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Questions tagged [probability-measures]

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0answers
39 views

How can we show that the total variation distance of $X_s$ and $Y_s$ is bounded by the distance of $(X_t)_{t\ge s}$ and $(Y_t)_{t\ge s}$?

Let $(X_t)_{t\ge0}$ and $(Y_t)_{t\ge0}$ be real-valued time-homogeneous Markov processes with a common transition semigroup $(\kappa_t)_{t\ge0}$. Let $\mathcal L(Z)$ denote the distribution of a ...
8
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1answer
153 views

On the existence of a particular type of finite measure on $\mathbb N$

Let $\mathbb N$ denote the set of all positive integers. Does there exist a countably additive measure $\mu : \mathcal P(\mathbb N) \to [0,\infty)$ such that $\mu(\mathbb N)<\infty$ and $\mu(\{nk: ...
3
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2answers
177 views

Example of measure for some algebra over N

$\mathcal F$ is set of events. Can you give an example of some algebra $\mathcal A$ over $\mathbb N$ and a non-zero finitely additive measure $\mu$ on this algebra, which has a countably additive ...
3
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1answer
102 views

Is there a coupling that induces a given coupling via a transition kernel?

Let $X,Y$ be two measurable spaces, $\mu,\nu$ two probability measures on $X$, and $\kappa$ a transition kernel from $X$ to $Y$. Define $\tilde\mu(dy)=\int_X\kappa(dy|x)\mu(dx)$ and $\tilde\nu(dy)=\...
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0answers
65 views

Distribution of dot product of two unit complex random vectors [duplicate]

Consider $u,v∈S^{M-1}\subset \mathbb{C}^M$ to be two independent unit norm random vectors on the $M−1$ dimensional complex sphere $S^{M−1}$. In addition, $u$ follows an isotropic distribution, i.e., $...
2
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1answer
66 views

How does a statistical divergence change under a Lipschitz push-forward map?

Let $\mu, \nu$ be two probability measures on a space $X$ (assume Polish space). $T: X \rightarrow Y$ is a Lipschitz-map that acts as a push-forward on these measures; let $\mu^\prime = T_{\#\mu}$ and ...
1
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1answer
111 views

A question on weak convergence of probability measures

Let $(\mu_{n})$ sequence of probability measures of $\mathbb{R}^{d}$ converging to the prob measure $\mu$. Then by definition we know that $\int f d\mu_{n} \longrightarrow \int f d\mu $ for f ...
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0answers
76 views

Schilder's theorem for brownian bridges

I am really not a probabilist and I apologize if my question is too naive or not appropriate, please feel free to migrate to SE. A bit of context: usually, Schilder's theorem tells us that the ...
1
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0answers
64 views

Upper bound for $\sup_{W_d(Q, P) \le \epsilon} Q(A)$, where $W_d$ is the Wasserstein metric

Let $X=(X,d)$ be a metric space and let $W_d$ denote the Wasserstein metric induced by this metric, on the space of probability distributions on $X$. Let $\epsilon \ge 0$, $A$ be a Borel subset of $X$...
1
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1answer
120 views

Wasserstein interpolation between two probability measures on a metric space

Question 1 Given probability measures $\mu$ and $\nu$ on the same metric space $X=(X,d)$, and $\alpha \in [0, 1]$, is it always possible to find another probability measure $\lambda_\alpha$ on $X$ ...
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0answers
37 views

Reformulate Wasserstein constraint optimization on product space in terms of marginal

Let $X = (X,d_X)$ be a metric space and $Y$ be an abstract set (with at least two elements). Consider the metric on $X \times Y$ defined by $$d((x,y),(x',y')) = \begin{cases}d_X(x,x'),&\mbox{ if }...
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0answers
23 views

Approximate in $W_1$ sense, an empirical distribution with restriction of true distribution on a set

Let $\mu$ be a probability distribution on a metric space $X=(X,d)$ (to avoid unnecessary complications, assume the full filtration $2^X$) and let $x_1,x_2,\ldots,x_N$ be a sample of size $N$ drawn i....
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0answers
77 views

