0
votes
0answers
24 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
0answers
24 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
41 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
17 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
37 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
21 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
18 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
35 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
46 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
14 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 $(g_k \in G)_{k\in \mathbb{N}}$ ...
1
vote
0answers
44 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
25 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
18 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
44 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
38 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
43 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
45 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
91 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
142 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
45 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
47 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
64 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
66 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 $X$. ...
0
votes
0answers
52 views

Exercise 3.2.5 Durrett

"Suppose $g, h$ are continuous with $g(x)>0$ and $|h(x)/g(x)|\rightarrow 0$ as $|x| \rightarrow \infty$. If $F_n =>F$ and $\int g(x) dF_n\leq C< \infty$ then $\int h(x)dF_n(x)\rightarrow ...
3
votes
1answer
60 views

Hilbert space - probability measure: st. norm. variables

I am considering the following homework. Let $\Omega=\ell_2$ be the Hilbert space of square summable sequences, $\mathcal A$ the Borel $\sigma$-algebra and $\{e_n: n\mbox{ natural}\}$ the natural ...
4
votes
1answer
44 views

continuity of a map on $M(\mathbb{R}^n)$

Let $M:=M(\mathbb{R}^n)$ be the space of probability measures on $\mathbb{R}^n$ with respect to the Borel $\sigma$-algebra. Let $K\subset M$ be a compact convex subset. $K$ carries a natural ...
6
votes
3answers
215 views

What is the relation between weak convergence of measures and weak convergence from functional analysis

To keep things simple, we assume $X$ to be a polish space (think of $X$ as $\mathbb{R}^n$ for example). Let's denote with $P(X)$ the space of all Borel probability measure on $X$. We say ...
2
votes
1answer
98 views

Has a probability density function a weak derivative

Assume I have a probability density function $\rho$ on $R$. (e.g $\rho \geq 0$ $\int \rho dx =1 $ $\rho \in L^1(R)$ ...). So $\rho$ is the density wrt the lebesgue measure. Now I try to understand if ...
5
votes
1answer
93 views

$P(X)$ is locally compact if $X$ is?

Let us assume, as in here Measurable structure on the space of probability measures that $X$ is a locally compact Polish space. Then can the same thing be said of $P(X)$, its probability measures ...
4
votes
1answer
87 views

Measurable structure on the space of probability measures

My advisor only half-jokingly mentioned that sometimes people like to consider the measurable structure on $P(X)$ where X is a locally compact polish space and $P(.)$ denotes the probability measures ...
4
votes
1answer
98 views

Kolmogorov continuity theorem for Banach space valued random processes

I am interested in the Kolmogorov continuity theorem. I would like to know if this theorem holds for Banach space valued random processes (probably separable Banach space). I cannot find a paper or a ...
2
votes
1answer
233 views

Measurability of supremum over measurable set

Consider a finite-valued function $f : \mathbb{R}^n \rightarrow \mathbb{R}$; and a closed-valued, measurable, set-valued mapping $S: \mathbb{R}^m \rightrightarrows \mathbb{R}^n$ . Measurability is ...
3
votes
0answers
147 views

What are some characterizations of the strong and total variation topologies on measures?

The Wikipedia article on convergence of measures defines three kinds of convergence: total variation, strong, and weak. For weak convergence, a number of equivalent formulations are given, but not for ...
2
votes
1answer
161 views

Prokhorov's theorem for finite signed measures?

Prokhorov theorem provides a useful characterization of relatively compact sets w.r.t. narrow topology (topology induced by narrow convergence) in the space of probability measure. Notation used ...
3
votes
0answers
86 views

Mathematics Courses for an Economist

I am an Economist and I am interested in further developing my mathematical knowledge and skills. I would like to get your opinions on the topics that I should cover and which are also important for ...
2
votes
0answers
74 views

Learn about reproducing kernel Hilbert spaces?

Why are reproducing kernel Hilbert spaces an important topic to learn? What is possibly achievable with that theory that is not reachable with just standard Hilbert space theory?
3
votes
1answer
87 views

functional analysis in probability theory, Feller processes, contraction semigroups

In an entire year of probability theory coursework at the graduate level, there was only one time when functional analysis seriously appeared. That was ergodic theory. Now that my self-studies have ...
3
votes
1answer
91 views

Measure dualization

What ways are known to correspond, or transfer, a Borel probability measure $\mu$ over some Banach space $X$ to a Borel probability measure $P$ over $X^{*}$, the dual space? Of course, if $X^{*} = ...
1
vote
0answers
50 views

Banach space :space of all adapted processes continuous equipped wih specific norm is complete

Let $\mathbb{B}$ be space of all adapted processes continuous equipped with the norm $\lVert Y\rVert_{\mathbb{B}}^2=E\left[\sup_{t\in [0,T]} |Y_{t}|^{2}\right] < \infty $, ...
2
votes
1answer
62 views

Application of Green's theorem to probability

I encountered this problem while reading a statistic text. Since I am not quite familar with the background knowledge. Wonder can someone help me to explain the details of the following proof? ...
3
votes
1answer
64 views

Question about a separation theorem

This theorem is called the Kreps-Yan theorem. I have just a small question about the proof. We have a probability space given. The statement is: Let $p\in [1,\infty]$, $q$ the conjugate. Suppose ...
1
vote
1answer
40 views

Supremum over dense subset

I'm interested about the following supremum: $$\sup_{g\in A}E[-gf]$$ where $A\subset\{g\in L^1:g\ge 0,E[g]=1\}$ and $f\in L^\infty$ is fixed. $E$ denotes the expectation, i.e. it is the integral ...
1
vote
0answers
59 views

How to represent $\limsup \cdot \liminf$ of Booleans

Let $X$ be some set, and let $A,B\subset X$. By $1_A(x)$ let us mean the indicator/characteristic function. Let $(x_n)_{n\in \Bbb N}$ be some sequence in $X$. I have an expression of the form $$ ...
3
votes
2answers
94 views

Riesz-Representation theorem for a special class of functions

The original riesz representation theorem states Let $X$ be locally compact hausdorff space. Then for any nonnegative functional $\Lambda$ on $C_c(X)$, there is a unique regular borel measure ...
1
vote
0answers
47 views

Total set of functions in $L^2(\Omega)$

Are the sets of functions $\{e^{\int_0^T h(s)dB_s -\frac{1}{2}\int_0^T h(s)^2 ds}\}$ and $\{e^{\int_0^T h(s)dB_s}\}$ total in $L^2(\Omega)$? What is the difference? What should one use to prove weak ...
0
votes
1answer
35 views

Continuous mapping theorem for convergence in $L^2$

This question is related to Analogue of continuous mapping theorem for convergence in $L^2$ but has narrower conditions. Is it true that: If 1) $g$- continuous $\mathbb{R}\to\mathbb{R}$ function, ...
2
votes
1answer
156 views

Analogue of continuous mapping theorem for convergence in $L^2$

Could you help please: Is there any analogue of continuous mapping theorem for convergence of sequence of random variables in $L^2$? I mean: If $g$ is a continuous function $\mathbb{R}\to\mathbb{R}$ ...
2
votes
0answers
46 views

Strictly monotone probability measure

Let $m$ be a probability measure on $X \subseteq \mathbb{R}^n$. Let $f: X \rightarrow \mathbb{R}$ be measurable. Assume that there exists $\epsilon > 0$ such $m\left( \{ x \in X \mid |f'(x)| \leq ...