SAMPLE PATH LARGE DEVIATIONS IN FINER TOPOLOGIES

SAMPLE PATH LARGE DEVIATIONS IN FINER TOPOLOGIES PETER EICHELSBACHER AND NEIL O'CONNELL Abstract. In this paper we present sucient conditions for sample path large deviation principles to be extended to ner topologies. We consider extensions of the uniform topology by Orlicz functionals and we consider Lipschitz spaces: the former are concerned with cumulative path behaviour while the latter are more sensitive to extremes in local variation. We also consider sample paths indexed by the half line, where the usual projective limit topologies are not strong enough for many applications.

1. Introduction Let Z be a topological space. A good rate function is a lower semi-continuous function I : Z ! lR+ , with compact level sets fz : I (z) g. A sequence of random variables (Zn )n2N , taking values in Z , is said to satisfy the large deviation principle (LDP) with rate function I if for all closed B Z , I (z); lim sup n1 log P(Zn 2 B ) ? zinf 2B n!1 and for all open G Z , 1 log P(Z 2 G) ? inf I (z): lim inf n n!1 n z2B Let (Xk )k2N be a sequence of random variables with values in lRd . For each 0 t 1 set [nt] X 1 Sn (t) = n Xk : k=1

Denote by L1 [0; 1] the space of (equivalence classes modulo equality a:e: of) bounded measurable functions on [0; 1], equipped with the uniform topology, by C [0; 1] the subspace of continuous functions, and by A[0; 1] the subspace of absolutely continuous functions on [0; 1] with (0) = 0. Denote by Ld1 [0; 1] the space of bounded measurable functions on [0; 1] with values in R d , equipped with the uniform topology. 1991 Mathematics Subject Classi cation. 60F10 (primary). Key words and phrases. Large deviations, partial sums, Orlicz-space, Lipschitz space. 1

2

P. EICHELSBACHER AND N. O'CONNELL

It is a classical result, originally due to Varadhan [13] and Mogulskii [9], that, if the (Xk )k2N are i.i.d. and () := log E exp(h; X1i) < 1 (1.1) for all 2 lRd , then the sequence fSn (); n 2 N g satis es the LDP in Ld1 [0; 1] with good rate function given by ( R 1 _ d (1.2) I () = 0 ()ds 2 A [0; 1]; 1 otherwise, where is the convex dual of : (x) = sup [hx; i ? ()]; 2Rd

and where _ denotes the derivative of , which exists almost everywhere for 2 Ad [0; 1]. The assumption that the moment generating function is nite has since been relaxed by Pinelis [11]; more recently, Dembo and Zajic [3] have relaxed the assumption of independence, replacing it with a variety of mixing and tightness hypotheses. We nd it convenient to consider, instead of Sn , its polygonal approximation: [ nt ] S~n (t) = Sn (t) + t ? n X[nt]+1: (1.3) Note that S~n () is continuous, and carries the same information as Sn (). The LDP for S~n () can typically be shown to hold under milder hypotheses, and in the Polish space C d [0; 1] (equipped with the uniform topology). In this paper we present sucient conditions for such an LDP to be extended to ner topologies. In Section 3 we consider extensions of the uniform topology by Orlicz functionals and in Section 4 we consider Lipschitz spaces; the former are concerned with cumulative path behaviour while the latter are more sensitive to extremes in local variation. We also consider sample paths indexed by the half line, where the usual projective limit topologies are not strong enough for many applications, particularly in queueing theory. This extends work of O'Connell [10] and work of Dobrushin and Pechersky [5]. 2. Some facts about Orlicz spaces What follows is mostly a summary of the relevant details from [1, Chapter 8]. A function C : lR+ ! lR+ is called an N -function if it is convex and satis es lim C (x)=x = 0; lim C (x)=x = 1: x!1 x!0+ Examples of N -functions are C (x) = xp ; (1 < p < 1); C (x) = ex ? x ? 1;

SAMPLE PATH LARGE DEVIATIONS

or

3

C (x) = (1 + x) log(1 + x) ? x;

Ca (x) = C (x ? a)1xa for any N -function C . We will write C for the convex dual of C . The Orlicz class KC is the set of all (equivalence classes modulo equality a:e: of) measurable functions f : [0; 1] ! lR satisfying Z

