On a Problem of Erdős and Taylor

Transcription

1 On a Problem of Erdős and Taylor By Davar Khoshnevisan, Thomas M. Lewis & Zhan Shi University of Utah, Furman University and Université ParisVI Abstract. Let {S n,né 0} be a centered d dimensional random walk d é 3 and con- sider the so called future infima process, J n =inf k é n S k. This paper is concerned with obtaining precise integral criteria for a function to be in the Lévy upper class of J. This solves an old problem of Erdős and Taylor 960, who posed the problem for the simple symmetric random walk on Z d, d é 3. These results are obtained by a careful analysis of the future infima of transient Bessel processes and using strong approximations. Our results belong to a class of Ciesielski Taylor theorems which relate d and d 2 dimensional Bessel processes. Keywords and Phrases. Brownian motion, Bessel processes, transience, random walk. Running Title. Transient random walk. A.M.S. Subject Classification 990. Primary. 60J65, 60G7; Secondary. 60J5, 60F05.

2 . Introduction Throughout this paper, we will adopt the following notation: given x =x,...,x d IR d, x will denote the Euclidean l 2 norm of x, i.e., x =x x2 d /2. Given x IR, we will write log e for the natural logarithm, lnx =ln x =log e x e and, for k é, ln k+ x =lnln k x. Let {S n,né 0} denote an IR d valued simple symmetric random walk. Erdős and Taylor 960 present a variety of results on the fine structure of the sample paths of {S n,né 0}. Especially noteworthy is their complete characterization of the number of returns to origin by time n for the simple planar random walk. When d é 3, the simple random walk is transient, and, as such, it has only a finite number of returns to the origin. Instead they study the behavior of the associated future infimum process: for all n é 0, let J n =inf k é S k. n In their Theorems 8 and 9 p. 54, Erdős and Taylor establish the following laws of the iterated logarithm for S n and J n : lim sup n lim sup n S n 2n ln2 n = d /2, a.s.,.a J n 2n ln2 n = d /2, a.s...b Let 0 <c<. Then.b demonstrates that for almost all ω there is an increasing sequence {n k,ké } of integers such that J nk ω = S nk ω é c 2n k ln 2 n k..2 In words, J n can be as large as S n when S n is near its upper envelope. A complete characterization in the sense of the integral test of Erdős 942 of the upper functions of {J n,né0} is left unresolved see the remarks on top of p. 55. The main purpose of this paper is to present this characterization for centered IR d valued random walks which

3 satisfy certain integrability conditions in particular, we make no assumptions about the support of the distribution of the increments. In this way we can find increasing functions f, g :[, with x gx fx increasing without bound such that for almost all ω there is an increasing sequence {n k,ké} and integer N such that a J n ω 6 fn for all n é N, while b S nk ω >gn k. In contrast with.2, this shows that whenever S n leaves a centered ball of radius gn, at the very least it must return to a ball of radius fn at some future time. Our approach consists of the following: first we solve the analogous problem for transient Bessel processes. Then, by way of a strong approximation argument, we solve the problem for random walks. Although we will comment on this at greater length in Section 2, our results demonstrate that the upper class of the future infimum of a d dimensional Bessel process d > 2 is identical to the upper class of a d 2 dimensionl Bessel process: this relates our work to a class of results sometimes referred to as Ciesielski Taylor theorems. Future infima processes such as the ones in this paper occur naturally in the more general setting of Markov processes. As a sample see Aldous 992, Burdzy 994, Chen and Shao 993, Khoshnevisan 994, Khoshnevisan et al. 994, Millar 997 and Pitman 975. In conclusion, we would like to add that in the case of the simple walk in dimension d 3 i.e., the problem of Erdős and Taylor, one can adapt the method of Section 3 to obtain a completely classical proof of the Erdős Taylor problem mentioned above. We will not discuss this approach, as it only seems to work for random walks which live on a sublattice of Z d. 2. Statements of results Let us first state our results for Bessel processes. Throughout this paper, X ={X t, té 0} will denote a d-dimensional Bessel process with d>2. This is a diffusion on [0, whose 2

4 generator, G, isgivenby Gfx = 2 f x+ d 2x f x. Since d>2, there is no need to specify a boundary condition at 0. The definition and properties of X can be found in Revuz and Yor 99. It should be noted that when d is an integer, the radial part of a standard d dimensional Brownian motion is a d dimensional Bessel process. Of particular importance to us is that the condition d>2 implies that X is transient. We will define two processes associated with X. First, we define the future infimum process: for all t é 0, I t =inf s é X s. t Next let L 0 = 0 and for r>0, define the escape process: L r =sup{s : X s 6 r}. I and L inherit scaling laws from X. For c>0, c /2 I ct and I t are equivalent processes, and c 2L cr and L r are equivalent processes. The processes I and L are inverses in the following natural sense: {ω : L r 6 t} = {ω : I t é r}. 2.a We point out in passing that the transience of X trivially implies that lim t L t = lim t I t =, almost surely. Properties of the processes I and L are explored in Getoor 979, Yor 992a,b and Khoshnevisan, Lewis and Li 994. Given a stochastic process {Z t,té 0} and a nondecreasing function f : [, 0,, we will say that f is in the upper class respectively lower class ofz if for almost all ω there is an integer N = Nω such that Z t ω 6 ft respectively Z t ω é ft for all t é N. We shall write this in the compact form, f UZ f LZ, respectively. Let ϕ :[, 0, be nonincreasing and let ψ :[, 0, be nondecreasing. Define, J ϕ = ϕ d/2 texp dt 2ϕt t J 2 ψ = ψ d 2 texp ψ2 t dt 2 t. 3

