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## coupling methods in probability theory

On the other hand if you are looking for original results proved using coupling arguments then have a look at: http://arxiv.org/abs/math/0404356. Writing out the formula directly is surprisingly messy. Not logged in There are plenty. to prove limit theorems, to derive inequalities, or to COUPLING METHODS IN PROBABILITY in Ellös, Sweden, June 14-19, 1999 The development in the last few decades of the coupling method in pure and applied probability has been remarkable. This algorithm tries to find a good configuration of variables by repeatedly resampling variables. Coupling is a powerful method in probability theory through which random variables can be compared with each other. Part of Springer Nature. Coupling has been applied in a broad variety of contexts, e.g. Its basic idea is to define two or more random elements on the same probability space, in order to draw conclusions about their distributions. What is the stationary distribution for the contact process on the half line? This paper discusses coupling ideas with focus on equivalences for exact coupling, shift-coupling and e-couplings of stochastic processes and the generalizations to random fields and topological transformation groups. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. Claim: $\mathbb P(X_n>k | X_0=j)$ is an increasing function of $j$. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Then $P(X_n > k)$ is an increasing function of $n$. Making statements based on opinion; back them up with references or personal experience. I'll suggest "Coupling, stationarity, and regeneration" by Hermann Thorisson. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. A coupling between two probability measures P1,P2 ∈ M(X) is a pair of X-valued random variables X1,X2 deﬁned on a common probability space and such that X1 P1 and X2 P2. the monographs by Lindvall (1992) and Thorisson (2000). To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Probabilistic Methods in Applied Mathematics, Volume 3 focuses on the influence of the probability theory on the formulation of mathematical models and development of theories in many applied fields. See page 312 on Durrett's book "probability: theory and examples". The present course is intended for master students and PhD students. In percolation, there is a simple proof that $p_c^{site} \geq p_c^{bond}$ on any graph, by coupling an exploration of the cluster of a point. ; Teen mothers who live with their parents are less likely to use marijuana than teen moms in other living arrangements. pp 319-339 | site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. Coupling methods for Markov processes. You perform a Markov chain. Google Scholar [21] G rimmett and S tirzaker (1992). Used primarily for estimates of total variation distances, the method also works well in establishing inequalities and has proven highly successful in the study of Markov and renewal process asymptotics. Cite as. Percolation theory: every edge in the lattice graph $\mathbb{Z}^2$ is connected with probability $p$ or disconnected with parobability $1-p$ (independently). the monographs by Lindvall (1992) and Thorisson (2000). As you get higher, you're more careful so that $p_n$ is a decreasing sequence. 185.32.188.76. rev 2020.11.24.38066, The best answers are voted up and rise to the top, MathOverflow works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us, Nice examples/arguments that illustrating the coupling method in probability theory, http://blameitontheanalyst.wordpress.com/2012/01/24/probabilistic-coupling/, “Question closed” notifications experiment results and graduation, MAINTENANCE WARNING: Possible downtime early morning Dec 2/4/9 UTC (8:30PM…, Regular Conditional Probability given a natural filtration of a stochastic process, Defining measures over frames in place of $\sigma$-algebras, Birkhoff Ergodic Theorem and Ergodic Decomposition Theorem for Continuous-Time Markov Processes, Stationary distribution of last passage percolation. I think arguments with coupling makes one think in a more probabulistic way. I want some examples that can best illustrate the idea/power/funny of the coupling argument in probability. Stationary distribution of Markov Chain with departure. In this paper the method is presented through a series of examples starting with correlation, domination and A coupling between two probability measures P1,P2 ∈ M(X) is a pair of X-valued 1,X2 1 P The method also works well in establishing inequalities, and it has proven highly successful in the study of Markov and renewal process asymptotics. Description. Download preview PDF. Which groups of variables to resample can depend in a complicated way on the previous state of the stochastic system. Applications in regeneration, Markov theory, Palm theory, ergodic theory, exchangeability and self-similarity are indicated and a set of general coupling references provided. For example one can show Liouville's theorem using a coupling argument (it is in Rogers&Williams second volume and here: http://blameitontheanalyst.wordpress.com/2012/01/24/probabilistic-coupling/). The coupling method has long been an important tool in probability theory, see e.g. Proof of the convergence to stationary measure in Markov chain theory, this is now the "classical" way followed by most text books. Math. Imagine a infinite ladder with rungs labelled by the positive integers. Over 10 million scientific documents at your fingertips.