Conditions for permanental processes to be unbounded

Abstract

An $\alpha$-permanental process $\{X_{t},t\in T\}$ is a stochastic process determined by a kernel $K=\{K(s,t),s,t\in T\}$, with the property that for all $t_{1},\ldots,t_{n}\in T$, $\vert I+K(t_{1},\ldots,t_{n})S\vert^{-\alpha}$ is the Laplace transform of $(X_{t_{1}},\ldots,X_{t_{n}})$, where $K(t_{1},\ldots,t_{n})$ denotes the matrix $\{K(t_{i},t_{j})\}_{i,j=1}^{n}$ and $S$ is the diagonal matrix with entries $s_{1},\ldots,s_{n}$. $(X_{t_{1}},\ldots,X_{t_{n}})$ is called a permanental vector.

Under the condition that $K$ is the potential density of a transient Markov process, $(X_{t_{1}},\ldots,X_{t_{n}})$ 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 $(X_{t_{1}},\ldots,X_{t_{n}})$ 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 $t_{1},\ldots,t_{n}\in T$, $A(t_{1},\ldots,t_{n}):=(K(t_{1},\ldots,t_{n}))^{-1}$ are $M$-matrices. The results in this paper are obtained by working with these $M$-matrices.