5 For future reference, we observe the relationship: J ψ 2 =J 2 ψ. 2.b Our first theorem characterizes the lower class of the exit process, L. Theorem 2.. Let ϕ : [, 0, be a nonincreasing function. Then t t 2 ϕt LL if and only if J ϕ <. Essentially due to 2.a and 2.b, we obtain our next theorem, which characterizes the upper class of I. Theorem 2.2. Let ψ : [, 0, be a nondecreasing function. Then t tψt UI if and only if J 2 ψ <. Remark Theorem 2.2 had been independently conjectured by Chen and Shao 993 and Khoshnevisan, Lewis and Li 994. In fact, Theorem 4.33 of the latter reference constitutes the so called easy half of Theorem 2.2. There it is shown that if ψ :[, is nondecreasing and J 2 ψ <, then t tψt UI. For purposes of comparison, let us recall the celebrated Kolmogorov Dvoretsky Erdős integral test see, e.g., Itô and McKean 965, p. 63: Theorem A. Let ψ : [, 0, be a nondcreasing function. Then t tψt UX if and only if ψ d texp ψ 2 t/2 dt t <. Theorems 2.2 and A together show that the upper class of I in dimension d is the same as the upper class of X in dimension d 2. It should be noted that when d =3, the comparison is stronger still; see Pitman 975 and Revuz and Yor 99 for details. Let τ 0 = 0 and, for all r>0, let τ r =inf{s >0:X s = r} : this defines the exit time of X 4

6 from the interval [0,r]. Then the lower class of L in dimension d isthesameasthelower class of τ in dimension d 2. Results relating d and d 2 dimensional Bessel processes are sometimes known as Ciesielski Taylor theorems and can be found, for example, in Ciesielski and Taylor 962, Jain and Taylor 973 and Yor 992b. Now let us state our theorems concerning random walks. Throughout let ξ, {ξ i,ié } be a sequence of independent and identically distributed IR d valued random variables, d é 3. Let S 0 = 0 and, for all n é, let S n = ξ ξ n. We will extend the definition of our random walk to continuous time by simply setting S t = S [t] for all t é 0. Here [t] is the greatest integer less than or equal to t. For all t é 0, let J t =inf u é S u. t Theorem 2.3. Let ξ have zero mean vector and identity covariance matrix and suppose there is a δ>0such that IE ξ 2+δ <. Let ψ :[0, 0, be an unbounded function such that t tψt is increasing. Then t tψt UJ if and only if J 2 ψ <. Remark It is easy to see how the above includes the promised solution to the Erdős Taylor problem. Indeed, suppose ξ is as in the statement of Theorem 2.3, except that its covariance matrix is σ 2 I,whereσ>0andI is the d d identity matrix. Obviously, Theorem 2.3 holds with S n and J n replaced with S n /σ and J n /σ, respectively. It is worth noting that the nature of the problem changes entirely when the random walk does not have identity covariance matrix; nonetheless, we can prove various laws of the iterated logarithm. Let d é 3 be an integer and Q denote a d d symmetric positive definite matrix with eigenvalues λ é λ 2 é...é λ d > 0. 5

7 Let W ={W t,té0} denote a d dimensional centered Brownian motion with covariance matrix Q and define the associated future infimum and escape processes by I Q t =inf s é W s and L Q r =sup{t é 0: W t 6 r}, t respectively. Likewise, let {S t : t é 0} denote a d dimensional centered random walk with covariance matrix Q, and define the associated future infimum and exit processes by J Q t =inf u é S u and K Q r =sup{t 0: S t r}, t respectively. Our theorem follows: Theorem 2.4. Suppose ξ, {ξ i,ié} form a sequence of i.i.d. random vectors taking values in IR d with d 3. We assume that ξ has mean vector zero, covariance matrix Q and that there is a δ > 0 such that IE ξ 2+δ <. Let {S t : t é 0} denote the associated random walk and let {W t,té0} be a d dimensional centered Brownian motion with covariance matrix Q. Then a lim inf t 2ln 2 t 2ln 2 t K t 2 Q t = lim inf L t t 2 Q t = a.s.; λ b lim sup t J Q t 2t ln2 t = lim sup t I Q t 2t ln2 t = λ a.s. An alternative approach is to view our processes as taking values in a certain Riemannian sub-manifold of IR d. To do this, we define a metric which is compatible with the underlying process. Because Q has orthogonal eigenvectors, there is an invertible matrix V such that Q = VV T. The standardized random vector V ξ has identity covariance matrix. For x IR d, let x V = V x, which defines a norm on IR d. Clearly, V and are equivalenorms and Theorem 2.3 holds if we measure distance by the norm V. The inner product on the aforementioned space is the obvious one, by polarization. Moreover, the intrinsic volume element is given by dm =detq /2 dx. This notion of geometry, however, is noatural to the study of the paths of random walk. 6