Upper-bound KL divergence between sub-gaussian variables with same variance

A random variable $X$ is said to be sub-gaussian with mean $\mu$ and pseudo-variance $\sigma^2$ iff $$\mathbb \log(E[\exp(t(X-\mu))]) \le \frac{t^2}{2\sigma^2},\;\forall t \in \mathbb R. $$ It's a ...
9
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1answer
466 views

Is KL divergence $D(P||Q)$ strongly convex over $P$ in infinite dimension

By KL divergence I mean $D(P||Q) = \int dP \log(\frac{dP}{dQ})$. I am looking for the conditions under which this strong convexity is true and possible references. I could not find an answer for ...
1
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1answer
83 views

are there measure preserving mapping in this case?

Suppose f and g are two Borel function on [0, 1]. The push-forward of the Lebesgue measure on [0,1] by f and by g are the same. Then are there some Borel measurable function from [0,1] to [0,1], ...
1
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1answer
89 views

Convergence of measurable functions in a locally compact space

Set $(X,\mathcal{B})$ a measurable space. If $f:X\rightarrow[0,\infty)$ is a measurable function then exists a sequence of simple functions $\{s_n\}_{n\geq1}$ such that $$0\leq s_1 \leq s_2\leq \...
3
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1answer
75 views

Antisymmetry of the stochastic order

An ordered topological space is a topological space $X$ equipped with a partial order $\leq$ which is closed as a subset of $X\times X$. By antisymmetry of $\leq$, it follows that the diagonal of $X$ ...
4
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1answer
221 views

A metric stronger than total variation

Let $P,Q$ be two distributions on a finite set $X$. Consider the following metric* ​$$ d(P,Q) = \frac12\max_{\emptyset\neq A\subseteq X} \|P​(\cdot\mid A)-Q(\cdot\mid A)\|_1. $$ Obviously, the total ...
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2answers
101 views

Spectrum of finite-band random matrices?

Let $X_n=(X_{ij})_{1 \leq i,j \leq n}$ such that : $$ \begin{cases} &X_{ij} = 0 \quad \text{if}\quad \vert i - j \vert > k\\ & X_{ij} \sim P_X \quad \text{otherwise} \end{cases}$$ And ...
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0answers
61 views

Conditonal convergence implies convergence?

Note : All measures below are probability measures. Let $\mu_n(X,Y)$ be a random probability measure on $\mathbb C$ depending on two random variables X and Y with values in $\mathbb{R}^N$. Actually,...
4
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1answer
216 views

An interesting Markov chain with uniform marginals

Consider the Markov chain $(\theta_n, \phi_n)$ on $S^1 \times S^1$ constructed in the following way. For $\xi_n$ a sequence of i.i.d. normal random variables and $\kappa > 0$ a fixed number, we set ...
1
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1answer
92 views

property of iid random variable

Let $ (\xi_i)_{i \ge 1} $ be independent identically distributed random variables, taking values in $ (1,3]$. Can we show: $P( \exists N \in \mathbb{N}, \text{ s.t. } \forall k \ge 0, \prod_{i=1}^{...
4
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1answer
291 views

Reference request: discretisation of probability measures on $\mathbb R^d$

Given a probability measures $\mu$ on $\mathbb R^d$ with finite first movement, i.e. $$\int_{\mathbb R^d}|x|\mu(dx)~~<~~+\infty.$$ My concern is to approximate $\mu$ some $\mu_n$ that is ...
4
votes
1answer
105 views

Weak convergence of measures on dense sets

We are given a complete (separable) metric space $X$ and a dense subset $D\subset X$. Consider a sequence of continuous functions $f_n\colon X\to \mathbb R$ such that $$\int\limits_D f_n \, {\rm d}\mu\...
1
vote
1answer
108 views

LLN large number law of Probability

I am studying the Law of large numbers for independent and identically distributed (i.i.d) random variables. Assume there are i.i.d variables $(\xi_k)_{k\ge 1}$ taking values in $(0,1)$. From the law ...
0
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0answers
28 views

Are there any non-asymptotic bounds for the minimum empirical risk vs theoretical risk?

