2015/11/16 by Marcus, Michael B., Rosen, Jay
#60G15 #60G17 #60J55 #FOS: Mathematics #Primary 60K99 #Probability (math.PR)
paper · doi:10.48550/arxiv.1511.05172
An \al-permanental process \X t,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|- \al is the Laplace transform of (X_t1,…,X_tn), where K( t1,…,tn) denotes the matrix \K(ti, tj)\i,j=1n and S is the diagonal matrix with entries s1,…,sn . (X_t1,…,X_tn) is 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 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.