8 The remainder of this paper is organized as follows. In Section 3, we develop the relevant probability estimates. In Section 4, we prove Theorems 2. and 2.2. In Section 5, we prove a strong approximation theorem which is then used to prove Theorem 2.3. Finally, in Section 6, we prove Theorem 2.4. Acknowledgements. This work was partially carried out while D. Khoshnevisan was visiting the Johns Hopkins University and T. Lewis was visiting the Center for Mathematical Sciences at the University of Wisconsin Madison. We would like to express our gratitude to Professors Dan Naiman, John Wierman and Colin Wu of the Johns Hopkins University and to Professors Tom Kurtz and Jim Kuelbs of the University of Wisconsin Madison for all of their assistance in arranging these visits. We would also like to extend our thanks to an anonymous referee for his careful reading of this paper. 3. Preliminary estimates The main purpose of this section is to present some lemmata which will be useful in the proof of the theorems. For all β é 0, let Hβ =IPL 6 β. The density of L is well known see, for example, Getoor 979, Yor 992b or Khoshnevisan, Lewis and Li 994: H x =γ d x d/2 exp d 2 Ix >0, where =2 d 2/2 Γ. 2x γ d 2 Thus, by an application of l Hôpital s rule, we obtain the following asymptotic estimate for the small ball probability of L : Hβ 2γ d β 2 d 2 exp /2β as β From this, it is clear that for any T>0, there exists a positive constant c, which depends only on T, such that c x2 d 2 exp /2x 6 Hx 6 cx 2 d 2 exp /2x for all x [0,T]

9 In the course of proving Theorem 2., it will be necessary to estimate the distribution of the increment L L s for 0 <s<. Our first lemma is a useful step in this direction. Lemma 3.. Let β, a > 0 and 0 <s<. Then IP L L s 6 β 6 Hβ + a Has 2. Proof. For any s 0,, and all β, a > 0, {ω : L L s β} {ω : L s a} {ω : L β + a}. By a theorem of Getoor 979, {L t,té0} has independent increments. The rest of the proof follows from the scaling law for L, since L s has the same distribution as s 2 L ; hence, IP L s 6 a =Has 2. Our next lemma is a direct application of Lemma 3.; it provides a useful estimate for the distribution of L L s when s is relatively small. Lemma 3.2. Fix λ>0. Then there exists a positive constant c, depending only on λ and d, such that IP L L s 6 β 6 chβ, for all 0 6 β 6 λ and 0 6 s 6 λβ. Proof. Since s IP L L s 6 β is increasing for all 0 6 s 6 λβ, it suffices to prove the lemma for s = λβ. To this end let µ to be the median of L, i.e., Hµ =/2. Applying Lemma 3. with a = µλ 2 β 2, we obtain: IP L L λβ 6 β 6 Hβ + µλ2 β 2 Hµ =2Hβ + µλ 2 β 2. Now fix c so that 3.2 holds for all 0 6 x 6 λ + µ. Since β + µλ 2 β 2 6 λ + µ, it follows that IP L 6 β + µλ 2 β 2 6 cβ + µλ 2 β 2 2 d 2 exp 8. 2β +2µλ 2 β 2

10 However, Moreover, β 2 d 2 if d é 4 β + µλ 2 β 2 2 d 2 6. β 2 d 2 + µλ 2 d 2 if d<4 exp 6 e /2β exp µλ 2 /2. 2β +2µλ 2 β 2 Finally by 3.2, c β2 d 2 e /2β 6 Hβ, which gives us the desired result. Our next lemma is an estimate for the distribution of the increment L L s when s is noecessarily small. In order to state this lemma, we will need the following definition. If d 6 4, let β =. If d>4, let β =sup{0 <x6e : 4d 4x lnx 6 /2}. Since x x lnx isincreasingon0,e, it follows that β > 0, which is all we will need. Lemma 3.3. and d, such that Let ε>0. Then there exists a positive constant c, depending only on ε IP L L s 6 β 6 c exp for all β 62ε β and 0 6 s 6 βε. s2, 4β Proof. First fix c > 0 such that 3.2 holds for all 0 6 x 6 2/ε. We will consider two cases: βε6 s 6 βε and 0 6 s 6 βε. In the first case, apply Lemma 3. with a = βs/ s and observe that β + a = β s 6 ε as 2 = β s + β s 6 2 ε 2β + a + s2 = 2as 2 2β β + a as 2 = s 9

11 Thus, by 3.2 and our choice of c, we obtain IP L L s 6 β 6 c 2 s2 d 2 exp s β In the second case, 0 6 s 6 βε. Then, by 3.3 and the monotonicity of r L r, we obtain IP L L s 6 β 6 IP L L βε 6 β 6 c 2 βε2 d 2 exp βε2. 2β However, since 0 <s6 βε, it is easy to see that exp βε2 6 e ε exp 2β s2. 2β Setting c 2 = c 2 ε 2 d 2 e ε, we obtain: IP L L s 6 β 6 c 2 β 2 d 2 exp s β If d 6 4, then we are done, since s and β are bounded by. If, however, d>4, then we need to consider two additional subcases. If /2 6 s 6 βε, then, by bounding s 2 d 2 by 2 d 2 +2 in 3.3, we obtain the desired result. If, however, 0 6 s 6 /2, then by 3.3 and 3.4 we obtain s2 IP L L s 6 β 6 c 2 exp 2β + d 4 2 lnβ. This last exponent may be expressed as: [ s2 d 4β ] lnβ 2β s 2. However, since 0 <s6 /2 andβ 6 β, it follows that which is what we wished to show. d 4β lnβ s 2 é 2, Remark Equations 3.2 and 3.4 already yield good estimates for the distribution of the increment L L s ; however, for our applications, it is better to place all of the 0

