1. Introduction Let fx(t) t 1g be a real-valued mean-zero Gaussian process, and let \kk" be a semi-norm in the space of real functions on [ 1]. In the

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1 Small ball estimates for Brownian motion under a weighted -norm by hilippe BERTHET (1) and Zhan SHI () Universite de Rennes I & Universite aris VI Summary. Let fw (t) t 1g be a real-valued Wiener process, starting from. For a large class of functions f which may vanish at, we obtain the exact asymptotics, as " goes to, of log ( <t1 jw (t)=f(t)j < "). Keywords. Small ball problem, Brownian motion Mathematics Subject Classication. 6J65. (1) I.M.R., Universite de Rennes I, Campus de Beaulieu, F{354 Rennes Cedex, France. berthet@maths.univ-rennes1.fr () Laboratoire de robabilites, Universite aris VI, 4 lace Jussieu, F{755 aris Cedex 5, France. zhan@proba.jussieu.fr 1

2 1. Introduction Let fx(t) t 1g be a real-valued mean-zero Gaussian process, and let \kk" be a semi-norm in the space of real functions on [ 1]. In the literature, asymptotic behaviours of (11) (1) log kxk > log kxk < " are studied due to various motivations. for! 1 for "! Despite their apparent resemblance, they are really very dierent types of estimates. roblems related to (1.1) are usually called \large deviations", which are studied by a large number of mathematicians and found many applications in mathematics and physics. Those related to (1.), often referred to as \small ball estimates", receive also much research interest, yet relatively little is known. As is demonstrated by a recent paper of Kuelbs and Li [14], one should not expect to see a general result for (1.) for an arbitrary Gaussian process X. It would therefore be preferable to study some \well-behaved" processes. Not surprisingly, the example of Brownian motion (as well as related processes, such as Brownian bridge and sheet) attracts the most contribution. Let us cite the recent works [3], [11], [14]{[17], [19]{[], [3]{[6], as well as the survey articles [18] and [1]. In the rest of the paper, fw (t) t 1g denotes a one-dimensional Wiener process, starting from, and we look at the asymptotic behaviour of log (kw k < ") for small ". A typical situation is (13) log kw k < "? c " b "! for nite constants b > and c >. Below are some known examples (i) uniform -norm kk 1 = t1 j(t)j b = and c = =8 (ii) L {norm kk = ( R 1 (t) ) 1= b = and c = 1=8 (iii) Holder norm kk H = s<t1 j(t)? (s)j=(t? s) (for < < 1=) b = =(1? ), but the value of c is unknown. (See, Chung [6], Cameron and Martin [5], Baldi and Roynette [3], and Kuelbs and Li [15]). We mention that the L p {norm (for 1 p < 1) is treated by Donsker and Varadhan [1] who get b = and an \exotic" value of c when p 6=. More general norms, such as Sobolev

3 or Besov norms, are also studied, and perhaps understandably, results are obtained with less accuracy concerning the values of b and especially c. We intend to study the log-probability in (1.3) for W under the weighted -norm (14) kw k f = <t1 f(t) for a large class of functions f. First, recall the following known results. Theorem A (Mogulskii []). If f is a piece-wise continuous function such that (H1) inf f(t) > t1 then (15) lim " log "! kw k f < " =? 8 f (t) Theorem B ([4]). If f(t) = t for some < 1=, then lim "! " log kw k f < " =? 8(1? ) Observe that condition (H1) is not satised by f(t) = t ( > ), nonetheless the constants agree in the theorems. Our aim is to characterize functions f for which (H1) fails, but for which (1.3) still holds with b =. art of the motivations is that weight functions vanishing at are of particular interest in limit theorems in probability and statistics, cf. the book of Csorg}o and Horvath [9]. Let us give some precisions about what kind of functions f is treated in the paper. In order that (1.4) be well-dened, we clearly have to limit ourselves to those functions f such that (16) inf f(t) > for all < 1 t1 In the literature, a function f satisfying (1.6) is called positive (cf. for example Csorg}o et al. [1]). There is no problem for the denition of kw k f under (H1). However, when this condition fails, the situation is somewhat delicate. Of course, kw k f < 1 implies (17) lim t! f(t) < 1 a.s. 3

