Conditions for permanental processes to be unbounded

• Michael B. Marcus ^[1] ; Jay Rosen ^[1]
  1. [1] City University of New York

     City University of New York

     Estados Unidos

     
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 45, Nº. 4, 2017, págs. 2059-2086
• Idioma: inglés
• DOI: 10.1214/16-AOP1091
• Enlaces
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• Resumen

  • An αα-permanental process {Xt,t∈T}{Xt,t∈T} is a stochastic process determined by a kernel K={K(s,t),s,t∈T}K={K(s,t),s,t∈T}, with the property that for all t1,…,tn∈Tt1,…,tn∈T, |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)}i,j=1n 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,…,tn∈Tt1,…,tn∈T, 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.