Skip to main content

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.

Filter by
Sorted by
Tagged with
0 votes
0 answers
27 views

Decoupling Linearly Coupled Wave Equations

I'm currently working numerically with wave equations and I was wondering if one can always decouple two wave equations, with potentials, which are linearly coupled. The system I'm talking about is ...
Afraxad's user avatar
2 votes
1 answer
63 views

Coupling of Random Variables Where Probability of Movement is Dependent on Position

Suppose a particle starts at position 0 and at each time period particle $i$ moves one position to the right with probability $p_{i, j}$ or moves to the left with with probability $1-p_{i, j}$, where $...
LRunner10's user avatar
0 votes
0 answers
24 views

Optimization problem with highly coupling objects and inequality constraints.

For a better vision, see this image: We have the optimization problem $$\begin{align} \min_{[a_1,a_2,\ldots,a_N]}&\prod_{i=1}^{N}\left[1-{A\over\prod_{n=1}^{R}\left(\lambda_n+{\sum_{u\ne i}|C_i\...
ShuchenSean's user avatar
4 votes
1 answer
125 views

Theorem about coupling and independence of random variables

I am reading a book of E. Rio and I found there a theorem (without a proof) about coupling. Please see below. Theorem: Let $(\xi_i)_{i \in \mathbb{Z}}$ be a sequence of random variables with values ...
Grigori's user avatar
  • 159
1 vote
2 answers
45 views

Solve 1-D coupled differential equations analytically

I'm currently going through an article where I came across these two 1-D coupled differential equations. $$\frac{dA}{dz} = a_1B(z)e^{-i\beta z} $$ $$\frac{dB}{dz} = a_2A(z)e^{i\beta z} $$ with these ...
Jerry Y's user avatar
  • 25
0 votes
0 answers
43 views

Trivial Optimal Couplings

I've been reviewing some basic optimal transport concepts (ref Peyré and Cuturi). I like the notation of the Kantorovich optimal transport problem. $$ L_C(a,b) = \min_{ P \in U(a,b) } \langle P , C \...
jeffery_the_wind's user avatar
0 votes
1 answer
56 views

Construction of a coupling of a sequence of Bernoulli random variables

Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of Bernoulli distributed random variables defined on a probability space $(\Omega,\mathcal{F},P)$ and $(\mathcal{F}_n)_{n\in\mathbb{N}}$ be a filtration such ...
yannik0103's user avatar
1 vote
0 answers
32 views

Questions about the proof of existence for maximal coupling

I'm trying to understand the proof of existence for the maximal coupling. That is: For any two probability measures $\mathbb{P},\mathbb{P}'$ on a measurable space $(E,\mathcal{E})$ there exists a ...
Almost-surely's user avatar
0 votes
0 answers
24 views

Will taking one more step from a different starting point lead to the same end point?

I am dealing with a total variance which can be represented as: $$d(t)=\|v_{t+1}-v_{t}||_{TV}$$ The two distributions have the following relation: $$v_{t+1}=W_{t}\times v_{t}=W_{t}W_{t-1}...W_{0}\...
Moon Traveler's user avatar
2 votes
2 answers
453 views

Distances in the space of probability measures and couplings of random variables

One-sentence summary: I want an intuitive explanation for why closeness of probability measures (in terms of, eg., TV or Lévy-Prokhorov) implies the existence of good couplings. Set up. Consider a ...
nootnoot's user avatar
0 votes
1 answer
156 views

Proof birth and death process

I need some help understanding the following proof: I don't really see how the first inequality is derived. I guess the authors are conditioning on whether $X_0^0 = X_0^p$ or $X_0^0 \neq X_0^p$, but ...
user675763's user avatar
2 votes
1 answer
78 views

When does the sum of dependent Bernoulli random variables stochastically dominate binomial distribution?

Suppose $B_1, ..., B_k$ are dependent Bernoulli random variables with $$P(B_i = 1) \geq p$$ for each $1 \leq i \leq k$. Let Binom(k,p) be a binomial random variable with parameters $k$ and $p$. Can we ...
baris_esmer's user avatar
1 vote
0 answers
37 views

Stochastic domination proof Ising model

Consider the 1-dimensional Ising model on a torus (or a line of which the endpoints are connected) of size N. Suppose that the configuration evolves according to the heat-bath dynamics, that is, each ...
user675763's user avatar
0 votes
1 answer
55 views

Coupled Resonators with sinusoidal coupling

I have two coupled resonators with complex resonance frequencies of $\beta_1$ and $\beta_2$. The coupling between them is varying sinusoidal with time. The time-evolution of their resonant mode field (...
SiPh's user avatar
  • 31
1 vote
0 answers
32 views

Total variation distance for dependent product measure using coupling

If $B_1, ..., B_k$ are $k$ random variables that are dependent. And $R_1, ..., R_k$ are independent variables, such that the total variational distance satisfies $d(B_i, R_i)\leq \epsilon_i$ for all $...
AspiringMat's user avatar
  • 2,457
1 vote
0 answers
23 views

Does there exist $f$ such that $f(X) \overset{D}{=} Q$?