4 According to a well-known integral test (cf. Csorg}o and Horvath [9, Corollary 4.1.1]), when a positive function f is nondecreasing in the neighbourhood of, (1.7) holds if and only if (18) exp??c f (t) < 1 for some c > t t p (A typical such function is t log log(1=t), with >, which is not surprising in view of the usual law of the iterated logarithm). By an abuse of notation, from now on, we call f a \weight function" if it satises (1.7) and (1.6). Here is the main result of the paper. Theorem 1.1. If a positive function f satises either (H1) or the following condition (H) f is nondecreasing in a neighbourhood of, then lim "! " log kw k f < " =? 8 f (t) Remark 1.. Condition (H) is of no surprise. If (H1) fails, i.e. if inf <t1 f(t) =, one has to use the test (1.8) to decide whether kw k f is well-dened, and this is where the monotonicity of f is needed. The problem of whether the monotonicity is necessary in this kind of test remains open (cf. Csorg}o et al. [1, p. 43]), to the best of our knowledge. Remark 1.3. An immediate consequence of Theorem 1.1 is that under the -norm k k f, the decay rate (1.3) is with b = if and only if (19) f (t) < 1 Remark 1.4. Since condition (1.8) is implicitly imposed in our study, compared with (1.8), it is easily seen that, from a practical point of view, (1.9) is veried by most \usual" weight functions of W, such as f(t) t (for?1 < < 1=), or f(t) t 1= (log(1=t)) (for > 1=), when t is p in the neighbourhood of. A typical weight function of W which disobeys (1.9) is f(t) t log log(1=t). Theorem 1.1 is proved in Section. Section 3 is devoted to some discussions of the rate of decay in (1.3) when (1.9) fails. In Section 4, we present some applications of Theorem 4

5 1.1 by giving Chung's functional iterated logarithm law for W and small ball estimates in higher dimensions. The situation for the Brownian bridge is also treated.. roof of Theorem 1.1 Notation for any deterministic function or stochastic process, we write, for brevity, (1) () (3) " (s t) = (u) sut # (s t) = inf (u) sut (s t) = j(u)j t > s sut and moreover, (4) (t) = ( t) = j(u)j t > ut Let f(t) t 1g be a standard one-dimensional Brownian bridge, which can be realized as fw (t)? tw (1) t 1g. Recall the well-known small ball probabilities for W and under the uniform -norm (5) (6) log W (1) < "? log 8" (1) < "? 8" "! Here is a basic heuristic explanation to the similarity between (.5) and (.6) when W is constrained to stay in a small ball, W (1) becomes very close to, which indicates that the Wiener process behaves somewhat like a Brownian bridge (note that log (jw (1)j < ") is negligible compared with 1=" ). roof of Theorem 1.1 the upper bound. ick n and let t k = t kn = k=n for k n. According to the notation in (.3) and (.1), kw k f < " = W (t k?1 t k ) < "f " (t k?1 t k ) for all 1 k n n \ A k 5

6 with obvious notation. Write F for the natural completed ltration of W. By conditioning on F tn?1, \ n h? A k = E 1l f\ n?1 A kg??ftn?1 A n i Consider the conditional probability (A n j F tn?1 ). Since t 7! W (t + t n?1 )? W (t n?1 ) is again a Wiener process, independent of F tn?1, by a well-known inequality of Anderson [] for Gaussian shifted balls, A n???ftn?1 xr ttn?tn?1 jw (t) + xj < "f " (t n?1 t n ) = W (t n? t n?1 ) < "f " (t n?1 t n ) = W (1) < "f " (t n?1 t n ) (t n? t n?1 ) 1= the last identity following from the Brownian scaling property. Accordingly, kw k f < " Iterating the same procedure, (7) kw k f < " Hence, in view of (.5), n?1 \ ny lim " log kw k f < "? "! 8 A k W (1) < "f " (t n?1 t n ) (t n? t n?1 ) 1= W (1) < "f " (t k?1 t k ) (t k? t k?1 ) 1= nx t k? t k?1 (f " (t k?1 t k )) This yields the upper bound in Theorem 1.1 by sending n to innity. tu The proof of the lower bound is based on a preliminary estimate and on Khatri's classical inequality. Lemma.1. If f satises (H1), lim inf "! " log t1 f(t) < "? 8 f (t) 6

