Problems of Information Transmission, Vol. 37, No. 2, 2001, pp. 120–139. Translated from Problemy Peredachi Informatsii, No. 2, 2001, pp. 40–61.
Original Russian Text Copyright
2001 by de La Fortelle.
Large Deviation Principle for Markov Chains
in Continuous Time
A. de La Fortelle
Received April 25, 2000; in ﬁnal form, October 18, 2000
be a homogeneous nonexplosive Markov process with generator R deﬁned
on a denumerable state space E (not necessarily ergodic). We introduce the empirical generator
and prove the Ruelle–Lanford property, which implies the weak LDP. In a fairly broad
setting, we show how to perform almost all classical operations (e.g., contraction) on the weak
LDP under suitable assumptions, whence Sanov’s theorem follows.
Large deviation theory has been widely studied and the range of objects for which the LDP holds
is nowadays very large (see [1, 2] and references therein). There are also several approaches [3–10]
by which one can demonstrate the principle—such as subadditivity, convex transforms, or change
The goal of this study is to propose the LDP for a new object, the empirical generator,and
show that it is a natural object from which lots of interesting corollaries can be derived. The
framework is constituted by continuous-time Markov processes on a countable state space, which
includes many applications, e.g., queueing networks.
In discrete time, this would be a “level-2.5” LDP, i.e., Sanov’s theorem for the pair empirical
measure (see  for the deﬁnition of the level). But in discrete time, the process (X
a Markov chain and therefore the level-2.5 LDP can be derived from the level-2 LDP, the classical
This is no longer true in continuous time since there is no such a thing like (X
the continuous-time entropy
has been recently introduced in  and a representation formula
of Sanov’s rate function has been proved using H
(see ) but there is no LDP. Our results prove
is really a rate function and the representation formula is just a contraction of the LDP.
Moreover, the validity is extended to a countable, instead of ﬁnite, state space.
A recent thorough search showed that the LDP
was in fact proved in  before the two works
previously cited [12, 13]. Surprisingly enough, neither this very interesting paper nor its author is
mentioned in the literature devoted to large deviations. But the idea of an empirical generator,
as well as the information-theoretic interpretation of the rate function, are already there. This
can be explained by a wall between diﬀerent mathematical ﬁelds: Hornik [14–16] was interested
in statistics (hence the idea of measuring the generator), namely, optimal tests, while Kesidis
and Walrand  and Baldi and Piccioni  (see also ) were rather focused on simulation
(importance sampling). Our point of view is probabilistic since further results exhibit a simple
relationship in queueing networks between the tails of stationary distributions and entropy function.
This variety of separate ﬁelds shows in itself the broad interest of such results.
We shall denote by the index c (respectively, d) the entropy function related to continuous (respectively,
The state space is assumed to be ﬁnite. The proof relies on the Perron–Frobenius theorem, so that our
approach diﬀers signiﬁcantly.
2001 MAIK “Nauka/Interperiodica”