Motivation: Let $X$ be a continuous real random variable and let $Q$ be a continuous distribution. Does there exist $f$ such that $f(X)\overset{D}{=}Q$? that is, transformation of $X$ has distribution ...
Albert Paradek's user avatar
1 vote
1 answer
32 views

Minorisation and Coupling in probability theory.

In probability theory, when studying the convergence of a stochastic process to equilibrium minorisation conditions can be exploited (see for instance Assumption 2.1. of https://www.sciencedirect.com/...
Monty's user avatar
  • 2,320
0 votes
1 answer
324 views

Decomposition of probability measures with a bounded total variation distance

Fix a probability space $(\Omega, \mathcal{F})$. Let $P$ and $Q$ be two probability measures on $(\Omega, \mathcal{F})$ such that there exist $\varepsilon$ and $\delta$ that for every $A \in \mathcal{...
MMH's user avatar
  • 714
0 votes
0 answers
37 views

What are some good resources to study probabilistic coupling?

I was reading a research paper where the author uses probabilistic coupling. The paper is very badly written, and a lot of details are just directly written without any justification. So, they define ...
Charles's user avatar
  • 371
0 votes
1 answer
49 views

Convergence in Probability - Hitting the Limit

Let $X_1, \ldots, X_\infty$ be discrete random variables taking values in a finite or countable set $E$. Suppose that for each $x\in E$, $$\mathbf{P}\left(X_n = x\right) \to \mathbf{P}\left(X_\infty = ...
The Substitute's user avatar
1 vote
1 answer
55 views

How to guarantee the existence of continuity sets?

The Skorohod's representation theorem says: Let $X, X_1, X_2,...$ be S-valued random elements, where (S, p) is a separable metric space, such that $X_n \xrightarrow{d} X$. Then there exists a common ...
NXWang's user avatar
  • 167
2 votes
0 answers
111 views

Biggest difference in the order statistics of a uniform random sample?

Let $U_1, \dots, U_n$ denote independent and identically distributed samples on the unit interval $[0, 1]$. Let $U_{(i)}$ denote the order statistics, so that $U_{(1)} \leq U_{(2)} \leq \cdots \leq U_{...
Drew Brady's user avatar
  • 3,774
1 vote
0 answers
39 views

What is the physical and mathematical meaning of nonsymmetric mass, damping and stiffness matrix of a linear fluid-solid interaction modeling problem?

Suppose the problem of the subsonic flow over a simply-supported rectangular plate as bellow. According to the assumptions of incompressible, inviscid and irrotational flow, and using perturbation ...
Mehdi's user avatar
  • 29
2 votes
1 answer
114 views

Summation of squared 3-j symbol

There is a property for the 3-j symbols as $$ \sum_{m_1 m_2 m_3} \begin{pmatrix} j_1 & j_2 & j_3 \\ m_1 & m_2 & m_3 \end{pmatrix}^2 = \Delta (j_1, j_2, j_3), $$ where $ \Delta (j_1, ...
AYBRXQD's user avatar
  • 123
1 vote
1 answer
63 views

Coupling for tail bounds

Let $X_1, X_2$ be two random variables such that $\mathbf{E}X_1=\mathbf{E}X_2=0$ and $$\forall x\geq 0, \quad \mathbf{P}[X_1\geq x]\leq \mathbf{P}[X_2\geq x].$$ Can we construct a coupling $(Y_1,Y_2)$ ...
charlus's user avatar
  • 905
1 vote
1 answer
460 views

Methods for "measuring dependence" in a stochastic process

Given a sequence of dependent random variables $(X_i)_i$, what are some ways to measure the amount of dependence in this process? The only clear example of this that I can refer to would be to couple ...
The Substitute's user avatar
2 votes
0 answers
49 views

Coupling of sequences of uniform random variables

Suppose that $(X_m)_{m=1}^k,(Y_m)_{m=1}^k$ are sequences of independent and uniform on $[0,1]$ random variables. I am trying to find a coupling of the sequences such that: $$\sum_{m=1}^nX_m<1\iff (...
Tair Galili's user avatar
  • 1,025
0 votes
1 answer
132 views

Can we generate the ratio of two unknown probabilities?

