is the plane created by ($\int f_0^u f_1^{1-u},\int f_1^u f_0^{1-u})$ continuous?

$f_0$ and $f_1$ are two continuous density functions on $\mathbb{R}$. I wonder if $$(x,y):=\Bigg(\int f_0^u f_1^{1-u},\int f_1^u f_0^{1-u}\Bigg)$$ for all $f_0$ and $f_1$ is complete for some ...
0
votes
1answer
25 views

Determining the semigroup and relating it to the generator via the functional calculus

In the theory of Hille Yoshida, we have a (definitely real) banach space $L$ and on it a contractive, strongly continuous semigroup of operators. ($T_tT_s=T_{t+s}$ and $T_0=I$.) We then define the ...
2
votes
2answers
94 views

Intuition behind convolution identity for Laplace transforms

Convolutions, relatively speaking, are fairly straightforward for simple systems (from an applied perspective), but I cannot, at all, find the intuition behind the Laplace identity for convolutions. ...
0
votes
0answers
15 views

L1 expected error for scale/translation in densities

This is an example given in the article about testability (Devroye and Lugosi 1999) (link: http://repositori.upf.edu/bitstream/handle/10230/1024/375.pdf?sequence=1 ) page 7. First I will introduce my ...
1
vote
0answers
36 views

Properties of the essential supremum

In the article about testability of densities (Devroye and Lugosi 1999, page 4) (link: http://repositori.upf.edu/bitstream/handle/10230/1024/375.pdf?sequence=1 ). They use some properties of the ...
3
votes
1answer
34 views

Local convexity of the topology of weak convergence of probability measures

Let $X$ be a Polish or standard Borel space, and $\mathcal P(X)$ be the space of all Borel probability measures on $X$ endowed with the topology of weak convergence. I am thinking of using ...
3
votes
1answer
81 views

Usual convex combination and the one with measure

Let $X$ be a Borel measurable subset of $\Bbb R^n$ and let $\nu$ be a probability measure on $X$. Can we always find an integer $m$, points $x_1,\dots,x_m\in X$ and coefficients $a_1,\dots,a_m \geq ...
2
votes
1answer
35 views

A question on linearity of inner product

The linearity of inner product on $(X,\langle.,.\rangle)$ is usually written as $$\langle x+\alpha y,z\rangle = \langle x,z\rangle + \alpha\langle y,z\rangle,\qquad \forall (\alpha,x,y,z)\in R\times ...
1
vote
0answers
13 views

Equivalent Gaussian measures

Let $\mu$ be a gaussian measure with eigenpair $\{e_k,2^{-k}\}$ and $\nu$ with eigenpair $\{ Te_k,2^{-k}\}$. Here, $T$ is the unitary operator given by $Tx = x - 2\left\langle x,v \right\rangle v$. ...
1
vote
0answers
41 views

Linear w.r.t. any measure

Let $X$ be a Banach space endowed with a Borel $\sigma$-algebra. How do we call a real-valued Borel function $f$ that satisfies for any Borel probability measure $\mu$ the following formula $$ ...
1
vote
0answers
47 views

Representation of a linear functional Lipschitz in total variation

Let $\Omega$ be a Borel space and let $\mathcal P(\Omega)$ be the space of all Borel probability measures on $\Omega$ endowed with the topology of weak convergence. Define the total variation metric ...
2
votes
1answer
65 views

Why the space of probability measures is a subset of the measure space

Consider $\mathcal M (X)$ the measure space of a metric, compact space $X$ allowed of the weak-* topology induced by the semi-norms $\mu \in \mathcal M (X) \mapsto |\int_X f ~d\mu| \in \mathbb R ...
1
vote
0answers
22 views

Spectral Representation for a real valued process

So I just finished reading a section in a book which discusses how every stationary stochastic process $\xi(t)$ can be expressed as $\xi(t)=\int_{\mathbb{R}}e^{it\lambda}\,dZ(\lambda)$ where ...
6
votes
1answer
156 views

Looking for different proofs of “Discrete Liouville's Theorem”.

Good day. There is a question I have already encountered twice, in very different contexts, that is relatively simple looking, but both solutions I know involve some pretty advanced theorems from the ...
2
votes
1answer
57 views

Solving $ \inf \left\{ F[\nu] : \nu \in L^2 , \nu \geq 0, \int _0 ^1 \nu=1\right\}$

Let $\phi \in \mathcal ( [0,1]^2)$ symetric , can we find a solution to the following minimisation problem? $$ \inf \left\{ F[\nu] : \nu \in L^2 , \nu \geq 0, \int _0 ^1 \nu=1\right\}$$ with $$ ...
1
vote
1answer
30 views

DKW-style $\ell_{\infty}$ bounds for sum of i.i.d. random functions: $\to [0,1]$

Let $\mathbf{G}$ be the set of (edit: convex) functions $g: X \to [0,1]$, where $X$ is a compact subset of $\mathbb{R}^d$ or something like that. Suppose I have a distribution $D$ on $\mathbf{G}$. ...
2
votes
0answers
43 views

