aa r X i v : . [ m a t h . P R ] N ov On the upper bound in Varadhan’s Lemma
H. M. Jansen , , M. R. H. Mandjes , K. De Turck , S. Wittevrongel July 11, 2018
Abstract
In this paper, we generalize the upper bound in Varadhan’s Lemma. Thestandard formulation of Varadhan’s Lemma contains two important el-ements, namely an upper semicontinuous integrand and a rate functionwith compact sublevel sets. However, motivated by results from queueingtheory, we do not assume that rate functions have compact sublevel sets.Moreover, we drop the assumption that the integrand is upper semicontin-uous and replace it by a weaker condition. We prove that the upper boundin Varadhan’s Lemma still holds under these weaker conditions. Addition-ally, we show that only measurability of the integrand is required whenthe rate function is continuous.
Keywords.
Varadhan’s Lemma ⋆ exponential integrals ⋆ large deviations prin-ciple ⋆ upper bound Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Sci-ence Park 904, 1098 XH Amsterdam, the Netherlands. TELIN, Ghent University, Sint-Pietersnieuwstraat 41, B-9000 Ghent, Belgium.
E-mail . { h.m.jansen|m.r.h.mandjes } @uva.nl , { kdeturck|sw } @telin.ugent.be Exponential integrals often play an important role in the proof of a large de-viations principle (LDP). Varadhan’s Lemma is a powerful generalization ofLaplace’s method for computing exponential integrals. Especially the upperbound in Varadhan’s Lemma turns out to be a very useful tool for provingLDPs. However, Varadhan’s Lemma is stated under somewhat restrictive con-ditions, which rule out many interesting cases. In particular, certain rate func-tions arising in queueing theory do not satisfy the conditions of Varadhan’sLemma. Motivated by this observation, we will generalize the upper bound inVaradhan’s Lemma. 1
Main result
Let X be a topological space and denote its Borel σ -algebra by B . Throughout,we will assume that { µ n } n ∈ N is a sequence of probability measures defined on B . We will say that the sequence { µ n } n ∈ N satisfies an LDP with rate function J if lim sup n →∞ n log µ n ( F ) ≤ − inf x ∈ F J ( x )for any closed set F ⊂ X andlim sup n →∞ n log µ n ( G ) ≥ − inf x ∈ G J ( x )for any open set G ⊂ X , where J : X → [0 , ∞ ] is a lower semicontinuous func-tion. Note that we do not assume that J has compact sublevel sets, i.e., we donot assume that J is a good rate function.An important goal of this paper is to prove the following lemma. Note thatthis is just the upper bound in Varadhan’s Lemma, but without the assumptionthat J is a good rate function. Moreover, the lemma states that a well knowntail condition is both necessary and sufficient for the upper bound to hold.Although this is not very surprising, it is never explicitly stated like this. Lemma 2.1.
Suppose that the sequence of measures { µ n } n ∈ N satisfies an LDPwith rate function J and let φ : X → R be an upper semicontinuous function.Then it holds that lim sup n →∞ n log Z X e nφ ( x ) µ n (d x ) ≤ sup x ∈X [ φ ( x ) − J ( x )] if and only if lim M →∞ lim sup n →∞ n log Z X e nφ ( x ) { φ ( x ) >M } µ n (d x ) ≤ sup x ∈X [ φ ( x ) − J ( x )] . This lemma is an immediate result from the following more general lemma,which is the main result of this paper. Its proof is inspired by the proof ofVaradhan’s Lemma given in [1]. As is customary, we define exp( −∞ ) = 0,log(0) = −∞ and exp( ∞ ) = log( ∞ ) = ∞ . Throughout, we will denote theclosure of a set A by cl A . Lemma 2.2.
Suppose that the sequence of measures { µ n } n ∈ N satisfies an LDPwith rate function J . Let φ : X → [ −∞ , ∞ ] be a Borel measurable function anddefine φ M = φ ∧ M for M ∈ R . Assume that at least one of the followingconditions is true:1. J is continuous;2. the superlevel set φ − ([ w, ∞ ]) is closed for every w ∈ R satisfying theinequality w ≥ lim M →∞ sup x ∈X [ φ M ( x ) − J ( x )] . hen it holds that lim sup n →∞ n log Z X e nφ ( x ) µ n (d x ) ≤ lim M →∞ sup x ∈X [ φ M ( x ) − J ( x )] if and only if lim M →∞ lim sup n →∞ n log Z X e nφ ( x ) { φ ( x ) >M } µ n (d x ) ≤ lim M →∞ sup x ∈X [ φ M ( x ) − J ( x )] . Proof.
