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Questions tagged [coupling]

Use this tag for questions about the proof technique that allows one to compare two unrelated random variables (distributions) X and Y by creating a random vector whose marginal distributions correspond to X and Y.

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27 views

coupling of distributions vs joint distributions

For continuoues variables, how is a coupling of two distributions different from their joint distribution? Are they the same concepts? Update: Coupling is the same as defining a joint distribution on ...
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1answer
41 views

Simple random walk on $\mathbb Z$; Coupling Argument

I am reading the proof of Theorem 3.1 from these notes and I am stuck at one point. Let $X_1, X_2, X_3, ...$ be i.i.d random variable valued in $\{1, -1\}$ each distributed uniformly. Let $S_n=\sum_{...
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29 views

Example/Reference needed for Laplace equation coupled with another equation

I have been trying to solve a heat-exchanger problem where two fluids are separated by a conducting wall between them and the fluids flow perpendicular to each other. So i need to consider two ...
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25 views

Question about Coupling of Measures

Let $\pi$ be a coupling of probability measures $\mu,\nu.$ For measurable sets $A,B$ we have $$ \pi(A\times B) \leq \mu(A) $$ and $$ \pi(A^c \times B^c) \leq \mu(A^c), $$ Therefore we have $$ \pi(A\...
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1answer
60 views

Find a modified coupling $((X_n,\tilde Y_n))_{n∈ℕ_0}$ with the same coupling time $τ$ and $\tilde Y_n=X_n$ for $n≥τ$ in the coupling lemma

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(E,\mathcal E)$ be a measurable space $(X_n)_{n\in\mathbb N_0}$ and $(Y_n)_{n\in\mathbb N_0}$ be independent $(E,\mathcal E)$-valued ...
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29 views

Diagonalization and numerical simulation of a 2x2 System of PDEs

I'm new here and looking for help - so, hello! I tried to find something related to this topic but couldn't find anything that fits my needs. I try to make it as easy, short and clear as possible. ...
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13 views

Reference request for general coupling between two probability distributions with densities

Consider two probability measures $\mu$ and $\nu$ on $\mathbb{R}^n$ with respective densities $\rho_\mu$ and $\rho_\nu$. Let $1 > \varepsilon \geq 0$ and $\eta$ another Borel probability measure ...
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1answer
136 views

Successful couplings and total variation convergence to equilibrium for time-homogeneous Markov processes

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $I=\mathbb N_0$ or $I=[0,\infty)$ $(E,\mathcal E)$ be a measurable space $\mu$ and $\nu$ be probability measures on $(E,\mathcal E)$ $...
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1answer
35 views

Give a coupling under which a random variable dominates another random variable

Let $U,V$ be random variables on $\mathbb{N}_0$ with pmf's \begin{equation} f_U(x) = \frac{1}{2} \mathbb{1}_{\{0,1\}}(x), f_V(x) = \frac{1}{3} \mathbb{1}_{\{0,1,2\}}(x) \end{equation} Give a ...
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1answer
101 views

How does Markov Chain coupling work?

I'm trying to understand the coupling argument used in proof of Theorem 5.2 in Finite Markov Chains and Algorithmic Applications by Häggström. My problem is actually more general than this particular ...
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1answer
30 views

Realize a coupling in the target space via a measure on the source space

Consider two product measurable spaces $\left(X \times Y,\mathcal{X} \otimes \mathcal{Y}\right)$, $\left(X' \times Y',\mathcal{X'} \otimes \mathcal{Y'}\right)$ with the usual product sigma-algebra, ...
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1answer
44 views

Coupling via push-forward from a source space

Consider a measurable function $g$ mapping a probability space $\left(\Omega,\mathcal{F},\mu\right)$ to a product measurable space $\left(T,\mathcal{T}\right)$ with cartesian product $T = X \times Y$ ...
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34 views

Maximal coupling of random variables

Let $X,Y$ be random variables with $$P(X=0)=P(X=1)=P(X=-1)=1/3$$ and $$P(Y=0)=P(Y=1)=P(Y=-1)=P(Y=2)=P(Y=-2)=1/5$$ What is a maximal coupling of $X$ and $Y$? Attempt: I know that for a maximal ...
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49 views