Representation of Stochastic Integrals as Lebesgue/Bochner Integrals

Just as the Riemann–Stieltjes integral can be equivalently defined as a Lebesgue integral with the corresponding Lebesgue–Stieltjes measure, I am looking for the corresponding results for the ...
4
votes
1answer
124 views

Convegence of regularized sequence in $L^2$

Let $(\rho_n)_{n \geq 0}$ be a standard regularizing sequence on $\mathbb R$. Let $P$ be a probability measure on $\mathbb R$ such that the sequence $(P*\rho_n)_{n \geq 0}$ is bounded in $L^2$. Then, ...
0
votes
1answer
18 views

Probability space problem

Given:$( \Omega ,F,\mu)$- probability space. $f:\Omega\rightarrow X'. X$-is a Banach space, such that the mapping $\Omega\ni\omega\mapsto\langle x, f(\omega)\rangle\in L^1(\Omega,\mu)$ for all $\ ...
1
vote
0answers
22 views

PDE-Based Triangle Inequality for Optimal Transportation

Suppose $\Omega$ is a suitably regular domain in $\mathbb{R}^n$ and $\rho_0,\rho_1\in\textrm{Prob}(\Omega)$. Benamou and Brenier showed that the $L_2$ transportation distance between $\rho_0$ and ...
2
votes
1answer
57 views

Weak convergence: equivalence of definitions

Consider a sequence of random variables $(X_n)_{n\geq 0}$ and a random variable $X$. How to prove that the two following definitions of weak convergence are equivalent? Def 1 $(X_n)_{n\geq 0} ...
0
votes
1answer
69 views

Wiener measure integration

Let $\mu_{x,y;t}$ be the Wiener measure generated by $\exp[t \Delta](x,y)$. Now I see in my book the following step: $\int dx\int d\mu_{x,x;t}(\omega) \phi(x) = \int dx\int d\mu_{x,x;t}(\omega) ...
1
vote
0answers
20 views

When are the marginals of an extremal invariant measure also extremal invariant?

Let's suppose that $X$ is a compact metric space, and thus as is $X \times X$. If given a Markov process on $X \times X$ with marginals that are Markov processes on $X$, then we know that the ...
0
votes
0answers
56 views

Is the martingale propertey preserved by taking weak$^*$-limits?

Let $(\Omega,\mathcal F)$ be a measurable space, $X:\Omega\rightarrow\mathbb R^d$ a $\mathcal F$-measurable map. Let $(\mathcal F_k)_{k\in\mathbb N}$ be a filtration of $\mathcal F$ such that ...
0
votes
1answer
66 views

Signed finite Radon measures with vague topology

If $X$ is a locally compact and $\sigma$-compact metric space. Let $M(X)$ be the space of signed finite Radon measures on $X$. (1) Show that measures with finite support is dense is $M(X)$ in the ...
1
vote
1answer
63 views

Weak*-convergence of probability measures

Let $(\Omega,\mathcal F)$ be a measurable space and $X:\Omega\rightarrow\mathbb R^d$ a $\mathcal F$-measurable, bounded map. Let $(\mathbb Q_n)_{n\in\mathbb N}$ be a sequence of probability measures ...
0
votes
1answer
23 views

Density transformation, distribution function

Suppose $X$ is a real-valued random variable with density $p_X$, so $$P(X\leq x) = \int_{-\infty}^x \, p_X(y) \, dy.$$ What conditions on a function $f$ are needed (typically?) to find the density ...
1
vote
0answers
9 views

invariant measures for the independently coupled Feller process

If I take countably many Markov processes and couple them independently, how does the collection of invariant measures for the coupled process relate to those of the marginals? I conjecture that it ...
1
vote
0answers
57 views

Expectation of $p$-norm under a Gaussian on the Hilbert space $L^2(S^1)$

Let $\mu$ be a centered Gaussian measure with (nondegenerate) covariance $Q$ on the Hilbert space $L^2(S^1;\mathbb R)$ where $S^1$ is the circle. We can take for example the covariance ...
1
vote
0answers
34 views

The infinitessimal generator of Brownian Motion

Background: I know, and can almost prove, that the infinitessimal generator of BM is the Laplacian/2. For me (and I have never heard it done differently) we have Feller Processes, of which BM is one ...
0
votes
0answers
23 views

Reflected borel sets are worse traps for Brownian paths.