12 dependency on ε and d into a constant. Although it complicates the proof of the lemma and weakens the result, it will make the application cleaner. Lemma 3.4. Suppose α é 0, ε,ε 2 0,, and 0 <β<2ε 2 β α. Then for all ε 6 s 6 βε 2 there is a constant c, depending only on ε,ε 2 and d, such that IP L s 6 s 2 α, L 6 β 6 chαexp s. 2β Proof. Choose c so that 3.2 holds for all 0 6 x 6. By the independence of increments and scaling we obtain: IP L s 6 s 2 α, L 6 β 6 IP L s 6 s 3 β+ips 3 β<l s 6 s 2 α, L L s 6 β s 3 6 IP L 6 βs+hα IP L L s 6 β s 3 We will estimate each of the terms on the right hand side. First, by 3.2, IP L 6 βs 6 c βs 2 d 2 e /2βs. Since 0 6 s<, it easily follows that s 2 d 2 6 ε 2 d 2 and e /2βs 6 e /2β exp Thus, by another application of 3.2 we see s 2β IP L 6 βs 6 c 2 ε 2 d 2 Hβexp s β Next we will apply Lemma 3.3 to estimate IP L L s 6 β s 3. The conditions of this lemma are satisfied; hence, IP L L s 6 β s 3 6 c 2 exp where c 2 depends only on ε 2 and d. However,. s2, 4β s 3 s 2 4β s 3 = s 4β + s + s 2 é s 2β.

13 Consequently, IP L L s 6 β s 3 6 c 2 exp s β To achieve the final form of the lemma, combine 3.5 and 3.6, noting that exp s 6 exp 2β s 2β and Hβ 6 Hα. 4. Proofs of Bessel process results Before proceeding to the proof of Theorem 2., let us make a few prefunctory remarks. Essential to the proofs will be the function x t x, which is defined for all real x by x t x =exp. 4. lnx We note that this function is increasing in x. Primarily we will be interested in the values of t x at the positive integers, but occasionally we will write, e.g., +lnn, whose meaning is given by 4.. By an application of the mean value theorem, it follows that + lim lnn =. n + Since + / approaches one as n tends to infinity, evidently the same limit is obtained upon replacing + by in the denominator. These considerations lead us to the following: there exist universal positive constants c and c 2 such that for all n é, c lnn c lnn, 4.2 c 2 lnn c 2 lnn. Recall from Khoshnevisan, Lewis and Li 994, the following law of the iterated logarithm for {I t,té 0}: lim sup t I t 2t ln2 t = a.s. 2

14 The above together with 2.a easily yield, lim inf t 2ln 2 t t 2 L t = a.s. 4.3 Let ϕ : [, 0, be nonincreasing and, for each positive integer n, let ϕ n = ϕ. In light of 4.3 with respect to the proof of Theorem 2. we may assume without any loss of generality that there exists a universal positive constant c such that c ln 2 t 6 ϕt 6 c 2ln 2 t Thus, without loss of generality, we may assume that there exists a universal positive constant c such that First we will prove c lnn 6 ϕ n 6 c lnn 4.4 J ϕ < = IP L t é t 2 ϕt, eventually =. 4.5 For each positive integer n, define the event F n ={ω : L tn 6 t 2 n+ ϕ n}. First we will demonstrate that n= IP F n <. To this end, observe that tn+ ϕ d/2 /2ϕt dt te t é ϕ d/2 n e /2ϕ n Thus, by 4.2 and 4.4, there exists a universal positive constant c such that for all n é, we obtain tn+ ϕ d/2 /2ϕt dt te t é c ϕ 2 d/2 n+ e /2ϕ n By scaling, 3. and some algebra, it follows that there exists a universal positive constant c such that IP F n 6 c+ / 2 ϕ n+ 2 d 2 e /2ϕ n+ exp /+ 2. 2ϕ n+ 3

15 This last exponent is easily estimated: observe that 2ϕ n+ t2 n 6 t 2 n+ ϕ n+ tn+ Now 4.2 and 4.4 show that this exponent is uniformly bounded in n. Thus, there exists a universal positive constant c 2 such that + IP F n 6 c 2 ϕ 2 d/2 n+ e /2ϕ n Combining 4.6 and 4.7 with J ϕ < we see that n= IP F n <. Thus, by the first Borel-Cantelli lemma, it follows that IP F n, i.o. = 0. Consequently, for almost all ω, there exists a random index N beyond which. L tn >t 2 n+ϕ. Finally, given n é N and s [,+ ] it follows that, with probability one, L s é L tn >t 2 n+ϕ é s 2 ϕs, wherewehaveusedthefactthats ϕs is nonincreasing. This demonstrates 4.5. Next we will show that J ϕ = = IP L t é t 2 ϕt, eventually = To this end, for each positive integer n, define the event E n ={ω : L tn 6 t 2 nϕ n }. To demonstrate 4.8, it suffices to show that IP E n, i.o. =. Since {E n, i.o.} is a 0- event, by Kochen-Stone 964, it is enough to show that there is a universal positive constant c such that IP E n = 4.9 n= N 2 IP E k E n 6 c IP E k k<n6 N k= 4

