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July 2017 Conditions for permanental processes to be unbounded
Michael B. Marcus, Jay Rosen
Ann. Probab. 45(4): 2059-2086 (July 2017). DOI: 10.1214/16-AOP1091

Abstract

An αα-permanental process {Xt,tT}{Xt,tT} is a stochastic process determined by a kernel K={K(s,t),s,tT}K={K(s,t),s,tT}, with the property that for all t1,,tnTt1,,tnT, |I+K(t1,,tn)S|α|I+K(t1,,tn)S|α is the Laplace transform of (Xt1,,Xtn)(Xt1,,Xtn), where K(t1,,tn)K(t1,,tn) denotes the matrix {K(ti,tj)}ni,j=1{K(ti,tj)}ni,j=1 and SS is the diagonal matrix with entries s1,,sns1,,sn. (Xt1,,Xtn)(Xt1,,Xtn) is called a permanental vector.

Under the condition that KK is the potential density of a transient Markov process, (Xt1,,Xtn)(Xt1,,Xtn) is represented as a random mixture of nn-dimensional random variables with components that are independent gamma random variables. This representation leads to a Sudakov-type inequality for the sup-norm of (Xt1,,Xtn)(Xt1,,Xtn) that is used to obtain sufficient conditions for a large class of permanental processes to be unbounded almost surely. These results are used to obtain conditions for permanental processes associated with certain Lévy processes to be unbounded.

Because KK is the potential density of a transient Markov process, for all t1,,tnTt1,,tnT, A(t1,,tn):=(K(t1,,tn))1A(t1,,tn):=(K(t1,,tn))1 are MM-matrices. The results in this paper are obtained by working with these MM-matrices.

Citation

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Michael B. Marcus. Jay Rosen. "Conditions for permanental processes to be unbounded." Ann. Probab. 45 (4) 2059 - 2086, July 2017. https://doi.org/10.1214/16-AOP1091

Information

Received: 1 January 2015; Revised: 1 November 2015; Published: July 2017
First available in Project Euclid: 11 August 2017

zbMATH: 06786076
MathSciNet: MR3693957
Digital Object Identifier: 10.1214/16-AOP1091

Subjects:
Primary: 60G15 , 60G17 , 60J55 , 60K99

Keywords: MM-matrices , Permanental processes

Rights: Copyright © 2017 Institute of Mathematical Statistics

Vol.45 • No. 4 • July 2017
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