1

0

C (jf (s)j)ds < 1;

the Orlicz space LC is de ned to be the linear hull of KC , and is a Banach space when equipped with the Luxemburg norm

kf kC = inf k > 0 :

1

Z

0

C (jf (s)j=k) ds 1 :

We have the following generalisation of Holder's inequality: Z

1

0

f (s)g(s)ds

2kf kC kgkC :

(2.1)

An N -function C is said to satisfy a 2 -condition near in nity if there exists a positive constant k > 0 such that for all x suciently large, C (2x) kC (x): This is a useful condition, because LC = KC if, and only if, C satis es a 2 condition near in nity [1, Lemma 8.8]; furthermore, LC is re exive, with dual LC , if, and only if, both C and C satisfy a 2 -condition near in nity [1, Theorem 8.19]. Note that if LC is re exive, then it is weakly complete and the unit ball is weakly compact. We will now record a lemma that will be useful for interpreting the results of the next section. A proof is given in Section 6.

Lemma 2.2. Suppose that C satis es a 2 -condition near in nity. A sequence

fn is weakly convergent to a function f in LC if, and only if, (i) kfn kC is bounded, Rand R (ii) for each t 2 [0; 1], 0t fn (s)ds ! 0t f (s)ds.

Note that all of the above remarks apply to Orlicz spaces of functions on any nite interval [0; t], which we denote by LC [0; t]. 3. The LDP in a topology extended by Orlicz functionals Consider the following Conditions: (H1) For each 2 lRd , the limit 1 log E exp(nh; S (1)i) (3.1) () = nlim n !1 n

4


exists as an extended real number. The sequence S~n () satis es the LDP in C d [0; 1] with good rate function given by

I () =

( R1

_ 0 ()ds

1

2 Ad [0; 1]; otherwise,

where is the convex dual of . For each j = 1; : : : ; d and 2 lR, set 1 j j () = nlim !1 n log E exp(n Sn (1)): Note that the existence of j follows from the existence of , and can be obtained by evaluating at multiples of the j th standard basis vector. (H2) (i) For each j = 1; : : : ; d, there exists an N -function Bj with lim sup Bj((jxx)j) < 1; jxj!1

j

and both Bj and its convex dual satisfy a 2 -condition near in nity. (ii) For some > 0, !

n X 1 lim sup n log E exp Bj (jXkj j) < 1; n!1 k=1 for each j = 1; : : : ; d.

(3.2)

Remark 3.3. The hypothesis (H1) is ful lled, if the sequence (Xk )k2N is i.i.d., see for example [4, Section 5.1]. But (H1) is also ful lled, if the assumptions (A-1) and (A2) in [3] are ful lled: by Theorem 2 in [3], the partial sums processes fSn (); n 2 N g satis es the LDP under (A-1) and (A-2) in Dd [0; 1], the space of cadlag functions on [0; 1] with values in R d , endowed with the uniform topology. By [3, Lemma 5], fSn (); n 2 N g and fS~n (); n 2 N g are exponentially equivalent in Dd [0; 1], and thus the LDP for fS~n (); n 2 N g in C d [0; 1] follows by [4, Lemma 4.1.5]. Moreover (A-1) implies (3.1) (see [3, Remark, page 201]). Thus some independent but not identically sequences (see [3, Corollary 1]), some Markov chains (see [3, Theorem 3]) and some stationary sequences with mixing conditions (see [3, Theorem 4 and 5]) come into consideration. Remark 3.4. The hypothesis (H2) is guaranteed in many situations. For example, if the (Xk )k2N are bounded, it holds. If the (Xk )k2N are i.i.d., then (i) implies (ii) (this follows from [4, Lemma 5.1.14]). The hypothesis (i) is certainly guaranteed if the eective domain of is nite. The hypothesis (ii), although it is essentially a mixing condition, appears to have no direct connection with the mixing conditions introduced in [3] and in [2].


