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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.

In probability theory, coupling is a proof technique that allows one to compare two unrelated random variables (distributions) X and Y by creating a random vector W whose marginal distributions correspond to X and Y. The choice of W is generally not unique, and the whole idea of coupling is about making such a choice so that X and Y can be related in a particularly desirable way.

To define coupling using standard formalism of probability, let X$_1$ and X$_2$ be two random variables defined on probability spaces (Ω$_1,$ F$_1,$ P$_1$) and (Ω$_2,$ F$_2,$ P$_2$). Then a coupling of X$_1$ and X$_2$ is a new probability space (Ω, F, P) over which there are two random variables Y$_1$ and Y$_2$ such that Y$_1$ has the same distribution as X$_1,$ and Y$_2$ has the same distribution as X$_2$.

107 questions
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Show $\alpha_{m+n}\leq \alpha_m\alpha_n$ for a Markov chain on a finite state space

This problem is from Durrett's Probability: Theory and Examples, 5/E, Exercise 5.6.4(or 6.6.3 in earlier editions). For any transition matrix $p$, define $$\alpha_n=\sup_{i,j}\frac{1}{2}\sum_k|p^n(i,k)-p^n(j,k)|.$$ The 1/2 is there because for any…
bellcircle
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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 dominated by the Poisson distribution…
Yuval Filmus
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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 $F(x) = \mu((-\infty,x])$. We consider the following inverse of…
forgottenarrow
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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 $Y$, such that $P(X \leq Y^{*})=1$ ?
Ewan Delanoy
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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 will get 3 dollar, the dealer and the other two…
martian03
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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 the Cartesian product space of the supports of…
user25004
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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 $\Omega_2$, $\mu_1 \times \mu_2 (x,y) = \mu_1(x)\mu_2(y)$ and $\nu_1 \times \nu_2…
The Hagen
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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 $p_{i,j}'=\mathbb{P}_2({X=i,Y=j})$ correspond to two matrices $(p_{i,j}),…
The Substitute
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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 in some Polish space $X$. Assume that $(\Omega,…
Grigori
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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 $$\mathcal C(\mu,\nu) = \{ \text{Law} (X,Y) : X\sim \mu, Y\sim \nu \} $$ Therefore, any $\pi \in…
Davi Barreira
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Random walk- minimizing expected distance to the origin

Given $\delta\in [0,1]$ and $n\in \mathbb{N}$, consider a (biased) random walk $S_n(\delta) = \sum_{i = 1}^n X_i$ where $\{X_i:1\le i\le n\}$ are i.i.d. and $X_i = 1$ with probabiltiy $(1+\delta)/2$ and $-1$ otherwise. I am wondering whether the…
ericswatch
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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 couplings of $X,Y$. How does the expected value…
Iced Palmer
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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 $\chi$ on: $$\text{Out}(N) =…
Henrique Augusto Souza
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Good introductory book coupling methods

I am very interested in coupling methods, can you recommend me a good introductory books on this subject? Thanks
tartartuna
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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: $$\sum_{m=1}^nX_m<1\iff (Y_m)_{m=1}^n \text{ is monotone}$$ Well, I am…
Tair Galili
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