16 The verification of 4.9 is straightforward. For all n sufficiently large, there exists a constant c such that tn+ ϕ d/2 /2ϕt dt te t 6 ϕ d/2 n+ e /2ϕ n + 6 c ϕ 2 d/2 n e /2ϕn, 4. where we have used 4.2. By scaling and 3.2, it follows that there exists a universal positive constant c 2 such that IP E n é c 2 ϕ 2 d/2 n e /2ϕ n. 4.2 Combining 4. and 4.2 with J ϕ = demonstrates 4.9. We are left to demonstrate 4.0. To this end, we introduce the following sets of indices: G N n = { k N : k lnn ln 2n} B N n = { k N : lnn k<lnn ln 2n} U N n = { k N : k<lnn} A N n j = { k N : { k N : j lnn j + } +k lnn à N j n j = lnn j + +k lnn In fact, we will prove that there exist universal positive constants c,c 2 and c 3 such that for all N é wehave }. N n= k Gn N N n= k Bn N N n= k Un N N 2 IP E n E n+k c IP E n 4.3 IP E n E n+k c 2 IP E n E n+k c 3 n= N n= N n= IP E n 4.4 IP E n, 4.5 which would suffice to verify 4.0. The verifications of 4.3 through 4.5 involve various counting arguments, which are contained in the following lemmas. 5

17 Lemma 4.. There exists universal constant c such that c +lnnln 2 n 6 lnn 6 c+lnnln2 n. Proof. Let q n = lnn. +lnnln2 n It suffices to show that lim n q n =. However, by some algebra, lnq n = ln + lnnln 2n n lnnln 2 n n n ln 2n+lnnln 2 n 2 n lnn +lnnln 2 n Since x ln + x asx 0, it follows that lnq n 0asn. Lemma 4.2. There exists a universal positive constant c such that for all k, né tn+k k c lnn. Proof. Observe that j t j lnj is increasing. From this and 4.2, it follows that k +k = +j+ +j j=0 k c lnn. k j=0 +j c lnn + j Solving for k, we obtain the lemma. Lemma 4.3. There exists some c>0 such that for all N n and all j 0, #B N n A N n j cj + 4. Proof. Let k B N n A N n j. We will show that k cj

18 By Lemma 4.2, But for k in question, tn+k k c lnn = c +k c +k lnn +k j + lnn. +k +lnn ln2 n c lnn, by Lemma 4.. Therefore, On the other hand, by the definition of the annulus, A N n j, k cj + lnn 3/ j + lnn +k +lnn e. Hence, for some c>0, j + c lnn. Equivalently, lnn cj + 2. By 4.7, we obtain 4.6 and hence the result. Lemma 4.4. There exists some c>0 such that for all N n and all j 0, #Un N ÃN n j cj +. Proof. Let k Un N ÃN n j. By Lemma 4.2, tn+k k c lnn = c lnn +k c +lnn j + cj k

19 Therefore #U N n ÃN n j cj +, thus proving the result. In the verification of 4.3 and 4.4, we will use the following inequality: for 06x < y and a, b > 0wehave IP L x 6 a, L y 6 b 6 IP L x 6 aip L y L x 6 b, 4.9 wherewehaveusedthefactthatt L t is nondecreasing and that the process {L t,té 0} has independent increments; see Getoor 979, for example. Verification of 4.3. By 4.9 and scaling we have IP E n E n+k 6 IP L tn 6 t 2 nϕ n IP L tn+ L tn 6 t 2 n+kϕ n+k =IPE n IP L L tn /+k 6 ϕ n+k. We wish to demonstrate the conditions of Lemma 3.2 prevail with s = /+k and β = ϕ n+k. To this end we need to show that there is a universal λ for which +k 6 λϕ n+k for all N é, né andk G N n. Since j t jϕ j is increasing, it is enough to show that lim sup <. 4.9 n +lnnln2 nϕ n+lnnln2 n This, however, follows immediately from 4.4 and Lemma 4.. By Lemma 3.2 and the definition of x Hx, we obtain, IP L L tn /+k 6 ϕ n+k 6 chϕ n+k = cip E n+k. In this way we have shown that there is a universal positive constant c such that IP E n E n+k 6 cip E n IP E n+k, for all n é andk Gn N. This verifies 4.3. Verification of 4.4. It suffices to prove the result for N sufficiently large. By 4.9 and Lemma 3.3, there exists n 0 such that for all N n n 0 and all k B N n, IP E n E n+k IP E n IP L L tn /+k ϕ n+k c IP E n exp 4 lnn /+k 2,