For notational convenience, define β M = sup x ∈X [ φ M ( x ) − J ( x )] for M ∈ R . Note that β M is well defined for each M ∈ R and that β M is nondecreasingin M . Hence, lim M →∞ β M is well defined.The statement is obviously true if lim M →∞ β M = ∞ , so in the remainder ofthis proof we will assume that lim M →∞ β M < ∞ .Fix any b ∈ R such that b > lim M →∞ β M and pick any w ∈ ( −∞ , b ] suchthat w ≥ lim M →∞ β M . For k ∈ N , define the measurable sets L ki = φ − b (cid:0)(cid:2) c ki − , c ki (cid:3)(cid:1) for i = 1 , . . . , k , where c ki = w − ik ( w − b )for i = 0 , . . . , k . Observe that c ki − c ki − = − w − bk and that L = φ − b ([ w, b ]) = ∪ ki =1 L ki for every k ∈ N .Obviously, it holds that lim sup n →∞ n log Z X e nφ b ( x ) µ n (d x ) =lim sup n →∞ n log (cid:18)Z L e nφ b ( x ) µ n (d x ) + Z L ∁ e nφ b ( x ) µ n (d x ) (cid:19) =max (cid:26) lim sup n →∞ n log Z L e nφ b ( x ) µ n (d x ) , lim sup n →∞ n log Z L ∁ e nφ b ( x ) µ n (d x ) (cid:27) and lim sup n →∞ n log Z L ∁ e nφ b ( x ) µ n (d x ) ≤ w. Now fix k ∈ N . We havelim sup n →∞ n log Z L e nφ b ( x ) µ n (d x ) ≤ lim sup n →∞ n log k X i =1 Z L ki e nφ b ( x ) µ n (d x )= max i =1 ,...,k lim sup n →∞ n log Z L ki e nφ b ( x ) µ n (d x ) ≤ max i =1 ,...,k lim sup n →∞ n log Z L ki e nc ki µ n (d x ) . i = 1 , . . . , k it holds thatlim sup n →∞ n log Z L ki e nc ki µ n (d x ) = c ki + lim sup n →∞ n log µ n (cid:0) L ki (cid:1) ≤ c ki + lim sup n →∞ n log µ n (cid:0) cl L ki (cid:1) ≤ c ki − inf x ∈ cl L ki J ( x )= sup x ∈ cl L ki (cid:2) c ki − J ( x ) (cid:3) . Suppose that the first condition is true. Thensup x ∈ cl L ki (cid:2) c ki − J ( x ) (cid:3) = sup x ∈ L ki (cid:2) c ki − J ( x ) (cid:3) , by continuity of J . But c ki − = c ki + w − bk , so c ki ≤ φ b ( x ) − w − bk for all x ∈ L ki .Hence, we get sup x ∈ L ki (cid:2) c ki − J ( x ) (cid:3) ≤ sup x ∈ L ki (cid:20) φ b ( x ) − w − Mk − J ( x ) (cid:21) ≤ sup x ∈X [ φ b ( x ) − J ( x )] − w − bk . Suppose that the second condition is true. Then we have cl L ki ⊂ cl φ − b (cid:0)(cid:2) c ki − , b (cid:3)(cid:1) and φ − b (cid:0)(cid:2) c ki − , b (cid:3)(cid:1) = φ − (cid:0)(cid:2) c ki − , ∞ (cid:3)(cid:1) is closed by assumption, so cl L ki ⊂ φ − b (cid:0)(cid:2) c ki − , b (cid:3)(cid:1) .We get c ki ≤ φ b ( x ) − w − bk for all x ∈ cl L ki andsup x ∈ cl L ki (cid:2) c ki − J ( x ) (cid:3) ≤ sup x ∈ cl L ki (cid:20) φ b ( x ) − w − Mk − J ( x ) (cid:21) ≤ sup x ∈X [ φ b ( x ) − J ( x )] − w − bk . Note that it does not matter which of the two conditions is true: we get thesame inequality in both cases. Consequently, for every k ∈ N it holds thatlim sup n →∞ n log Z L e nφ b ( x ) µ n (d x ) ≤ max i =1 ,...,k lim sup n →∞ n log Z L ki e nc ki µ n (d x ) ≤ sup x ∈X [ φ b ( x ) − J ( x )] − w − bk ≤ w − w − bk , so lim sup n →∞ n log Z L e nφ b ( x ) µ n (d x ) ≤ w. n →∞ n log Z X e nφ b ( x ) µ n (d x ) =max (cid:26) lim sup n →∞ n log Z L e nφ b ( x ) µ n (d x ) , lim sup n →∞ n log Z L ∁ e nφ b ( x ) µ n (d x ) (cid:27) ≤ w. Because this holds for all w ∈ ( −∞ , b ] with w ≥ lim M →∞ β M , it follows imme-diately thatlim sup n →∞ n log Z X e nφ b ( x ) µ n (d x ) ≤ lim M →∞ sup x ∈X [ φ M ( x ) − J ( x )]for all b ∈ R .Now observe that for each b ∈ R we have lim sup n →∞ n log Z X e nφ ( x ) µ n (d x ) ≤ lim sup n →∞ n log (cid:20)Z X e nφ b ( x ) µ n (d x ) + Z X e nφ ( x ) { φ ( x ) >b } µ n (d x ) (cid:21) =max (cid:26) lim sup n →∞ n log Z X e nφ b ( x ) µ n (d x ) , lim sup n →∞ n Z X e nφ ( x ) { φ ( x ) >b } µ n (d x ) (cid:27) ≤ max (cid:26) lim M →∞ sup x ∈X [ φ M ( x ) − J ( x )] , lim sup n →∞ n Z X e nφ ( x ) { φ ( x ) >b } µ n (d x ) (cid:27) , which implies the statement of the lemma. References [1] Frank den Hollander.