Coupling two random variables

Let $X$ and $Y$ be random variables with density $f_X(x)=1_{[0,1]}(x)$ and $f_Y(x)=\frac{1}{2}1_{[0,2]}(x)$. How do we couple these such that $X\leq Y$? Attempt: To couple these we must find new ...
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1answer
370 views

Proof for total variation distance for product measure using coupling

If $\mu_1$ and $\nu_1$ are probability distributions on the finite state space $\Omega_1$, $\mu_2$ and $\nu_2$ are probability distributions on the finite state space $\Omega_2$, $\mu_1 \times \mu_2 (...
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1answer
174 views

Trying to Understand Coupling Arguments Through a Simple Markov Chain Example

Let $X=(X_t)_{t=0}^\infty$ be an irreducible and aperiodic Markov chain on a finite state space $\Omega$. We know that $X$ admits a unique stationary state $\pi$. For each $x\in \Omega$, let $p_x^{(t)}...
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64 views

Showing that the balls respect to Wasserstein metric with equal radius in $\mathcal{P}_{p}(\Xi)$ and $\mathcal{P}(\Xi)$ respectively are equal.

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 $W_{p}(\mu,\nu)$ ...
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1answer
53 views

How to uncouple and reduce/solve a system of 2nd order PDEs

I have the following system of 2nd order PDEs in cylindrical coordinates, $\frac{1}{r} \frac{\partial }{\partial \theta} \left( \frac{h}{3} a - \frac{\ell^2}{2} \nabla^2 a \right) + \frac{\partial }{...
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1answer
101 views

Clue for solving problem about Coupling of Random Variables.

Just have been trying to approach this problem from Resnick's book on probability but have got no clue so far. The problem is like this: We are giving two random variables X, Y on the same space $(\...
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1answer
33 views

Markov kernels between distributions

Let $(P_i,Q_i)$ be couples of distributions, $i = 1,...n$. Suppose they share the same Markov kernel K, that is, $\forall i\in\{1,...n\}$ : $$ Q_i(dy) = \int K(x,dy)P_i(dx) $$ Now let $(X_i,Y_i)$ be ...
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1answer
96 views

Moving from one Coupling to another

Let $X,Y$ be two discrete random variables. Two joint mass distributions (couplings) with marginals $X$ and $Y$ and with entries $p_{i,j}=\mathbb{P}_1(X=i,Y=j)$ and $p_{i,j}'=\mathbb{P}_2({X=i,Y=j})$ ...
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1answer
54 views

Combining Couplings of Random Variables

Given $n$, let $(A_i){i\le n}, (B_i)_{i\le n}$ be sequences of distinct random variables, where $(B_i)_i$ is an independent process. For each $i$, I have a coupling of $A_i,B_i$ with some desired ...
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1answer
153 views

Question on Wasserstein Metric

Consider two random variables $X,Y$. Given some metric $d(X,Y)$, the Wasserstein distance, with respect to $d$, is $$d_W(X,Y)=\inf_{\text{couplings}}\mathbb{E}(d(X,Y))$$ where the infimum is over all ...
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230 views

proof check: bound on total variation distance of probability measures

Let $\mu_1,\mu_2$ be two probability measures on a discrete set $X$. Define the total variation distance between $\mu_1,\mu_2$ in terms of coupling as $$d(\mu_1,\mu_2) = \inf_{\gamma \in \Gamma(\mu_1,...
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1answer
205 views

What is the difference between the coupling and the product of two measures?. How is the law of Total probability for couplings?

Let $(\Xi,\mathcal{E})$ be a measurable space and $\xi$ and $\xi'$ random variables with distributions $\mu$ and $\vartheta$ respectively in this space. We say that the measure $\Pi$ in $\Xi^{2}$ is ...
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0answers
91 views

Couple nonlinear system of equations with linear system of equations

I have a linear system of equations with a variable dependent on a nonlinear system of equations: The linear system, solving for $\mathbf{x}_{n+1}$, is: $$ \mathbf{A}(\boldsymbol\mu_{n+1})\mathbf{x}...
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3 vectors function multipole expansion

Normally a function depends on 2 vectors can be expanded in term of spherical harmonics as $v(|\vec{r_1}-\vec{r_2}|)=\sum_L v_L(r_1,r_2) \sum_M Y^M_L(\vec{r_1})Y^{M*}_L(\vec{r_2})$. So for a ...
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Suppose $p>q$, $X\sim \text{Bernoulli}(p)$, $Y\sim \text{Bernoulli}(q)$. Couple $X$ and $Y$ to maximise $P(X=Y)$.