20 since by 4.4, if n 0 is large enough, for all N n n 0 and all k B N n, +k +lnn 2e c ln2n c 2ϕ n+k. By 4.20 and Lemma 4.3, for all N n n 0, N N IP E n E n+k IP E n exp c lnn /+k 2 n=n 0 j=0 k Bn N AN n j n= k Bn N n 0 +lnn 0 ln 2 n 0 IP E n n= N #Bn N AN n j IP E n exp c j 2 +c 2 n= j=0 n= N c IP E n, n= N IP E n verifying 4.4. Verification of 4.5. It suffices to prove the result when N is sufficiently large. Take k U N n. We want to use Lemma 3.4 with s = /+k,α= ϕ n and β = ϕ n+k. By 4.2 and 4.4, as n, s s +lnn e + lnn cϕ n. Thus there exist ε > 0, ε 2 > 0andn 0, such that for all N n n 0 and all k G N n, ε <s< ε 2 β. By scaling Hϕ n =IPE n. Hence, by Lemma 3.4, for all N n n 0 and all k U N n, IP E n E n+k cip E n exp t n. 2ϕ n +k 9

21 Therefore by 4.2 and Lemma 4.4, for all N n n 0, N N IP E n E n+k n= k Un N n=n 0 j=0 n 0 +lnn 0 IP E n n= n= j=0 n= k U N n ÃN n j IP E n exp c/ϕ n /+k + N IP E n exp c lnn /+k + c 2 n= j=0 k Un N ÃN n j N #Un N N ÃN n je c j IP E n +c 2 IP E n N c IP E n. n= N IP E n n= This verifies 4.5 and finishes the proof of Theorem 2.. Finally, we offer the following: Proof of Theorem 2.2. Recall 2.b; since the case J 2 ψ < is contained in the result of Khoshnevisan, Lewis and Li 994, we need only consider the case J 2 ψ =. Let ϕt =ψ 2 t. Then ϕ satisfies the conditions of Theorem 2.. Moreover, by exactly the same proof as in Theorem 2.: L ψn+ + /ψ n t2 n ψ 2 n+, i.o. 4.2 where =expn/ lnn and ψ n = ψ. Since t ψt is nondecreasing, the interval [t 2 n/ψ 2 n+,t 2 n+/ψ 2 n] is not empty. On the other hand, 4.2 is equivalent to the existence of a random and infinite set, N, such that I t 2 n /ψn+ 2 t ψ n+ n+, for all n N. ψ n Notice that for all n N and all s [t 2 n /ψ2 n+,t2 n+ /ψ2 n ], I s I t 2 n /ψn+ 2 t ψ n+ n+ ψ n s /2 ψ n+ s /2 ψs, which is what we wished to show. 20

22 5. Proofs of random walk results Throughout this section, let d é 3 be an integer and let W ={W t,té0} denote a standard d dimensional Brownian motion. Given a collection {ξ i, é } of i.i.d. IR d valued random vectors, let S 0 = 0 and, for all n é, let S n = ξ + + ξ n. We will extend the random walk to continuous time by setting S t = S [t] for all t é 0, where [t] denotes the greatest integer which is less than or equal to t. For real numbers 0 6 t 6 T, let J t =inf S u, u t I t =inf W u, u t J t,t = inf t 6 S u ; u T I t,t = inf W u. t u T The main result of this section is the following strong approximation theorem: Theorem 5.. Let ξ be an IR d valued random variable with zero mean vector, identity covariance matrix and IE ξ 2+δ <, for some δ>0. Then on a suitable probability space we can reconstruct a sequence of i.i.d. random vectors {ξ i,ié} with Lξ =Lξ and a Brownian motion W such that for some q = qδ 0, /2, almostsurely, I t J t = ot /2 q. Remark 5... With more work, the condition IE ξ 2+δ < can be relaxed to the following: there exists some p> such that IE Kξ <, where Kx = x 2 lnx 2 d d 2 2p ln2 x d 2. The proof of Theorem 5. relies on a strong approximation result of Einmahl 987, which is implicit in the following lemma. Given δ>0, let η = 2 2+δ. 5. 2

23 Lemma 5.2. Under the hypothesis of Theorem 5. almost surely sup S u W u = ot /2 η, 0 6 u 6 t where η is given by 5.. Proof. Since t sup 0 6 u 6 t S u W u and t t /2 η are nondecreasing, it suffices to prove the result for integral values of t. Le be a positive integer. By the triangle inequality, sup 0 6 u 6 n S u W u 6 sup S u W [u] u 6 n = max 0 6 k 6 S k W k + max n 0 6 k 6 n = I n + II n, sup W u W [u] 0 6 u6n with obvious notation. We will estimate each of these terms in order. max k 6 s 6 W s W k. k+ A direct consequence of Theorem 2 of Einmahl 987 is the following: on a suitable probability space, one can reconstruct the sequence {ξ i,ié } and a sequence of independent standard normal random variables, {g i,ié }, such that max k n S k G k = on 2 η, a.s., 5.2 where G k = k j= g j. To construct the Brownian motion W, enlarge the probability space by introducing product spaces so that it contains independent processes, {B i t; 0 t } i, where B i is a standard Brownian motion starting at G i conditioned to be G i+ at time an interesting construction of such processes appears in Pitman 974. On this extended probability space, define W t = [k,k+ tb k t k. k=0 It is easy to see that W is a standard Brownian motion and that W k = G k ;consequently, 5.2 can be written as follows: as n, I n = max 0 6 S k W k = o n 2 η, a.s. 5.3 k n 22