7 Fact. (Khatri [13]). Let (Y 1 Y n ) be a mean-zero Gaussian vector. For all positive numbers a k (1 k n), \ n ( jy k j < a k ) ny ( jy k j < a k ) By assuming Lemma.1 for the moment, we can prove the lower bound in Theorem 1.1. Observe that Lemma.1 is basically Mogulskii's Theorem A, except that we do not assume piece-wise continuity. To illustrate our approach, we include the proof of the lemma, which is postponed to the end of the section. roof of Theorem 1.1 the lower bound. Let f be a positive function, in the sense of (1.6). Of course, we only have to treat the case R 1 =f (t) < 1 (otherwise the upper bound suces for the conclusion). If (H1) holds, Lemma.1 is nothing else but the lower bound in Theorem 1.1. So the remaining situation to be discussed is f nondecreasing in a neighbourhood of, say ( ). Let < a <. Dene g(t) = f(at)=3 p a for t [ 1]. Then g is nondecreasing over [ 1]. Let us bound below the probability 1 = kw k g < 3" jw (1)j < " where > is a xed constant. For j, write s j =?j and n F j = F?1 = jw (s j+1 )j < g(s j+1 )" n o jw (1)j < " o Observe that (8) where 1 h 1\ j= h 1\ j= W (s j+1 s j ) < 3g(s j+1 )" \ j (s j+1 s j ) < g(s j+1 )" \ 1\ j=?1 1\ j=?1 F j i F j i j (t) = W (t)? W (s j+1 )? t? s j+1 W (s j )? W (s j+1 ) s j? s j+1 7

8 Since f j (s j+1 + (s j? s j+1 )t)= p s j? s j+1 t 1g j is a sequence of independent Brownian bridges, and independent of (F j ) j, the term on the right hand side of (.8) equals h 1Y j= (1) < g(s i j+1)" 1\ p sj? s j+1 j=?1 F j Going back to (.8), and applying Khatri's inequality (cf. Fact.), we obtain 1 h 1Y j= (1) < g(s j+1)" p sj? s j+1 i 1Y j=?1 (F j ) Since F j fw (s j+1 ) < g(s j+1 )"g (for j ) and (F?1 ) "= (for " < " (), where " () is a small constant depending only on ), it follows from the Brownian scaling property that 1Y j=?1 (F j ) " 1Y j= W (1) < g(s j+1)" p sj+1 By (.5){(.6), there exists a nite universal constant > such that W (1) < x exp?? x and for all x >. Hence, for < " < " (), 1 " exp? " 1X " exp? 4 " exp? 4 ta j= " g (t) 1 j+1 g (?j?1 )? " = " g (t) (1) < x which, by the denition of g, means (9) < " jw (a)j < p a " " f(t) exp?36 1X j=? exp? x 1 j+1 g (?j?1 ) for all < a < and all < " < " (). Now write = inf at1 f(t) > and x < < 1. It is noted that kw k f < " = ta ta f(t) f(t) < " at1 f(t) < " jw (a)j < " jw (t)? W (a)j at1 f(t) 8 Z a < " f (t) < (1? )"

9 Since the Wiener process has independent increments, by scaling, the probability on the right hand side is equal to ta f(t) < " jw (a)j < " t1 f((1? a)t + a) < (1? )" p 1? a Applying, respectively, (.9) to = = p a, and Lemma.1 to t 7! f((1? a)t + a), we have lim inf "! " log kw k f < "? 36 Z a Z " f (t)? 1 8(1? ) f (t) Letting a and tend to gives the desired lower bound in Theorem 1.1. The rest of the section is devoted to the proof of Lemma.1. roof of Lemma.1. Let n and t k = k=n (for k n). ick a small number > such that inf <t1 f(t) 3n (which is possible, in light of (H1)). Then where kw k f < " W (t k?1 t k ) < "f # (t k?1 t k ) jw (t k )? W (t k?1 )j < " for all 1 k n k(t k?1 t k ) < (f # (t k?1 t k )? n)" jw (t k )? W (t k?1 )j < " for all 1 k n k (t) = W (t)? W (t k?1 )? t? t k?1 W (t k )? W (t k?1 ) t k? t k?1 Observe that f (t k?1 t k )g 1kn and fw (t k )?W (t k?1 )g 1kn are mutually independent variables, and that for each k, (t k?1 t k ) is distributed as p t k? t k?1 (1). Accordingly, h Y n kw k f < " (1) < (f #(t k?1 t k )? n)" i (t k? t k?1 ) 1= ny jw (1)j < Since lim "! " log (jw (1)j < ") =, by (.6), this implies lim inf "! " log kw k f < "? 8 9 nx a " (t k? t k?1 ) 1= t k? t k?1 (f # (t k?1 t k )? n) tu

