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I have a random walk (RW) (=Markov process) on Z^d. Then the empirical
distributions of this RW satisfy a large deviation principle (LDP) on
the space of probability measures on the state space of the RW, i.e.
Z^d. If I now want to apply Varadhan's lemma I get into troubles
because the rate function is just a weak one (or as others say: not a
good one = non-compact level sets). Thus I employ a compactification
trick by considering the RW on the torus [-N,N)^d \cap Z^d with
periodic boundary conditions. That is how it is done in the book by den
Hollander (Large deviations) on page 87 and the following pages (search
for
"den Hollander we kill the SRW"
in books.google.de to get the document). But why on earth is the inf
appearing in the variational expressions which is given by Varadhan's
lemma taken over all prob. measures on the torus which attribute zero
mass to the boundary of the torus and not over all defective (mass <=1)
probability measures on the torus attributing zero mass to the boundary
(cf. ninth display on page 88)?
Best regards
pkg
link for the book
http://books.google.de/books?vid=ISBN0821819895&id=UnCK7qs1oQcC&pg=PA87&lpg=PA88&printsec=8&dq=den+Hollander++we+kill+the+SRW&sig=ezDu9xtyuzk3YSJQ93QGZ3GuGsc
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