24 Finally, by Theorem.7. of Csörgő and Révész 98, as n, lnn II n = max sup W 0 6 k 6 s W k = O, a.s. n k 6 s 6 k+ which, in light of 5.3, proves the lemma in question. A small but important step in proving Theorem 5. is the following: Lemma 5.3. Let θ 0,η, where η is given by 5.. Then, with probability one, I t t 2 θ and J t t 2 θ for all t sufficiently large. Proof. Let f :IR + IR + be nondecreasing. Then a real-variable argument shows that the following are equivalent: i J t ft, eventually I t ft, eventually ii S t ft, eventually W t ft, eventually. Let θ 0 > 0. Then, either by direct calculation using the Borel-Cantelli lemma and the explicit density of I, given in Khoshnevisan, et al. 994 or by the theorem of Dvoretsky and Erdős see Motoo 959 on the rate of escape of W t, it can be shown that eventually, W t t 2 θ 0, a.s. 5.4 Now the lemma follows by Lemma 5.2, i, ii, 5.4 and the choice of θ. Proof of Theorem 5.. Construct a probability space in accordance with Lemma 5.2. Given δ>0, ηis given by 5.. Now, choose θ, ε IR + such that 0 <θ<ηand 0 <ε< η θ 2η, and set Significantly, ρ = +2ε 2θ. ρ 2 θ = + ε, ρ 2 η <

25 Our first task is to show that the infimum of W s for s é t is actually attained for s [t, t ρ ]forallt sufficiently large. Indeed, by Lemma 5.3 and 5.5, it follows that I t ρ ét ρ /2 θ é t /2+ε, a.s., for all t sufficiently large. However, by the ordinary law of the iterated logarithm, I t 6 W t <t /2+ε, a.s., for all t sufficiently large. Since I t = I t,t ρ I t ρ, these considerations yield: with probablity one, I t = I t,t ρ, 5.7 for all t sufficiently large. Likewise, with probability one, J t = J t,t ρ, 5.8 for all t sufficiently large. Finally, it is easy to see that J t,t ρ I t,t ρ 6 sup 0 6 u6t ρ S u W u. Thus, by 5.7, 5.8 and Lemma 5.2 we obtain: as t, J t I t 6 sup 0 6 u6t ρ S u W u = o t ρ/2 η, a.s. By 5.6, this proves the theorem. Lemma 5.4. Let ψ : [, 0, be nondecreasing and unbounded. For all t é, let { ψ U t = ψt+ψt if ψt é 2 if 0 <ψt <. { ψ L t = ψt ψt if ψt é 2 if 0 <ψt < 2. Then ψ U and ψ L are nonegative and nondecreasing and J 2 ψ, J 2 ψ U and J 2 ψ L converge or diverge together. 24

26 Proof. That ψ U and ψ L are nonegative and nondecreasing follows trivially from the fact that the mappings x x + x and x x x are nonnegative and increasing on x é and x é 2, respectively. Since ψ is unbounded, it is easy to see that, as t, ψ d 2 t ψ d 2 U t ψd 2 t L Moreover, there exist positive constants c,c 2,c 3 and c 4 such that which proves the lemma. c exp ψ 2 t/26 exp ψ 2 U t/2 6 c 2exp ψ 2 t/2 c 3 exp ψ 2 t/26 exp ψ 2 L t/2 6 c 4exp ψ 2 t/2, Proof of Theorem 2.3. Let ψ :[, 0, be nondecreasing. As is customary, we will assume, without loss of generality, there there exists a positive constant c such that ln2 t 6 ψt 6 c ln 2 t. c If J 2 ψ <, by Lemma 5.4, J 2 ψ L <. Consequently,fort sufficiently large, I t < tψ L t 6 t tψt c ln 2 t, By Theorem 5., J t 6 I t + t /2 q, where 0 <q</2. From this it follows that, for t sufficiently large, J t < tψt, Likewise, if J 2 ψ =, by Lemma 5.4, J 2 ψ U =, as well. Consequently, there exists a random increasing and unbounded sequence {,né } for which I tn é ψ U é ψ +c a.s. ln 2, However, by Theorem 5., J t é I t t /2 q, where 0 <q</2, from which it follows that a.s. a.s. J tn é ψ, a.s. 25

27 This proves the theorem in question. 6. The general covariance case Throughout this section, d é 3 will be an integer and Q will denote a d d symmetric positive definite matrix with eigenvalues λ é λ 2 é...λ d > 0 and corresponding orthonormal eigenvectors ζ,...,ζ d. Let B,...,B d denote d independent standard dimensional Brownian motions. Then for all t é 0, W Q t = d λi B i tζ i i= defines a centered Brownian motion with covariance matrix Q. From this, it is clear that W Q t 2 = d λ i Bi 2 t i= = ΛW t 2, where W t =B t,...,b d t is a standard d dimensional Brownian motion, and Λ is the d d diagonal matrix λ λ Λ = λd This demonstrates that it is sufficient to verify Theorem 2.4 for L Q r =sup{t é 0: ΛW t 6 r} and I Q t =inf ΛW s. s é t Our first task is to derive a large deviation estimate for I Q, which, in turn, will yield the small ball probability of L Q. The following geometric argument will be useful in this regard. For every t é 0, let us define the ellipsoid E t ={x IR d : Λx 6 t}. 26

