2019/08/12 by Marcus, Michael B., Rosen, Jay
#60F20 #60G17 #60J27 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1908.04155
Let U=\Uj,k,j,k∈ \mathbb N\ be the potential of a transient symmetric Borel right process X with state space \mathbb N. For any excessive function f=\f_k,k∈ \mathbb N\ for X , \widetilde U=\\widetilde Uj,k,j,k∈ \mathbb N\, where \widetilde Uj,k= Uj,k +f k, j,k∈ \mathbb N, is the kernel of an α-permanental sequence \widetilde Xα=(\widetilde Xα, 1 ,…) for all α>0. The symmetric potential U is also the covariance of a mean zero Gaussian sequence η=\ηj,j∈ \mathbb N\. Conditions are given on the potentials U and excessive functions f under which, \limsupj→ ∞\frac ηj( 2 ϕj)1/2 =1 a.s. ⇒ \limsupn→ ∞\frac\widetilde Xα, jϕj =1 a.s., for all α>0, and sequences ϕ=\ϕj\ such that fj=o(ϕj). The function ϕ is determined by U. Many examples are given in which U is the potential of symmetric birth and death processes with and without emigration, first and higher order Gaussian autoregressive sequences and Lévy processes on \mathbf Z.