5

Let (Bj ) be the N -functions of (H2), and write B for the lRd -valued function with coordinates (Bj )dj=1 . Recall that if is absolutely continuous, then _ exists almost everywhere. De ne a space XB = f 2 Ad [0; 1] : _ j 2 LBj ; j = 1; : : : ; dg: We will consider a variety of topologies on XB : the uniform topology, which we Q will denote by u ; the topology inherited from the product space j LBj , which we will denote by B ; the corresponding weak topology, Bw ; and the smallest topology containing both u and Bw , which we will denote by B . The topology B is the smallest extension of the uniform topology for which the functions Z 1 7! _ j (s)g(s)ds 0

are continuous for all g 2 LBj , and for each j = 1; : : : ; d. By Lemma 2.2, a sequence n converges in this topology if, and only if, it converges uniformly and the sequences k_ jn kBj are bounded. Theorem 3.5. If (H1) and (H2) are satis ed then fS~n (); n 2 N g satis es the LDP in (XB ; B ), with good rate function given by

I () =

Z

1

0

(_ )ds:

Proof. It follows from (H1) and [4, Lemma 4.1.5] that fS~n (); n 2 N g satis es the LDP in the space XB when equipped with the relative uniform topology: the eective domain of the rate function I (), denoted by DI , is a subspace of Ad [0; 1]. To get the LDP on the subspace XB we have to check that DI XB . This follows from (H2) and the fact that LB = KB . To strengthen this to the space (XB ; B ) we appeal to the inverse contraction principle [4, Theorem 4.2.4, Corollary 4.2.6], by which it suces to prove that the sequence fS~n (); n 2 N g is exponentially tight in this space (XB ; B ). It is convenient to consider the natural imbedding of the weak topology Bw in the space C d [0; 1]. This is complete by the 2-condition. Goodness of the rate function in (H1) implies exponential tightness in the space C d [0; 1], when equipped with the uniform topology, since this is a Polish space (see, for example, [4, Exercise 4.1.10]). Thus, by Lemma 6.1, we need only check that fS~n (); n 2 N g is exponentially tight in (XBd ; Bw ). For each > 0, set K = f 2 XB : max k_j kBj g: j

By the hypothesis (H2), the spaces LBj are re exive and so K is compact in (XBd ; Bw ). For each > 0, set

Z

C = 2 Ad [0; 1] : max j 0

1

Bj (j_ j j)ds :

6


Lemma 3.6. For suciently large, C K . Proof of Lemma 3.6. For suciently large we have, by convexity of the Bj , that

max j

Z

1

0

Bj (j_ j j)ds ) max j Z

Z

1

0 1

Bj (j_ j j=)ds

B (j_ j j=)ds 1 ) max j 0 j ) max k_ j kBj : j The exponential tightness of S~n () in (XB ; Bw ), and hence the statement of the theorem, is thus established by the following lemma.

Lemma 3.7. If (H2) is satis ed, then

1 log P?S~ () 2 C c = ?1: lim lim sup n !1 n n!1

Proof of Lemma 3.7. By Chebyshev's inequality we have, for ; > 0, ! n X 1 B (jX j j) > P(S~ () 2 C c ) = P max n

j

?

n k=1 j

k

exp ?n max E exp j

n X k=1

!

Bj (jXkj j) :

The result is obtained by choosing > 0 as in (H2), considering the normalised logarithmic limit as n ! 1, and then letting ! 1. Thus the Theorem is proved. To get the LDP for fSn (); n 2 N g in (XB ; u) we have to assume the following exponential tail condition in addition to (H1): (H3) For each > 0 +m

kX 1

(3.8) Xi < 1: log E exp an ( ) = sup m k;m2N;k2[0;n m] i=k+1 This is a special case of Assumption (A-2) in [3].

Remark 3.9. As already been remarked in [3, Remark on page 201], the assumption (H3) becomes a0( ) < 1 for every xed < 1, when (Xi)i2N is a stationary sequence. For an i.i.d. sequence (Xi)i2N , this assumption holds as soon as kX k has a nite logarithmic moment generating function, which is the standard condition for deriving the LDP of fSn (1); n 2 N g.

Corollary 3.10. If (H1) and (H3) are satis ed then fSn (); n 2 N g satis es the LDP in (XB ; u), with good rate function given as in Theorem 3.5.