I'm trying to see if there's any bounds on the difference between $f_{ERM}$ and $f^{*}$. For now, define $\mathcal{F}$ to be a function class. Let $P$ be a probability measure and $\hat{P_n}$ be the ...
1
vote
1answer
283 views

A version of the Portmanteau theorem - reference request

I am trying to find peer-reviewed references to the following version of the Portmanteau theorem: Let $M$ be a metric space and let $(\mu_n)_{n\in\mathbb N}$ be a sequence of Borel probability ...
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0answers
61 views

In Wasserstein, what is the relationship between $W_2 (\widehat{\mathbb{P}}_{N},\mathbb{P})$ and $W_2 (\widehat{\mathbb{P}}_{N}^{x},\mathbb{P}^{x})$?

Before presenting my question (which I already formulate in the title of this post) is important to establish the context of my problem: Definition: The $p$-Wasserstein metric $W_{p}(\mu,\nu)$ ...
3
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2answers
168 views

The disintegration of the convolution of two probability measures

Let $G$ be a topological group with all the topological conditions in order that some form of the disintegration theorem be applicable (for instance, take $G$ metrizable). Let $N$ be normal and closed,...
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0answers
61 views

Showing that $b$ is a inner point of $\mathcal{G}$ where $\mathcal{G}$ is a subset of $\mathbb{R}^{N+3}$ determined by $\mathcal{M}^{+}$

Let $(\Xi,\mathscr{E})$ be a measurable space, $(\mathbb{R_{+}},\mathfrak{B})$ other measurable space where $\mathfrak{B}$ a $\sigma$-algebra. We consider the measurable space $(\Xi\times\Xi\times\...
2
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1answer
248 views

Explicitly representing a random variable in terms of indicator functions

Motivation: I want to compute $$E[g(X)] := \int_{\Omega} g(X(\omega)) d\mathbb{P}(\omega) \tag{*}$$ without needing change of variable formula. I want to prove the change of variable formula (you ...
1
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1answer
111 views

Is this function measurable?

Let $(B, \Sigma_B)$ and $(C, \Sigma_C)$ be standard Borel spaces and let $\mu$ be a sub-probability measure on $C$. Given $Y\in \Sigma_{B\times C}$, I would like to use the following function: $$ f:...
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2answers
307 views

Conditional Expectation for $\sigma$-finite measures

Someone knows of some definition or reference of how to define conditional expectation for a measure space with $\sigma$-finite measure. I think it should be as follows: Let $(X,\mathcal{B},\nu)$ ...
2
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1answer
77 views

Total variation and relative $\ell_\infty$ metric

Let $$D_{tv}(P,Q) = \frac{1}{2}\sum_{a \in A}|P(a)-Q(a)|$$ and $$D_{\infty}(P,Q) = \sup_{a \in A} \log \max\{\frac{P(a)}{Q(a)}, \frac{Q(a)}{P(a)}\},$$ where $P$ and $Q$ denote probability measures on ...
0
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1answer
135 views

Weak-convergence of probability measures implies the convergence of the measure of a continuity set

Let $\Omega$ be a Polish space and $\mathcal{B}(\Omega)$ be its Borel $\sigma$-algebra. Let $\{\mu_n\}$ be a sequence of probability measures on $\mathcal{B}(\Omega)$ such that $\mu_n$ weak-converges ...
0
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1answer
78 views

Upper bound for KL divergence on compact space

Let $(\Omega, \mathcal{F}, \mu)$ be a probability space and let $Q$ be the uniform distribution on $(\Omega, \mathcal{F})$ such that $q = dQ / d\mu$ exists. Then the KL-divergence for some probability ...
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1answer
71 views

transformation of two measures on different space

Let $\{e_1,e_2,...,e_n\}=E $ be the standard bases of $\mathbb{R}^n$, and $U\subset\mathbb{R}^n$ be a linear space generated by $\{e_1,e_2,...,e_n\}$. Let $\Sigma_U$ be the smallest $\sigma-$ field ...
3
votes
1answer
142 views

