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Conditions for permanental processes to be unbounded

An $\al$-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 $, $ |I+K( t_{1},\ldots,t_{n}) S|^{- \al} $ 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.