7

Proof. It follows from (H1) that fS~n (); n 2 N g satis es the LDP in XB , equipped with the relative uniform topology. With Lemma 4.1.5(a) in [4] we obtain the LDP in Dd [0; 1], the space of all maps continuous from the right and having left limits, equipped with the uniform topology. Observing that supt2[0;1] kSn (t) ? S~n (t)k = max1kn kXk k=n, it follows that for all > 0 and every > 0 ? P sup kSn (t) ? S~n (t)k > n exp(? n) max E exp( kXk + 1k) : 0kn

t2[0;1]

In view of (H3), the exponential equivalence of fS~n (); n 2 N g and fSn (); n 2 N g is obtained in the uniform topology by considering the logarithmic limit as n ! 1 and then letting ! 1. Remark 3.11. There is no hope to get a LDP for fSn (); n 2 N g in (XB ; B ), even when we change (H2) by the stronger assumption that (3.2) holds true for every > 0. The picewise constant paths of Sn () come not into play. To see why, we only like to give some sketchy arguments: Notice that one can proof that if 1 and 2 are both complete metric topologies on a space Y , and the sequences fZn ; n 2 N g and fZ~n ; n 2 N g are exponentially equivalent with respect to both topologies (see de nition [4, De nition 4.2.10]), then they are exponentially equivalent with respect to 1 _ 2 (the smallest topology containing 1 and 2). Thus one have only to check that fS~n (); n 2 N g and fSn (); n 2 N g are exponentially equivalent in (XBd ; Bw ). One obtains for every g 2 LBj : Z

P

1 0

X Z i+1 Z 1 n

1 n _ _ j j Sn (s) g(s) ds? S~n (s) g(s) ds

> = P

n Xi i g(s) ds

:

0

i=1

n

But this probability is not exponentially small for each > 0. (Consider d = 1, B (t) = tp =p with 1 < p < 1 and g(t) = n). Remark 3.12. In [6], the LDP for partial sums U -processes was considered in a topology extended by Orlicz functionals.

4. The LDP in Lipschitz spaces For 0 we denote by Lip [0; 1] the space of -Holder continuous functions on [0; 1], equipped with the norm (4.1) kk = sup j(jtt)??sj(s)j : s6=t Set Rj (x) = (x) ^ (?x). We will begin by recording some hypotheses. Note that (H5) follows automatically from (H1) if the (Xk )k2N are stationary. (H1a) (H1) holds with lower semi-continuous and dierentiable at the origin. (H4) For each j , there exists 0 j 1 such that lim inf 0 0 and for each pair 1 ; 2 2 lR, lim sup n1 log P(kS~nj ()k j r) n!1 lim sup sup n1 log P(jSnj (t) ? Snj (s)j > r(t ? s) j ) n!1 0 0, [ 1 lim sup n log P fS~n [0; t] 2= K(t)g ?: n!1 t>1

(5.4)

For ; t > 0, de ne c() := exp(exp()) and (

d (t) =

2 t c() p ? 1 ( log log t) t > c()

and consider the sets

D :=

d \ j =1

2Y:

j (t) (t)

d (t); for all t; [0; t] 2 K (t) for all t > 1 :

The exponential tightness of fS~n (); n 2 N g in (Y ; k k ) will be established by the following two lemmas.

Lemma 5.5. For each > 0, D is compact in (Y ; k k ). Proof. Let n be a sequence in D . By Tychono's theorem, the set \t>1 K(t) is compact in Y when equipped with the topology of uniform convergence on compact intervals, so there exists a subsequence n(k) such that n(k) converges to some 2 \t>1 K (t) in this topology. It follows that, for each T > 0, and for each j , j n(k) (t) lim sup k!1 tT (t)

j ? ((tt)) = 0:

Note that this implies, for each t and j , j (t) (t) d (t); and so 2 D . Now for each > 0 (suciently small), we have for k suciently large, j (t) sup 2 n (k(t)) tc(1= ) j (t) + sup 2 n (k(t)) t>c(1= )

kn(k) ? k

j (t)

? (t)

j ? ((tt)) + 2d(c(1=2 )) = 3:

(5.6)

The set D is therefore sequentially compact, and hence compact, in (Y ; kk ).