Suppose we have two coins $A$ and $B$, where coin $A$ comes heads up with unknown probability $p_A(0<p_A<1)$, and coin $B$ comes heads up with probability $p_B(0<p_B<1)$, and we would like ...
wcysai's user avatar
  • 3
0 votes
0 answers
48 views

Under which conditions two different random variables belong to same distribution

I think I understand the idea of coupling given here from page 18 - 20.I specially looked at the example. They are discussing the cases when given random variables have the same distribution with ...
False Equivalence's user avatar
0 votes
1 answer
190 views

Eigenvalues of decoupled system (diagonalised matrix)

A question I'm looking at has a matrix, M, composed of sub-matrices A, B, C and D, \begin{equation} M = \begin{bmatrix} A \, \,& B \\ \hline C & D \end{bmatrix} \end{...
HWWW's user avatar
  • 33
0 votes
0 answers
65 views

Understanding the concept of "dynamical coupling" between measures in the Villani's book

I am currently studying the geodesics in the space of probability measures over a Polish space (complete and separable metric space). In the "Optimal Transport - Old and new" book of C. ...
Mathemachicken's user avatar
0 votes
0 answers
151 views

Kantorovich duality with distance cost function: any direct proof of the characterization with $\sup$ over Lipschitz functions?

Let $X$ be a Polish space, $d$ be a lower semicontinuous metric on $X$ and $\mu,\nu$ be Radon probabilities on $X$. Denote by $Lip(\cdot)$ the Lipschitz constant of a function and by $\Pi(\mu,\nu)$ ...
LuaLua's user avatar
  • 131
0 votes
1 answer
44 views

Example and intuition on natural couplings

So I am following Barrera, Högele and Pardo's paper, about cutoff thermalization in the Wasserstein distance (you can find it here) and they prove the shift linearity property that goes: For $p\geq 1$...
leplata's user avatar
  • 549
1 vote
1 answer
67 views

Why is $P(X_t\in A_t)\le P(|S_t-\mu_t|\ge n/4)$?

I'm having problems understanding one inequality for one proof. The assumptions given are the following: Let $S_t$ be the $(p-q)$-biased lazy random walk on $\mathbb{Z}$ and $\mu_t=\frac{t(p-q)}{2}$. ...
Alex's user avatar
  • 147
0 votes
1 answer
111 views

The equivalence between two expressions of asymmetric coupling-based distance

Why is $$ C(\mathbb{Q}, \mathbb{P}):=\inf _{\mathbb{M}} \sqrt{\int \sum_{i=1}^{n}\left(\mathbb{M}\left[Y_{i} \neq x_{i} \mid X_{i}=x_{i}\right]\right)^{2} d \mathbb{P}(x)} $$ where the infimum ranges ...
Zhao Zhao's user avatar
2 votes
0 answers
130 views

Construct a (conditional) independent strong approximation

It is well known that for the sum of independent r.v., the strong approximation can be achieved using Yurinskii's coupling: Theorem (Yurinskii, 1978). Let $\xi_{1},\ldots,\xi_{n}$ be independent ...
Q9y5's user avatar
  • 1,404
0 votes
1 answer
232 views

What does nonlocal coupling mean in the context of oscillators?

I am trying to understand some papers on chimera type behaviour but I don't understand what nonlocal coupling means in the context "... in nonlocal coupling oscillator system". This is the ...
ThreeOrangeOneRed's user avatar
6 votes
1 answer
247 views

Sampling from an arbitrary distribution on Polish spaces

Let $U\sim \text{Unif}(0,1)$, and let $\mu \in \mathcal{P}(\mathbb{R})$ be an arbitrary probability measure on $\mathbb{R}$. Then from $\mu$, we can derive an associated CDF $F(x) = \mu((-\infty,x])$. ...
forgottenarrow's user avatar
1 vote
0 answers
34 views

3 dimensional couplings

Suppose we have two probability measures on $R^2$ $\mu_1,\mu_2$. I am looking to construct a coupling $\pi$ on $R^3$ such that the $(x,y)$-marginal is $\mu_1$ and the $(y,z)$ marginal is $\mu_2$. Can ...
user593295's user avatar
0 votes
1 answer
44 views

