Conditions for permanental processes to be unbounded

An .-permanental process {Xt,t.T} is a stochastic process determined by a kernel K={K(s,t),s,t.T}, with the property that for all t1,.,tn.T, |I+K(t1,.,tn)S|.. is the Laplace transform of (Xt1,.,Xtn), where K(t1,.,tn) denotes the matrix {K(ti,tj)}ni,j=1 and S is the diagonal matrix with entries s1,.,sn. (Xt1,.,Xtn) is called a permanental vector. Under the condition that K is the potential density of a transient Markov process, (Xt1,.,Xtn) 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 (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 K is the potential density of a transient Markov process, for all t1,.,tn.T, A(t1,.,tn):=(K(t1,.,tn)).1 are M-matrices. The results in this paper are obtained by working with these M-matrices.