10 Lemma.1 is proved by rst sending to and then letting n tend to innity. tu 3. Critical case Theorem 1.1 conrms that (1.3) holds for kw k f with b = for most of the weight functions f of W (this means that the decay of the small ball probability is of the same rate for many f's). However, if condition (1.9) fails, we should no longer expect to see (1.3) in general instead of a power of ", additional terms (typically logarithms) appear. If R 1 =f (t) = 1, we call it the \critical" case, since this ruins the rate of decay b = in (1.3). R 1 Theorem 3.1. For any positive function f satisfying (H), such that =f (t) = 1,? lim "! " F (") log kw k f < " 8 where F (") = " f (t) roof. First, we replace (.5) by a rened estimate (cf. Chung [6]) (W (1) < x) exp(? =8x ) for x >. By (.7), for all " ( 1=), > " such that f is nondecreasing on ( ), and n > =", kw k f < " ny [n] Y h exp? (t k? t k?1 ) i 8" (f " (t k?1 t k )) h exp? (t k? t k?1 ) i 8" (f " (t k?1 t k )) = exp [n] log? 8" [n] X Recall that t k = k=n. Since f is nondecreasing on ( ), [n] X Take n = [3="], then t k? t k?1 (f " (t k?1 t k )) Z [n] log 3 " 1=[n] log 8" Z " f (t) = Z "= 1 " t k? t k?1 (f " (t k?1 t k )) f (t) + Z " f (t) 8" Z " 1=[n] 1=[n] f (t) f (t)

11 where in the second inequality, we have used the monotonicity of f together with the fact lim t! + f(t) =. Consequently, Z kw k f < " exp? 8" " f (t) proving Theorem 3.1. tu Remark 3.. The choice of n in the proof of Theorem 3.1 is not optimal if in addition, f(t) is greater than (a multiple of) t 1= in the neighbourhood of (which, in practice, is what happens for a \typical" weight function f). In this case, one can replace F (") in Theorem 3.1 by e F (") = R 1 " =f (t). R Example 3.3. Consider two \typical" weight p 1 functions f of W for which =f (t) = 1. According to Theorem 3.1, for f 1 (t) = t log(1=t) (in the neighbourhood of ), (31) lim "! " log log(1=") log kw k f1 < "? 8 whereas for f (t) = p t log log(1=t) (in the neighbourhood of ), lim "! " log log(1=") log(1=") log kw k f < " < Remark 3.4. We do not have a lower bound in general for log (kw k f < ") when R p 1 =f (t) = 1. Let us consider the particular example f 1 (t) = t log(1=t) treated above. Let " ( 1=) and = (") = exp(?1=" 4 ). By scaling, (3) t p t log(1=t) < " = s1= jw (s)j p s log(1=s) < " Since for all " ( 1=), log(1=s) p log(1=s) p log(1=) p log(1=s) =", and since test (1.8) conrms that s1= jw (s)j=s 1= (log(1=s)) 1=4 is a well-dened variable, it follows that the probability term on the left hand side of (3.) is bounded below by a positive absolute constant, uniformly for " ( 1=). This playing the role of (.9), and applying the same argument as in the proof of Theorem 1.1, we arrive at the following estimate (33) lim inf "! " log(1=") log kw k f1 < " >?1 11