Lemma 5.7. Under the assumptions of Theorem 5.2 we get 1 log P?S~ 2= D = ?1: lim lim sup n !1 n n!1


13

Proof. We have for some > 0 and for each j : P

[

tc()

fjS~nj (t)j > c() (t)g

P

n[ )]?1 ?1 [c([

n

fjS~nj (k + i=n)j > (k)c()g

i=0 k=0 [c(X )]?1

C ()e?n (k)c()

k=0 nc()e?n (3)c() :

(5.8)

Here we have used Chebyshev's inequality. It follows that [ 1 j lim sup n log P fjS~n (t)j > (t)c()g ? (3)c(): n!1 tc()

We also have, for each j and some > 0, P

[

t>c()

fjS~nj (t)j > (t)d (t)g P

n[ ?1

n

1 [

(5.9)

fjS~nj (k + i=n)j > (k)d(k)g

i=0 k=[c()] 1 X k=[c()]

C ()e? n (k)d (k)

nC ()De?n

pc()?1=2

:

(5.10)

Here we have used Chebyshev's inequality, and the inequality X

kk0

p p e? k De? k0 ?1=2:

It follows that [ p 1 j ~ fjSn (t)j > (t)d (t)g ? c() ? 1=2: lim sup n log P n!1 t>c()

(5.11)

The statement can now be obtained from (5.4), (5.9) and (5.11), via the principle of the largest term. This concludes the proof of the Theorem 5.2. We will now conclude this section with the observation that the LDP holds true with respect to an even ner topology. Consider the space \ YB = f 2 Y : _ j [0; t] 2 LBj [0; t] for all t > 0g; j

where Bj are the N -functions of hypothesis (H2)(i). Denote by Bw (t) the product of Q the weak topologies on j LBj [0; t], and by Bw (1) the projective limit, as t ! 1, of the Bw (t). The Rrestriction of Bw (1) to YB is the weakest topology for which Q t _ the mappings 7! 0 g(s)(s)ds are continuous, for all t > 0 and g 2 j LBj [0; t].

14


Now write !B for the smallest topology on YB containing the topology of the norm k k and the restriction of Bw (1). Theorem 5.12. If in the i.i.d. case (H2)(i) is satis ed, then fS~n (); n 2 N g satis es the LDP in (YB ; !B ) with good rate function given by (5.1). Proof. It suces to prove that fS~n (); n 2 N g is exponentially tight in (YB ; Bw (1)). This follows from Lemma 6.1, as in the proof of Theorem 3.5: Bw (1) can be thought of as a projective limit of weak topologies, each of which is complete by (H2), and so the natural embedding of Bw (1) on Y is also complete; by the previous theorem, we have exponential tightness in Y , which is Polish. The sets \ K = 2 YB : sup (1t) k_ j [0; t]kBj t

j

are compact in (YB ; Bw (1)), by Tychono's theorem. It follows (c.f. Lemma 3.6) that, for suciently large, the sets Z t d \ 1 2 YB : sup (t) Bj (j_ j (s)j)ds C = t 0 j =1 are precompact. Now, by (H2), we have for some > 0 and for each j : [ X [nt] Bj ( P

t k=1

jXkj j) > n (t)

n[ +i=n)] )]?1 [n(kX ?1 [[ Bj ( P

n

i=0 k=0 [X ]?1

k=1

jXkj j) > (k) n

(5.13)

C ()e?n (k)

k=0 ne?n (3) :

Analogously to the calculations in (5.13) and (5.10) we obtain [nt] [ X Bj ( P

t k=1

jXkj j) > n (t)

p nC ()D exp(?n ? 1=2)

(5.14)

for some constants C () and D. For suciently large, the result follows as in the proof of Lemma 5.7. 6. Abstract large deviations result, convergence in LC Lemma 6.1. If 1 and 2 are both complete metric topologies on a space Z , and the sequence Zn is exponentially tight with respect to both topologies, then it is exponentially tight with respect to 1 _ 2 (the smallest topology containing 1 and 2).


