## A Different Construction Of Gaussian Fields From Markov Chains: Dirichlet Covariances (2002)

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Venue: | Ann. I. H. Poincaré |

Citations: | 3 - 1 self |

### BibTeX

@ARTICLE{Diaconis02adifferent,

author = {Persi Diaconis and Steven and Steven N. Evans},

title = {A Different Construction Of Gaussian Fields From Markov Chains: Dirichlet Covariances},

journal = {Ann. I. H. Poincaré},

year = {2002},

volume = {38},

pages = {863--878}

}

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### Abstract

. We study a class of Gaussian random fields with negative correlations. These fields are easy to simulate. They are defined in a natural way from a Markov chain that has the index space of the Gaussian field as its state space. In parallel with Dynkin's investigation of Gaussian fields having covariance given by the Green's function of a Markov process, we develop connections between the occupation times of the Markov chain and the prediction properties of the Gaussian field. Our interest in such fields was initiated by their appearance in random matrix theory. 1.

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Citation Context ...s of Un have eigenvalues on the unit circle T in the complex plane. The study of the distribution of these eigenvalues under Haar measure makes up a chapter of random matrix theory (see, for example, =-=[Meh91]-=-). Let H 1 2 2 denote the space of functions f 2 L 2 (T) such that kfk 2 1 2 := X j2Z j f j j 2 jjj ! 1; and define an inner product on H 1 2 2 by hf; gi 1 2 := X j2Zsf jsg j jjj: GAUSSIAN FIELDS FROM... |

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Citation Context ... 1 2 = 1 16 2 ZZ (f(OE) \Gamma f(`)) (g(OE) \Gamma g(`)) sin 2 i OE\Gamma` 2 j d` dOE (see Equations (1.2.18) and (1.2.21) of [FOT94]). In independent work, Johansson [Joh97] and Diaconis-Shahshahani =-=[DS94]-=- proved the following result which was extended in [DE01]. Theorem 1.3. Choose M 2 Un from Haar measure. For f in H 1 2 2 let W f (M ) = P n j=1 f(e i` j ), where ` 1 ; : : : ; ` n are the eigenvalues... |

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Citation Context ...ion space and it coincides with the Besov space B 1 2 2;2 , the Sobolev--Lebesgue space F 1 2 2;2 and the Lipschitz space 1 2 2;2 (see Equations (18) and (19) in x3.5.4 and Equation (13) in x3.5.1 of =-=[ST87]-=-). However, for our purposes the interesting observation is that the space H 1 2 2 equipped with the inner product h\Delta; \Deltai 1 2 is nothing other than the Dirichlet space and Dirichlet form of ... |

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Citation Context ...sin 2 i OE\Gamma` 2 j d` dOE (see Equations (1.2.18) and (1.2.21) of [FOT94]). In independent work, Johansson [Joh97] and Diaconis-Shahshahani [DS94] proved the following result which was extended in =-=[DE01]-=-. Theorem 1.3. Choose M 2 Un from Haar measure. For f in H 1 2 2 let W f (M ) = P n j=1 f(e i` j ), where ` 1 ; : : : ; ` n are the eigenvalues of M . Then, as n tends to infinity, for any finite coll... |

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Citation Context ...s of elds other than Gaussian ones have also been studied: a recent paper with an extensive bibliography is [Bol01]. Here are two lesser known alternative appearances of this connection. Bhattacharya =-=[Bha82]-=- establishes general results that specialise in our nite setting to the following. Suppose that the chain X is ergodic. For f R in the range of Q, T 0 f(Xs) ds converges in distribution as T !1to a Ga... |

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Citation Context ...d` dOE ! 1; and, moreover, hf; gi 1 2 = 1 16 2 ZZ (f(OE) \Gamma f(`)) (g(OE) \Gamma g(`)) sin 2 i OE\Gamma` 2 j d` dOE (see Equations (1.2.18) and (1.2.21) of [FOT94]). In independent work, Johansson =-=[Joh97]-=- and Diaconis-Shahshahani [DS94] proved the following result which was extended in [DE01]. Theorem 1.3. Choose M 2 Un from Haar measure. For f in H 1 2 2 let W f (M ) = P n j=1 f(e i` j ), where ` 1 ;... |

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Citation Context ...aussian field with covariance given by the associated Green's function was discovered independently by several people. In physics, there is work of Symanzik [Sym69] followed by work of Brydges et al. =-=[BFS82]-=-. In statistics, Ylvisaker [Ylv87] gives references to Hammersley's [Ham67] work on harnesses as followed up by Williams [Wil73], Kingman [Kin85, Kin86] and Dozzi [Doz81, Doz89]. Variants of Dynkin's ... |

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Citation Context ...eatures (densities and correlations). A review of the literature on this conjecture and its connections to de Finetti's theorem -- an early joint interest of Bretagnolle and Dacunha-Castelle -- is in =-=[DF81]-=-. 1.1. Continuous time and Dirichlet forms. As we noted above, it will be more convenient to work with continuous time Markov chains. To this end, let X now be a continuous time Markov chain on the fi... |

