Abstract
Permanental processes are a natural extension of the definition of squared Gaussian processes. Each one-dimensional marginal of a permanental process is a squared Gaussian variable, but there is not always a Gaussian structure for the entire process. The interest to better know them is highly motivated by the connection established by Eisenbaum and Kaspi, between the infinitely divisible permanental processes and the local times of Markov processes. Unfortunately the lack of Gaussian structure for general permanental processes makes their behavior hard to handle. We present here an analogue for infinitely divisible permanental vectors, of some well-known inequalities for Gaussian vectors.
Highlights
A real-valued positive vector is a permanental vector if its Laplace transform satisfies for every (α1, α2, ..., αn) in Rn+1 E[exp{− 2 n αiψi}] = |I + Gα|−1/β i=1 (1.1)where I is the n × n-identity matrix, α is the diagonal matrix diag(αi)1≤i≤n, G = (G(i, j))1≤i,j≤n and β is a fixed positive number
Permanental vectors represent a natural extension of squared centered Gaussian vectors
The recent extension of Dynkin isomorphism theorem [5] to non necessarily symmetric Markov processes suggests that the path behavior of local times of Markov processes should be closely related to the path behavior of infinitely divisible permanental processes
Summary
A real-valued positive vector (ψi, 1 ≤ i ≤ n) is a permanental vector if its Laplace transform satisfies for every (α1, α2, ..., αn) in Rn+. Where I is the n × n-identity matrix, α is the diagonal matrix diag(αi)1≤i≤n, G = (G(i, j))1≤i,j≤n and β is a fixed positive number Such a vector (ψi, 1 ≤ i ≤ n) is a permanental vector with kernel (G(i, j), 1 ≤ i, j ≤ n) and index β. The recent extension of Dynkin isomorphism theorem [5] (reminded at the beginning of Section 2) to non necessarily symmetric Markov processes suggests that the path behavior of local times of Markov processes should be closely related to the path behavior of infinitely divisible permanental processes.
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