My professor's solution to this is as follows: "Create a 2 x 2 matrix with the first row (corresponding to $X=0$) summing to $P(X=0)=1-p$, the second row summing to $P(X=1)=p$, the first column ($Y=0$)...
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1answer
97 views

Coupling of “Max of two uniform random variable” & “root of an uniform one”

Consider $X,Y,Z \in [0,1]$ be three independent uniform random variables. It's easy to show that $A=max\{X,Y\}$ and $B=\sqrt{Z}$ have same distribution. But I want a direct proof that completely ...
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1answer
348 views

The Wasserstein distance on $\mathbb{R}$

I was reading "On choosing and bounding probability metrics" and in it there was a remark, that I couldn't quite understand. Take the Wasserstein distance, defined by $W(\mu,\nu):=\inf\limits_{\pi\in\...
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Marginals for measures on product of (Polish) spaces.

Suppose $S$ to be a Polish space and $P(S \times S)$ the set of probability measures on the space $S \times S$. If $\nu$, $\mu$ are in $P(S)$, we can construct the probability measure $\gamma$ on $S \...
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1answer
70 views

Weak convergence of couplings of probability measures

I have the following problem: I am given a Polish space $(\mathcal{X},d)$ and a weakly convergent sequence of probability measures $ (\mu_k)_{k\in\mathbb{N}} $ with limit $\tilde{\mu}$. Given another ...
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52 views

Given a graph bipartit G = (S, T ; E)

Given a graph G = (S, T ; E). For every vertices's $v = st, \forall v \in E, s \in S, t \in T$. Prove the fact that there is a coupling that uses all the elements of S. I see the fact that $ |S| \...
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213 views

Coupling showing that $\operatorname{Bin}(n,\frac{1}{n+1})$ stochastically dominates $\operatorname{Bin}(n-1,\frac{1}{n})$

The classical inequality $$ \left(1-\frac{1}{n}\right)^{n-1} > \frac{1}{e} $$ has a probabilistic generalization: the binomial distribution $\operatorname{Bin}(n-1,\frac{1}{n})$ is stochastically ...
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1answer
140 views

Formal Definition of a Coupled System

I'm working through a paper on interconnected systems, and the main result relies on the assumption that the systems are weakly coupled. Intuitively, I understand what weak coupling means, but I am ...
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37 views

Coupling of a size biased reordered sequence

Given a finite sequence of positive real numbers $\mathbf w = (w_1,w_2,\ldots,w_n)$, the size biased reordering of $\mathbf w$ is a random vector ${\mathbf w}_o = (w_{r(1)},w_{r(2)},\ldots,w_{r(n)})$ ...
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107 views

Coupling Boundary Condition of one PDE with source term of another PDE

We have a system of equations, wherein the BC of one PDE is coupled with the source term of another PDE. We have a regular 2D unit grid in x and y. There are two PDEs to be solved The first PDE (...
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1answer
683 views

Solve a nonlinear system of coupled differential equations

I have this system of differential equations which describes the motion of a missile launcher model with 5 degrees of freedom: (1)$$(m_w +m_v +m_p)\ddot{y}_w - (m_v + m_p)h_v\ddot{\vartheta}_w \sin(\...
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1answer
50 views

Group extension that doesn't realize a coupling

Let $E$ be an extension of $N$ by $G$: $$N \hookrightarrow E \twoheadrightarrow G$$ If $N$ is abelian, then $E$ uniquely defines an action of $G$ on $N$. More generally, it defines a unique class $\...
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1answer
613 views

Decoupling coupled differential equations with time dependent coefficients

Consider the following system of coupled differential equation. $$\left[ \begin{array}{c} \frac{dc_1}{dt} \\ \frac{dc_2}{dt} \end{array} \right] = \begin{bmatrix} -B & -V(t) \\ -V(t) & B \end{...
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1answer
101 views

Good introductory book coupling methods

I am very interested in coupling methods, can you recommend me a good introductory books on this subject? Thanks
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584 views

Creating a new random variable with the same distribution

Let $X,Y$ be two random variables taking real values. Suppose that $P(Y\leq t) \leq P(X\leq t)$ for any $t\in{\mathbb R}$. Is there always a random variable $Y^{*}$ with the same distribution as $Y$, ...