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Citation Context ...killing occurs), then P y Q(x; y) = 0 for all x 2X and = 0. A standards4 PERSI DIACONIS AND STEVEN N. EVANS reference for Dirichlet forms is [FOT94], but we nd the original paper by Beurling and Deny =-=[BD58]-=- useful and readable. It follows from (1.1) that (1.2) (x; y) :=, (x)Q(x; y) is a positive semi-de nite self{adjoint matrix. Hence is the covariance of a mean zero Gaussian eld Z = fZxgx2X indexed by ... |

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Citation Context ...ling occurs), then P y Q(x; y) = 0 for all x 2 X ands= 0. A standard 4 PERSI DIACONIS AND STEVEN N. EVANS reference for Dirichlet forms is [FOT94], but we find the original paper by Beurling and Deny =-=[BD58]-=- useful and readable. It follows from (1.1) that (1.2) \Sigma(x; y) := \Gamma(x)Q(x; y) is a positive semi-definite self--adjoint matrix. Hence \Sigma is the covariance of a mean zero Gaussian field Z... |

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Citation Context ...s, there is work of Symanzik [Sym69] followed by work of Brydges et al. [BFS82]. In statistics, Ylvisaker [Ylv87] gives references to Hammersley's [Ham67] work on harnesses as followed up by Williams =-=[Wil73]-=-, Kingman [Kin85, Kin86] and Dozzi [Doz81, Doz89]. Variants of Dynkin's isomorphism have been established by Eisenbaum [Eis95], as well as by Marcus and Rosen in the papers cited above. Markov chain r... |

11 |
Euclidean quantum field theory
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Citation Context ...lationship between a Markov chain and the Gaussian field with covariance given by the associated Green's function was discovered independently by several people. In physics, there is work of Symanzik =-=[Sym69]-=- followed by work of Brydges et al. [BFS82]. In statistics, Ylvisaker [Ylv87] gives references to Hammersley's [Ham67] work on harnesses as followed up by Williams [Wil73], Kingman [Kin85, Kin86] and ... |

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Citation Context ...95], as well as by Marcus and Rosen in the papers cited above. Markov chain representations of fields other than Gaussian ones have also been studied: a recent paper with an extensive bibliography is =-=[Bol01]-=-. Here are two lesser known alternative appearances of this connection. Bhattacharya [Bha82] establishes general results that specialise in our finite setting to the following. Suppose that the chain ... |

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Citation Context ...nces to Hammersley's [Ham67] work on harnesses as followed up by Williams [Wil73], Kingman [Kin85, Kin86] and Dozzi [Doz81, Doz89]. Variants of Dynkin's isomorphism have been established by Eisenbaum =-=[Eis95]-=-, as well as by Marcus and Rosen in the papers cited above. Markov chain representations of fields other than Gaussian ones have also been studied: a recent paper with an extensive bibliography is [Bo... |

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Citation Context ...h the "edges" (x; y) such that Q(x; y) ? 0. Generating realisations of Gaussian fields on grids or graphs with general covariances can be a complex enterprise. A useful review of the literat=-=ure is in [GS89]-=-. In Section 3 we show how the problem of using the observations fZ y g y2B to predict Z x for x = 2 B ae X is intimately related to the properties of the occupation times of the Markov chain X. In Se... |

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Citation Context ...ght enough so that refined knowledge of Gaussian fields (e.g. continuity of sample paths) can be carried over to develop fine properties of Markov processes (e.g., continuity of local time). Sheppard =-=[She85]-=- gave a proof of the Ray--Knight theorem on the Markovianity of local times of one-dimensional GAUSSIAN FIELDS FROM MARKOV CHAINS 7 diffusions that used Dynkin's result and the obvious Markovianity of... |

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Bhattacharya.On the functional central limit theoremand the law of the iterated logarithm for Markov processes
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Citation Context ...of fields other than Gaussian ones have also been studied: a recent paper with an extensive bibliography is [Bol01]. Here are two lesser known alternative appearances of this connection. Bhattacharya =-=[Bha82]-=- establishes general results that specialise in our finite setting to the following. Suppose that the chain X is ergodic. For f in the range of Q, 1 p T R T 0 f(X s ) ds converges in distribution as T... |

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1 |
E-mail address: diaconis@math.Stanford.EDU
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Citation Context ...anity of the associated Gaussian field. We do not see such depth for our construction, but find the parallels tantalising. Dynkin's construction has been used in statistical applications by Ylvisaker =-=[Ylv87]-=-. He used the Gaussian fields as Bayesian priors for prediction and design problems. Dynkin's fields only have positive correlations while the fields we construct have negative correlations; using ind... |

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1 |
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Citation Context ...relationship between a Markov chain and the Gaussian eld with covariance given by the associated Green's function was discovered independently by several people. In physics, there is work of Symanzik =-=[Sym69]-=- followed by work of Brydges et al. [BFS82]. In statistics, Ylvisaker [Ylv87] gives references to Hammersley's [Ham67] work on harnesses as followed up by Williams [Wil73], Kingman [Kin85, Kin86] and ... |