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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 $\alpha$-permanental process $\{X_{t},t\in T\}$ is a stochastic process determined by a kernel $K=\{K(s,t),s,t\in T\}$, with the property that for all $t_{1},\ldots,t_{n}\in T$, $\vert I+K(t_{1},\ldots,t_{n})S\vert^{-\alpha}$ is the Laplace transform of $(X_{t_{1}},\ldots,X_{t_{n}})$, where $K(t_{1},\ldots,t_{n})$ denotes the matrix $\{K(t_{i},t_{j})\}_{i,j=1}^{n}$ and $S$ is the diagonal matrix with entries $s_{1},\ldots,s_{n}$. $(X_{t_{1}},\ldots,X_{t_{n}})$ is called a permanental vector.

Under the condition that $K$ is the potential density of a transient Markov process, $(X_{t_{1}},\ldots,X_{t_{n}})$ is represented as a random mixture of $n$-dimensional random variables with components that are independent gamma random variables. This representation leads to a Sudakov-type inequality for the sup-norm of $(X_{t_{1}},\ldots,X_{t_{n}})$ 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 $K$ is the potential density of a transient Markov process, for all $t_{1},\ldots,t_{n}\in T$, $A(t_{1},\ldots,t_{n}):=(K(t_{1},\ldots,t_{n}))^{-1}$ are $M$-matrices. The results in this paper are obtained by working with these $M$-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

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.45 • No. 4 • July 2017
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