Radon-Nikodym derivative of the group action on the Furstenberg-Poisson boundary of lamplighter groups

Let $G_d$ be the Lamplighter group $G_d = \mathbb{Z}^d \wr \mathbb{Z}_2 $ and $\Gamma =\{(\bar{\eta},\tilde{0}),(\bar{0},\tilde{e_1}), \cdots,(\bar{0},\tilde{e_d})\}$ be the generator set of $G_d$ (...
1
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1answer
124 views

Time Interval of Existence of an SDE solution with Locally Lipschitz Drift

Consider the stochastic ODE $$ dX = F(X)dt + dB $$ where $B$ is Brownian motion. If the drift $F$ is locally Lipschitz, then the solution exists and is unique over $[0,T]$ where $T$ is an "almost ...
3
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1answer
183 views

Measurable functions in product space

I am reading a book by Billingsley (convergence of probability measures) and he makes a footnote on page 27 which I am struggling to understand. I'll explain the setup below. Suppose $(X_n,Y_n)$ are ...
8
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1answer
169 views

Tail bound of a distribution

Let $X_1, X_2, \ldots, X_n, Y_1, Y_2, \ldots, Y_n$ be independent binary random variables each being $1$ with probability $\frac{1}{k}$. Let $Z = X_1(Y_1 + \cdots + Y_k) + X_2(Y_2 + \cdots + Y_{k+1})...
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0answers
80 views

Expected value of parametrized Gibbs distribution w.r.t another probability distribution

Let $\mu$ be a compactly supported probability measure on a finite-dimensional euclidean space (for simplicity) $\mathbb E$, and suppose $\mu$ has density. For a random point $x \sim \mu$, consider ...
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0answers
156 views

Can a topological vector space be probabilistic metric space too? [closed]

Is it possible to have probabilistic metric space (S,F,T) be a topological vector space too? In specific way, the probabilistic metric space is Menger and does not have a norm, however with Menger ...
2
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1answer
103 views

Reference Request for Couplings with Conditions

I have two discrete (integer-valued) random variables $A,B$, with $1\le A\le n$ and $1\le B$. A coupling is a joint distribution of $A,B$ with marginal distributions $A,B$. I know there are several ...
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1answer
101 views

How to characterize Radon Nikodym's derivative of a coupling with respect to any measure in the product space?

In my math essay of thesis I have defined the probability coupling as follows $$\Pi(\mu,\nu)=\left\lbrace \pi \in \Omega \left\vert \begin{matrix} \pi(A\times\mathcal{Y})=\mu(A) \\ \pi(\mathcal{X} \...
2
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0answers
28 views

Orthogonal polynomial expansion for bivariate noncentral chi-square and bi-variate noncentral student t distribution

This is a research question for which I am not able to find any existing reference. So, I am reaching out for help. The project is related to studying the sequence of rejections in multiple hypothesis ...
3
votes
1answer
127 views

Measurability of a particular set generated by discrete probability measures

Suppose that $(S,\Sigma)$ is a measurable space with $S$ Polish and $\Sigma$ its Borel sigma algebra. Let $\mathcal{C}$ be the collection of discrete probability measures on $S$ having countably ...
8
votes
3answers
408 views

Question about Wasserstein metric

Let $\mu$ and $\nu$ be two probability measures on $\mathbb R^n$ with finite first moment. Denote by $d:=W_1(\mu,\nu)$, where $W_1(\cdot,\cdot)$ stands for the Wasserstein distance of order $1$. My ...
2
votes
1answer
116 views

Measure space for trees and other algebraic datatypes

Given a measure space $\mathcal M$, I am wondering what kind of measure space $\mathcal T(\mathcal M)$ one could associate to the set of binary trees with elements from $\mathcal M$ at each node. The ...
1
vote
1answer
101 views

Diffuse measure space as a product of $[0;1]$ and another diffuse measure space

The title speaks of itself. How far is an arbitrary finite diffuse measure space from being almost isomorphic to a product of $[0;1]$ with another diffuse measure space? What would be reasonable ...