# Construction of coupling and maximal coupling (probability theory)

• TaPaKaH
In summary, the conversation discusses a homework problem involving random variables U and V with given probability density functions. The task is to find a coupling of U and V such that U is greater than or equal to V with probability 1, and a maximal coupling which minimizes the difference between the distributions of U and V. The attempt at a solution involves constructing a random variable (u,v) with the desired properties, but the author is having difficulties finding a suitable distribution for u and v.
TaPaKaH

## Homework Statement

Let U, V be random variables on $[0,+\infty)$ with probability density functions $f_U(x)=2e^{-2x}$ and $f_V(x)=e^{-x}$.
1. Give a coupling of U and V under which $\{U\geq V\}$ with probability 1.
2. Give a maximal coupling of U and V.

## Homework Equations

Cumulative distribution functions (probability measures) for U and V are:
$P_U([a,b])=P(u\in[a,b])=e^{-2a}-e^{-2b}$,
$P_V([a,b])=P(v\in[a,b])=e^{-a}-e^{-b}$.

## The Attempt at a Solution

I'm having difficulties constructing coupling required in exercise 1. I tried introducing third variable W and letting U=max{V,W}, U=V+W, U=V*W and U=(V+W)/2 but in none of the four cases I could come up with a 'good' cumulative distribution function for W.

Last edited:
TaPaKaH said:

## Homework Statement

Let U, V be random variables on $[0,+\infty)$ with probability density functions $f_U(x)=2e^{-2x}$ and $f_V(x)=e^{-x}$.
1. Give a coupling of U and V under which $\{U\geq V\}$ with probability 1.
2. Give a maximal coupling of U and V.

## Homework Equations

Cumulative distribution functions (probability measures) for U and V are:
$P_U([a,b])=P(u\in[a,b])=e^{-2a}-e^{-2b}$,
$P_V([a,b])=P(v\in[a,b])=e^{-a}-e^{-b}$.

## The Attempt at a Solution

I'm having difficulties constructing coupling required in exercise 1. I tried introducing third variable W and letting U=max{V,W}, U=V+W, U=V*W and U=(V+W)/2 but in none of the four cases I could come up with a 'good' cumulative distribution function for W.

What do you mean by a "coupling" of U and V? Do you mean a bivariate density function with marginals ##f_U, f_V##, or do you mean a probability space ##\Omega## in which ##U## and ##V## are functions ##U(\omega), V(\omega)##? And: what do you mean by a "maximal" coupling?

Ray Vickson said:
What do you mean by a "coupling" of U and V? Do you mean a bivariate density function with marginals ##f_U, f_V##...
Yes, I need a random variable ##(\hat{u},\hat{v})## on ##[0,+\infty)\times[0,+\infty)## such that ##u\overset{D}{=}\hat{u}## and ##v\overset{D}{=}\hat{v}##.
Easy case is to put ##\hat{u}## and ##\hat{v}## independent with marginals above, second example I thought of is ##\hat{u}=\min\{\hat{v},w\}## with ##\hat{v}## and w being independent 'copies' of v, but this gives the opposite of needed ##\hat{\mathbb{P}}(\{\hat{u}\geq\hat{v}\})=1## (might as well be a typo in the exercise).

Maximal coupling is such ##(\hat{u},\hat{v})## that the total variation norm ##\|\mathbb{P}_U-\mathbb{P}_V\|_{tv}=2\hat{\mathbb{P}}(\{\hat{u} \neq \hat{v}\})## while ##u\overset{D}{=}\hat{u}## and ##v\overset{D}{=}\hat{v}##.
The total variation norm is 1/2, computed by hand.

## 1. What is a coupling in probability theory?

A coupling in probability theory is a mathematical technique used to compare two different probability distributions. It involves constructing two random variables that are linked in some way, such as sharing the same underlying sample space, and then analyzing their behavior in relation to each other. Couplings are often used to prove theorems and establish inequalities between probability distributions.

## 2. How is a coupling different from a maximal coupling?

A maximal coupling is a special type of coupling where the two random variables are constructed in such a way that they are as closely linked as possible. This means that the two random variables have a high probability of being equal to each other. In other words, a maximal coupling minimizes the difference between the two distributions being compared, making it a useful tool for proving convergence and other properties of probability distributions.

## 3. What is the purpose of constructing a coupling?

The main purpose of constructing a coupling is to compare two probability distributions and establish relationships between them. Couplings can be used to prove convergence, establish inequalities, and analyze the behavior of random variables. They are also useful for understanding the similarities and differences between different probability distributions.

## 4. How is the construction of a coupling related to stochastic domination?

Stochastic domination is a concept in probability theory that describes the relationship between two random variables in terms of their probability distributions. In particular, it refers to the situation where one random variable has a higher probability of being larger than the other. The construction of a coupling can be used to prove stochastic domination by constructing a coupling that maximizes the probability of the larger random variable being larger than the smaller one.

## 5. What are some applications of couplings in probability theory?

Couplings have numerous applications in probability theory, including proving convergence theorems, establishing stochastic domination, and analyzing the behavior of random variables. They are also used in fields such as statistical physics, where they can be used to study the behavior of complex systems with many interacting components. Additionally, couplings have practical applications in fields such as finance and engineering, where they can be used to model and analyze risk and uncertainty.

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