12 There is a clear gap between (3.1) and (3.3). 4. Some applications 4.1. Chung's functional law The estimate in Theorem 1.1 allows to establish Chung-type functional iterated logarithm laws for the increments of the Wiener process W. Let S be Strassen's set dened by S = n g g absolutely continuous, with Lebesgue derivative _g, such that g() = and J(g) = _g (t) 1 In the language of probability functional analysis, S is the unit ball of the reproducing kernel Hilbert space pertaining to the Wiener measure on C[ 1], the space of all continuous function on [ 1] endowed with the uniform topology. Our rst application gives the rate of convergence in Strassen's functional law of the iterated logarithm for W. Theorem 4.1. If f is a positive function satisfying either (H1) or (H), such that (19) holds, then for all g S with J(g) < 1, p T log log T? g lim inf T!1 (log log T )?????? W (T )?? = f o 4 p 1? J(g) c f a.s. where c f = 1= f (t) Remark 4.. In case f = 1 and g =, this is Chung's classical law of the iterated logarithm. For f = 1 and general g, Theorem 4.1 is rst discovered by Csaki [7], and ultimately extended by many other mathematicians, cf. for example de Acosta [1], Baldi and Roynette [3], Kuelbs and Li [15] among others. roof of Theorem 4.1. From Theorem 1.1 and \standard" arguments, (41) lim " log kw? g "! " k f r" =? (c f )? J(g) 8r 1

13 for all r > and g S (for details, cf. for example de Acosta [1]). On the other hand, thanks to the conditions in Theorem 4.1, there exists a nite constant C >, depending only on f, such that for all < < 1, (4) kg(1 ^ ((1 + ) ))? gk f C p The estimates (4.1){(4.) together imply Theorem 4.1 as is pointed out by Deheuvels and Mason [11]. tu More generally, from Theorem 1.1, we can get local and global Chung's functional iterated logarithm laws for the increments of W. For more details, cf. Berthet [4]. 4.. Higher dimensions Let fw d (t) t 1g denote standard d{dimensional Brownian motion (d 1), and \k k" the usual Euclidean modulus in R d. Here is the analogue of Theorem 1.1 in dimension d. Theorem 4.3. If f is a positive function satisfying either (H1) or (H), then lim " log "! <t1 kw d (t)k f(t) < " =? j (d?)= f (t) where j (d?)= is the smallest positive root of the Bessel function J (d?)=. Since j?1= = =, Theorem 4.3 is in agreement with Theorem 1.1 when d = Brownian bridge In the theory of empirical processes, the Brownian bridge plays an important role. Let f d (t) t 1g denote a standard R d -valued Brownian bridge, with d 1. Theorem 4.4. Assume that t 7! f(t) and t 7! f(1? t) are both nondecreasing in a neighbourhood of. Assume that inf atb f(t) > for all < a b < 1. Then lim " log "! <t<1 k d (t)k f(t) < " =? j (d?)= where j (d?)= is as before the smallest positive root of J (d?)=. 13 f (t)

14 To see how Theorem 4.4 can be applied to weighted empirical processes, we refer to Csaki [8] General moving boundaries Let h 1 and h be Borel functions on [ 1]. Following Mogulskii [], we consider the probability of the event when " goes to. E " = n o " h 1 (t) W (t) " h (t) for all t [ 1] Theorem 4.5. Assume that h? h 1 is a positive function satisfying either (H1) or (H), such that h 1 + h is absolutely continuous whose Lebesgue derivative is square integrable over [ 1]. Then (43) lim "! " log (E " ) =? (h (t)? h 1 (t)) When h? h 1 satises (H1) such that h 1 and h are piece-wise continuous, Theorem 4.5 is due to Mogulskii []. To prove (4.3), observe that E " = f kw? " gk f < " g, with g = (h 1 + h )= and f = (h? h 1 )=. The upper bound in (4.3) clearly follows from the same argument as in Section. Its lower bound is a consequence of the Cameron{Martin formula (this is where the condition upon h 1 + h comes in) and of our Theorem 1.1. Acknowledgements We are grateful to Endre Csaki for having pointed out Mogulskii's paper [] to us, and for his constant encouragements during the preparation of the manuscript. 14