28 Lemma 6.. Then Let r = λ. λ d λ E {x IR d : x + re 6 r +/ λ }, where e is the unit vector, 0,...,0. Proof. In the case d =2, parameterize the boundary of E by x = cosθ and x 2 = sinθ λ λ2 where θ [0, 2π]. Now the proof follows by some elementary algebra and trigonometry. If d é 3andx E, then where y é 0and λ x 2 + λ d y 2 6, y 2 = λ 2 x 2 2 λ + + λ d x 2 d d λ + x2 d. d Then, by the two dimensional case and the ordering of the eigenvalues, x + re 2 =x + r 2 + x x2 d 6x + r 2 + y 2 6r +/ λ 2, as was to be shown. Our next result is the aforementioned large deviation estimate for I Q. Lemma 6.2. a b lim t 2 ln IP I Q t é t =. 2λ lim ε ln IP L Q 6 ε =. ε 0 + 2λ Proof. let Since IP L Q 6 ε =IPI Q é / ε, it is enough to prove a. Given t>0, D ={x : x + tre 6 tr + λ }. 27

29 Observe that if x E t, then x/t E. Thus, by Lemma 6., E t D. Consequently, IP I Q é t =IPW s / E t for all s é é IP W s / D for all s é. Now pick α>/ λ and let H be the following half-space H ={x IR d : x é αt}. Then we have IP W s / D for all s é é IP W s / D for all s é W = xip W dx. 6. H Notice that each of the conditional probabilities is the probability that W, started from a point in the exterior of the ball D, never hits D. Consequently, by a gambler s ruin calculation, IP W s / D for all s é W = x = d 2 tr +/ λ x + rte Since x H, we know that x + rte é tr + α. Consequently, we have the uniform lower bound: IP W s / D for all s é W = x é r +/ λ r + α d 2 = κ. 6.2 Combining 6. and 6.2, we obtain: IP W s / D for all s é é κip W H = κip B é αt, where, in our notation, B is the first coordinate of W ; hence, a standard one dimensional Brownian motion. Simple considerations yield: lim inf t t 2 ln IP I Q é t é α2 2. The lower bound is achieved by sending α to / λ. The upper bound is easy: I Q 6 ΛW and, by a result of Zolotarev 96, lim t ln IP ΛW é t =. t2 2λ 28

30 This proves the lemma in question. Proof of Theorem 2.4. It is sufficient to prove a, since the results of part b follow by inversion and strong approximation respectively. The proof of a is standard. n é, Let B>andlet = B n for all n é. Let 0 <ε< and define the events: for all A n ={ω : L Q 6 εt 2 n /2λ ln 2 }. By scaling and Lemma 6.2, it follows that as n, ln IP A n ε lnn Thus n IP A n < and, by the easy half of the Borel Cantelli lemma, IP A n i.o. = 0. It follows that for almost all ω there is an integer N such that for all n é N we have Hence for n é N and 6 s<+ we obtain: 2λ ln 2 L t 2 Q é ε. 6.3 n 2λ ln 2 s ε L s 2 Q s é, B 2 where we have used 6.3 and the fact that t L Q t is nondecreasing. This demonstrates that lim inf s 2λ ln 2 s s 2 L Q s é, a.s.. Let ε>α>0 be chosen. Let t = e and, for all n é, let + = exp n α. It follows that ln n+α +α, ln 2 + αlnn. 6.4a Before proceeding with Borel Cantelli argument, let us observe that W Q t é λ d W t for all t é 0. Moreover, by a theorem of Dvoretsky and Erdős see Motoo 959, almost surely W t é 29 t lnt 3,

31 for all t sufficiently large. Thus by a real variable argument and inversion, it follows that with probability one there is a positive constant c such that L Q t 6 ct 2 lnt 6, 6.4b for all t sufficiently large. For every n é, let A n ={ω : L Q + L Q 6 + εt 2 n+ /2λ ln 2 + } {ω : L Q εt 2 n+ /2λ ln 2 + }. Hence by Lemma 6.2 and 6.4a we obtain: as n, lim inf n ln IP A n lnn é +α +ε >, from which it follows that n IP A n=. Since increments of the escape process are independent, the events {A n,né } are independent. Thus, by the independence half of the Borel Cantelli lemma, P A n i.o. =. Consequently, for almost all ω there is an infinite N Z + such tha N implies 2λ ln 2 + t 2 n+ From 6.4a and 6.4b, it follows that From this, we obtain lim n L Q ε+ 2λ ln 2 + t 2 L Q. 6.5 n+ ln 2 + t 2 L Q =0, a.s.. n+ lim inf t 2λ ln 2 t t 2 L Q t 6 + ε, a.s.. The proof is completed upon sending ε to 0. 30

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33 DAVAR KHOSHNEVISAN THOMAS M. LEWIS ZHAN SHI Department of Mathematics Department of Mathematics L.S.T.A.- C.N.R.S. URA 32 University of Utah Furman University Université ParisVI Salt Lake City, UT 822 Greenville, SC Tour e étage U.S.A. U.S.A. 4 Place Jussieu davar@math.utah.edu Lewis Tom/furman@furman.edu Paris Cedex 05 FRANCE shi@ccr.jussieu.fr