15

Proof. Let d1 and d2 be a pair of complete metrics that respectively generate the topologies 1 and 2. By hypothesis, for every > 0, there exist i -compact sets Ki for which lim sup 1 log P (Zn 2= Ki ) ?:

n It follows from the principle of the largest term that lim sup n1 log P (Zn 2= K1 \ K2 ) ?: n!1 Recall that a set in a complete metric topology is precompact if, and only if, it is totally bounded. The sets K1 \ K2 are thus totally bounded in both metrics, and hence in the metric d1 _ d2 , which generates the topology 1 _ 2 . The statement follows. n!1

Proof of Lemma 2.2. Suppose fn converges weakly to f in LC . In other words, for every g 2 LC , Z

1

0

fn (s)g(s)ds !

Z

1

0

f (s)g(s)ds:

(6.2)

Then (i) follows from general principles (see, for example, [7, Theorem 4.10.7]) and (ii) follows from (6.2). Now suppose that (i) and (ii) hold. To deduce that 1

Z

0

fn (s)g(s)ds !

1

Z

0

f (s)g(s)ds;

for every g 2 LC , we need the following lemma.

Lemma 6.3. If A is an N -function satisfying a 2-condition near in nity, then any h 2 LA can be written as an almost everywhere limit of step functions hn with khn ? hkA ! 0. Proof. Without loss of generality we can assume that h is non-negative. (For general h simply split it into its positive and negative parts.) Then h can be written as a monotone increasing sequence of simple functions (see, for example, [7, Theorem 2.2.5]), which therefore converge in the mean and [1], since A satis es a 2 -condition near in nity, in LA. Furthermore, each simple function can be written as an almost everywhere limit of step functions, which also converge in the mean, by bounded convergence, and hence in LA .

Now (ii) implies that

Z

fn g ?

Z

fg

! 0;

for all step functions g. For general g 2 LC , write g as a limit of step functions gm, as in the above lemma, so that kg ? gm kC ! 0. Now we have, applying (2.1)

16


for each m,

Z lim sup n

=

fn g ?

Z lim sup n

Z

fg Z

Z

Z

Z

fn g ? fn gm + fn gm ? fgm + fgm ? Z C +

limnsup 2kfn kC kg ? gm kC + 2kf kC kg ? gm k 2( + kf kC )kg ? gmkC

Z

fg Z

fn gm ?

fgm

(6.4) where = supn kfn kC , and since this holds for all m we can let m tend to in nity for the result. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

References R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. W. Bryc and A. Dembo, Large deviations and strong mixing, Ann. Inst. H. Poincare. Probab. Statist. 32 (1996), no. 4, 549{569. A. Dembo and T. Zajic, Large deviations: From emprical mean and measure to partial sums process, Stochastic Process. Appl. 57 (1995), 191{224. A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, Jones and Bartlett, Boston, MA, 1993. R. L. Dobrushin and E. A. Pechersky, Large deviations for random processes with independent increments on in nite intervals, preprint, 1995. P. Eichelsbacher, Partial sums U -processes large deviations in stronger topologies, Stat. and Prob. Letters (1998), (to appear). A. Friedman, Foundations of Modern Analysis, Holt, Rinehart and Winston, New York, 1982. J. A. Johnson, Banach spaces of lipschitz functions and vector valued lipschitz functional, Trans. Amer. Math. Soc. 149 (1970), 147{169. A. A. Mogulskii, Large deviations for trajectories of multidimensional random walks, Theory Probab. Appl. 21 (1976), 300{315. N. O'Connell, A large deviation principle with queueing applications, BRIMS technical report 97-05, 1997. I. F. Pinelis, A problem on large deviations in the space of trajectories, Th. Probab. Appl. 26 (1981), 69{84. R. T. Rockafeller, Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970. S. R. S. Varadhan, Asymptotic probabilities and dierential equations, Comm. Pure Appl. Math. 19 (1966), 261{286. (Peter Eichelsbacher) Fakultat fur Mathematik, Universitat Bielefeld, Universitats-

strae 1, D-33501 Bielefeld, Germany

E-mail address, Peter Eichelsbacher:

[email protected]

(Neil O'Connell) BRIMS, Hewlett-Packard Labs, Bristol, Filton Road, Stoke Gifford,

Bristol BS12 QZ, UK

E-mail address, Neil O'Connell:

[email protected]