15 References [1] de Acosta, A. (1983). Small deviations in the functional central limit theorem with applications to functional laws of the iterated logarithm. Ann. robab {11. [] Anderson, T.W. (1955). The integral of symmetric unimodular functions over a symmetric convex set and some probability inequalities. roc. Amer. Math. Soc. 6 17{176. [3] Baldi,. and Roynette, B. (199). Some exact equivalents for the Brownian motion in Holder norm. robab. Th. Rel. Fields {484. [4] Berthet,. (1997). Inner rates of clustering to Strassen type sets by increments of empirical and quantile processes. Stoch. roc. Appl. (to appear) [5] Cameron, R.H. and Martin, W.T. (1944). The Wiener measure of Hilbert neighborhoods in the space of real continuous functions. J. Math. hys {9. [6] Chung, K.L. (1948). On the maximum partial sums of sequences of independent random variables. Trans. Amer. Math. Soc. 64 5{33. [7] Csaki, E. (198). A relation between Chung's and Strassen's laws of the iterated logarithm. Z. Wahrscheinlichkeitstheorie verw. Gebiete 54 87{31. [8] Csaki, E. (1994). Some limit theorems for empirical processes. In Recent Advances in Statistics and robability (roc. 4th IMSIBAC, eds. J.. Vilaplana and M.L. uri) pp. 47{54. VS, Utrecht. [9] Csorg}o, M. and Horvath, L. (1993). Weighted Approximations in robability and Statistics. Wiley, Chichester. [1] Csorg}o, M., Shao, Q.-M. and Szyszkowicz, B. (1991). A note on local and global functions of a Wiener process and some Renyi-type statistics. Studia Sci. Math. Hung. 6 39{59. [11] Deheuvels,. and Mason, D.M. (1998). Random fractal functional laws of the iterated logarithm. (Invited paper). Studia Sci. Math. Hungar {1. [1] Donsker, M.D. and Varadhan, S.R.S. (1977). On laws of the iterated logarithm for local times. Comm. ure Appl. Math. 3 77{753. [13] Khatri, C.G. (1967). On certain inequalities for normal distributions and their applications to simultaneous condence bounds. Ann. Math. Statist {1867. [14] Kuelbs, J. and Li, W.V. (1993). Metric entropy and the small ball problem for Gaussian measures. J. Funct. Anal {157. [15] Kuelbs, J. and Li, W.V. (1993). Small ball problems for Brownian motion and the Brownian sheet. J. Theoretical robab {

16 [16] Kuelbs, J., Li, W.V. and Shao, Q.-M. (1995). Small ball probabilities for Gaussian processes with stationary increments under Holder norms. J. Theoretical robab {386. [17] Kuelbs, J., Li, W.V. and Talagrand, M. (1994). Lim inf results for Gaussian samples and Chung's functional LIL. Ann. robab. 1879{193. [18] Ledoux, M. (1996). Isoperimetry and Gaussian analysis. In Ecole d'ete de Saint- Flour XXIV. Lecture Notes in Mathematics 1648 pp. 165{94. Springer, Berlin. [19] Li, W.V. (199). Lim inf results for the Wiener process and its increments under the L {norm. robab. Th. Rel. Fields 9 69{9. [] Li, W.V. and Shao, Q.-M. (1997). Small ball estimates for Gaussian processes under Sobolev type norms. J. Theoretical robab. (to appear) [1] Lifshits, M.A. (1998). Asymptotic behavior of small ball probabilities. In robability Theory and Mathematical Statistics (roceedings of the 1998 Vilnius Conference, eds. B. Grigelionis, et al.). (to appear) [] Mogulskii, A.A. (1974). Small deviations in space of trajectories. Th. robab. Appl {736. [3] Shao, Q.-M. and Wang, D. (1995). Small ball probabilities of Gaussian elds. robab. Th. Rel. Fields 1 511{517. [4] Shi, Z. (1996). Small ball probabilities for a Wiener process under weighted norms, with an application to the remum of Bessel local times. J. Theoretical robab {99. [5] Stolz, W. (1996). Some small ball probabilities for Gaussian processes under nonuniform norms. J. Theoretical robab {63. [6] Talagrand, M. (1994). The small ball problem for the Brownian sheet. Ann. robab. 1331{

1. A remark to the law of the iterated logarithm. Studia Sci. Math. Hung. 7 (1972)

1 PUBLICATION LIST OF ISTVÁN BERKES 1. A remark to the law of the iterated logarithm. Studia Sci. Math. Hung. 7 (1972) 189-197. 2. Functional limit theorems for lacunary trigonometric and Walsh series.