CONDITIONS FOR PERMANENTAL PROCESSES TO BE UNBOUNDED

• 成果类型: Article
• 署名作者: Marcus, Michael B.; Rosen, Jay
• 署名单位: City University of New York (CUNY) System; City College of New York (CUNY); City University of New York (CUNY) System; College of Staten Island (CUNY)
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/16-AOP1091
• 发表日期: 2017
• 页码: 2059-2086
• 关键词: markov-processes; local-times; continuity

摘要

An alpha-permanental process {X-t, t is an element of T} is a stochastic process determined by a kernel K = {K (s, t), s, t is an element of T}, with the property that for all t(1),..., t(n) is an element of T, vertical bar I + K(t(1),..., t(n)) S vertical bar(-alpha) is the Laplace transform of (X-t1,...,X-tn), where K (t(1),...,t(n)) denotes the matrix {K(t(i), t(j))}(i,j)(n) = 1 and S is the diagonal matrix with entries s(1),...,s(n). (X-t1,..., X-tn) s called a permanental vector. Under the condition that K is the potential density of a transient Markov process, (X-t1,..., X-tn) 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-t1,....,X-tn) 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 Levy processes to be unbounded. Because K is the potential density of a transient Markov process, for all t(1),...,t(n) is an element of T, A(t(1),...,t(n)) := (K(t(1),..., t(n)))(-1) are M-matrices. The results in this paper are obtained by working with these M-matrices.