This is for a research project. I am trying to prove that given a borel set, it's reflected version will have a lower Wiener measure of brownian paths intersecting it. In this paper they elaborate ...
2
votes
1answer
57 views

Skorohod convergence (space of right continuous functions with left limit)

If $f_n$ is a sequence of functions of the Skorohod Space $D([0,\infty),E)$, where $E$ is a separable Banach space, such that $f_n \to f$ in the Skorohod topology. Is it possible that there exists a ...
0
votes
1answer
63 views

Need a proof for an assumption on conditional probability density function based on probability theory

While reading book "Elements of Information Theory", I came across an assumption used in a proof on page 33. The assumption is as follows. Let $(X,Y)\sim p(x,y)=p(x)p(y|x)$. "If $p(y|x)$ is fixed, ...
1
vote
0answers
16 views

$L_1$ mean ergodic theorem for the action of compact group

Let $X$ be a Polish space with Borel probability measure $\mu$. A compact group $G$ acts on X continuously. It is right that for any $f\in C_b(X)$ exists a sequence $(g_k \in G)_{k\in \mathbb{N}}$ ...
2
votes
0answers
86 views

How to define a probability distribution over a function space?

What is the mathematically rigorous way of defining a probability distribution over some function space e.g. $L^1[0,1]$? Edit: After reading about the basics of measure theory, I realized that the ...
0
votes
0answers
31 views

Average over all positive functions on the unit interval whose Lebesgue integral is one

I want to average over all positive functions on the unit interval whose Lebesgue integral is one. Formally, I want to compute the mean of the following probability distribution defined over function ...
0
votes
0answers
29 views

Are there “necessary” conditions for a solution to the multivariate, truncated Hausdorff moment problem?

I am looking for NECESSARY conditions for a solution to the multivariate, truncated Hausdorff moment problem (i.e., conditions under which a given finite sequence of numbers is the sequence of first ...
1
vote
0answers
18 views

Continuity of covariance kernels

Let $I$ be a locally compact Hausdorff (LCH) topological space. Let $c : I \times I \to \mathbb R$ be a covariance kernel, that is, a symmetric, nonnegative-definite function. Does it follow that $c$ ...
1
vote
1answer
58 views

Continuity of integral w.r.t. Lebesgue measure

Let $f:\Bbb R^n\times \Bbb R^m\to \Bbb R$ be a bounded measurable function, and $\mu$ be a probability measure on $\Bbb R^m$ which is absolutely continuous w.r.t. Lebesgue measure $\lambda$, e.g. $m ...
3
votes
1answer
43 views

Pointwise Ergodic Theorem - one particular estimate

I am struggling with an estimate in a proof of the pointwise ergodic theorem discussed in T. Ward and M. Einsiedler: Ergodic Theory with a view towards Number Theory, which is left as an exercise. I ...
3
votes
0answers
46 views

Is it reasonable to think of the expectation of an infinite-dimensional vector?

Given a probability space $(\Omega, \mathcal{F}, P)$, a random vector is an $\mathcal{F}$-measurable mapping $X: \Omega \rightarrow \mathbb{R}^{k}: X(\omega) = (X_{1}(\omega), ...
1
vote
1answer
49 views

supremum norm and convergence.

Suppose $f:[0,\infty) \rightarrow \mathbb{R}$ is continuous. Suppose that for some $\epsilon$ > 0, $\max_{t \in [0,n]} |f(t)|$ < $\epsilon$ for all $n \in \mathbb{N}$. Is it then true that ...
2
votes
2answers
112 views

Measurability of a total variation metric

Let $X$ be a standard Borel space and let us denote by $\mathcal P(X)$ the space of Borel probability measures on $X$ endowed with the topology of weak convergence. Define $d:\mathcal P(X)\times ...
8
votes
1answer
147 views

Maximum of measures over sets and functions

Let $(X,\mathcal A)$ be any measurable space and denote by $\mathrm b\mathcal A_1$ the set of all real-valued $\mathcal A$-measurable functions $f$ satisfying $\|f\|:=\sup_{x\in X}|f(x)|\leq 1$. Let ...
0
votes
1answer
50 views

A concrete example of a non-pure-point measure

I am considering $P(\mathbb{R})$ the space of probability measures on the real line. We can regard this as within the space of continuous linear functionals on the space of continuous functions ...
2
votes
0answers
56 views

Generalizing the ptwise or $L^1$ ergodic theorem

I was able to write a proof based on the hardy-littlewood maximal function of the following statement: Let $(X, \Sigma, \mu)$ be a $\sigma$-finite measure space, and $d=n \geq 1$ be fixed. Let ...
0
votes
0answers
68 views

Exchangability of inner product and integral in bochner spaces

For the linear operator $e \in \mathcal{L}(V,V^{*})$, and sufficiently small $\delta s \in V := L^2(0,T,L^2(D))$ and $p \in V$ we have; $$ E[\langle e^{*}p, \delta s \rangle_{V}] = \langle E[e^{*} p ] ...
3
votes
1answer
69 views

How to prove that $P^n(x,\cdot)\stackrel{w}{\rightarrow} \pi$ for all $x$, implies the equicontinuity of $\{P^n f\}_n$ on compact sets?

Let $X$ be a separable and locally compact metric space and let $P\colon X\times \mathcal{B}(X) \rightarrow \mathbb{R}$ be the transition probabability kernel of a homogeneus Markov chain on $X$. ...