Coupling defined upon marginals with different mass

Let us say we define a coupling of two measures $\mu,\nu$ (reference : http://websites.math.leidenuniv.nl/probability/lecturenotes/CouplingLectures.pdf). It seems that a coupling can't be defined if $\...
Marine Galantin's user avatar
2 votes
1 answer
353 views

Prove stochastic domination via coupling of a one-point perturbation

Let $S_n$ be the usual Simple Symmetric Random Walk on $\mathbb Z$. That is, $X_n \sim \mbox{Ber}_{\pm 1}(\frac 12)$ are iid with $X_0 = 0$ and $S_n = X_0+...+X_n$. (Where $\mbox{Ber}_{\pm 1}(p)$ for $...
Sarvesh Ravichandran Iyer's user avatar
0 votes
0 answers
44 views

How can one prove that $\int \langle x,y\rangle d\gamma’ = \int \langle x-\tau,y-\tau' \rangle d\gamma$

I'm reading the book Computational Optimal Transport, by Peyré and Cuturi. In the book, in Remark 2.19 the authors make a claim regarding the translation of measures. While trying to prove the remark, ...
Davi Barreira's user avatar
2 votes
1 answer
139 views

Given marginal of $U$ and $V$, how can we know if there is joint such that $\mathbb E[U|V]=V$

Suppose we are given two random variable $U$ and $V$ on the interval $[0,1]$ with respective probability measure $p_U$ and $p_V$. When is there a coupling measure $p_{U,V}$, that respect the ...
P. Quinton's user avatar
  • 6,076
2 votes
1 answer
102 views

Couplings and existence of a dominating measure

Here's a simple but interesting coupling / measure theory question: Say we have measures $P_1, P_2$ and probabilities $Q_1, Q_2$ on a nice measurable space $(\mathcal{X},\mathscr{F})$, with $P_i(A) \...
John O's user avatar
  • 23
5 votes
2 answers
394 views

a coupling probability problem and random walk game

There are 3 players and one dealer in a casino. The dealer chooses a player randomly($p_1=\frac{1}{3}$). The chosen player tosses a coin($p_2=\frac{1}{2}$). If the coin lands head, the chosen player ...
martian03's user avatar
1 vote
2 answers
201 views

Problems in understanding of markov chains in context of coupling from the past

I have read about the coupling from the past algorithm that is used for perfect sampling from the stationary distribution of a discrete markov chain. My question is not exactly about this algorithm, ...
S. M. Roch's user avatar
4 votes
3 answers
154 views

Given a coupling $\pi(\mu,\nu)$, show that $E_\mu f- E_\nu f= E_\pi [f(X) - f(Y)]$

In the lecture notes by for High-Dimensional Probability by Handel, the following is affirmed: Let $\mu$ and $\nu$ be probability measures, then $$\mathcal C(\mu,\nu) = \{ \text{Law} (X,Y) : X\sim \mu,...
Davi Barreira's user avatar
0 votes
1 answer
63 views

Implications of $\inf_{\left\|x\right\|,\left\|y\right\|\le c}\sup_{\gamma\text{ coupling of }δ_xκ,δ_yκ}\gamma\left(\{\rho<δ\}\right)>0$

Let $E$ be a $\mathbb R$-Banach space, $d$ be a complete separable metric on $E$, $\delta$ denote the Dirac kernel on $(E,\mathcal B(E))$ and $\kappa$ be a Markov kernel on $(E,\mathcal B(E))$. Assume ...
0xbadf00d's user avatar
  • 13.9k
1 vote
1 answer
223 views

How to construct a joint pdf, other than the independent coupling, given two marginal pdf's?

I am trying to find joint densities for which the marginal densities are $f_U(u) = 2\exp(-2u), u\geq 0$ and $f_V (v) = \exp (-v), v \geq 0 $. Of course I can take the joint pdf $f(u,v) = f_U(u) \cdot ...
JanTinbergen1991's user avatar
0 votes
1 answer
127 views

If $\gamma$ is a coupling of $\delta_x$ and $\delta_y$, can we show that $\int f\:{\rm d}\gamma=f(x,y)$?

Let $(E,\mathcal E)$ be a measurable space, $\pi_i$ denote the projection of $E^2$ onto the $i$th coordinate, $\delta_x$ denote the Dirac measure on $(E,\mathcal E)$ at $x$ for $x\in E$ and $\gamma$ ...
0xbadf00d's user avatar
  • 13.9k