An -permanental process {Xt,t∈T} is a stochastic process determined by a kernel K={K(s,t),s,t∈T}
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, with the property that for all
t1,…,tn∈T
,
|I+K( t₁,…,t_n) S|⁻
is the Laplace transform of
(Xt1,…,Xtn)
, where
K(t1,…,tn)
denotes the matrix
{K(ti,tj)}i,j=1n
and
S
is the diagonal matrix with entries
s1,…,sn
.
(Xt1,…,Xtn)
is called a permanental vector. Under the condition that
K
is the potential density of a transient Markov process,
(Xt1,…,Xtn)
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
(Xt1,…,Xtn)
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