AN EXTREME-VALUE ANALYSIS OF THE LIL FOR BROWNIAN MOTION 1. INTRODUCTION

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1 AN EXTREME-VALUE ANALYSIS OF THE LIL FOR BROWNIAN MOTION DAVAR KHOSHNEVISAN, DAVID A. LEVIN, AND ZHAN SHI ABSTRACT. We present an extreme-value analysis of the classical law of the iterated logarithm LIL for Brownian motion. Our result can be viewed as a new improvement to the LIL.. INTRODUCTION Let Bt t be a standard Brownian motion. The law of the iterated logarithm LIL of Khintchine 9 states that lim t 2t lnln t /2 Bt = a.s. Equivalently,. With probability one, Bs as t. 2s lnln s The goal of this note is to determine the rate at which this convergence occurs. We consider the extreme-value distribution function Resnick, 987, p. 8,.2 Λx = exp e x x R. Also, we place L k x or L k x in favor of the k-fold, iterated, natural logarithm, ln lnx k times. Then, our main result is as follows: Theorem.. For all x R,..4 lim P t lim P t 2L 2 t 2L 2 t Bs 2sL2 s 2 L t + L 4 t + ln Bs 2sL2 s 2 L t + L 4 t + ln 2 2 The preceding is accompanied by the following strong law: Theorem.2. With probability one, L 2 t.5 lim t L t Bs 2sL2 s = 4. 2 x = Λx, x = Λx. This should be compared with the following consequence of the theorem of Erdős 942: L 2 t Bs.6 lim t L t 2sL2 s = a.s. 4 [Erdős s theorem is stated for Bernoulli walks, but applies equally well and for the same reasons to Brownian motion.] Date: November 6, Mathematics Subject Classification. 6J65, 6G7, 6F5. Key words and phrases. law of the iterated logarithm, Brownian motion, exteme value, limit theorem. The research of D. Kh. was ported by a grant from the National Science Foundation.

2 2 D. KHOSHNEVISAN, D. A. LEVIN, AND Z. SHI Theorem. is derived by analyzing the excursions of the Ornstein Uhlenbeck process,.7 X t = e t/2 Be t t. Our method is influenced by the ideas of Motoo PROOF OF THEOREM. An application of Itô s formula shows us that the process X satisfies the s.d.e, 2. X t = X + expt s dbs 2 X s ds. The stochastic integral in 2. has quadratic variation expt s ds = t. Therefore, this stochastic integral defines a Brownian motion. Call the said Brownian motion W, to see that X satisfies the s.d.e., 2.2 dx = dw 2 X dt. In particular, the quadratic variation of X at time t is t. This means that the semimartingale local times of X are occupation densities Revuz and Yor, 999, Chapter VI. In particular, if L t X t denotes the local time of X at zero, then 2. L t X = lim ε 2ε X s ε ds a.s. See Revuz and Yor 999, Corollary.6, p Let τt t denote the right-continuous inverse-process to L X. By the ergodic theorem, τt/t a.s. converges as t diverges. In fact, τt 2.4 lim = 2π a.s. t t To this, note that τt/t t/l t X a.s. But another application of the ergodic theorem implies that L t X E[L t X ] a.s. The assertion 2.4 then follows from the fact that E[L t X ] = t/ 2π Horváth and Khoshnevisan, 995, Lemma.2. Define X s 2.5 ε t = t e. 2ln s s τt Lemma 2.. Almost surely, 2.6 ε M j n = O 2ln j ln n L 2 n n where 2.7 = X s s [τj,τj +] j. n, Proof. According to 2.4, 2.8 lnτj ln s 2ln j L 2 j j ln j s [τj,τj +] j.

3 KHINTCHINE S LIL AND EXTREMES On the other hand, according to. and 2.4, almost surely, 2.9 = O lnτj + = O ln j j. The lemma follows from a little algebra. Lemma 2., and monotonicity, together prove that Theorem. is equivalent to the following: For all x R, 2ln n 2ln j 2 L 2n + L n + ln 2 x = Λx. 2. lim n P We can derive this because: i By the strong Markov property of the OU process X, M,M 2,... is an i.i.d. sequence; and ii the distribution of M can be found by a combining a little bit of stochastic calculus with an iota of excursion theory. In fact, one has a slightly more general result for Itô diffusions i.e., diffusions that solve smooth s.d.e. s at no extra cost. Proposition 2.2. Let Z t t denote the regular Itô diffusion on, which solves the s.d.e. 2. dz t = σz t dw t + az t dt, where σ, a C R, σ is bounded away from zero, and w t t is a Brownian motion. Write θ t t for the inverse local-time of Z t t at zero. Then for all λ >, 2.2 P Z t λ t [,θ ] Z f = = exp 2 f λ f. Proof. The scale function of a diffusion is defined only up to an affine transformation. Therefore, we can assume, without loss of generality, that f = and f = ; else, we choose the scale function x f x f /f instead. Explicitly, Z t t has the scale function Revuz and Yor, 999, Exercise VII f x = x exp 2 Owing to Itô s formula, N t := f Z t satisfies y au σ 2 u du d y. 2.4 dn t = f Z t σz t dw t = f f N t σ f N t dw t, and so is a local martingale. According to the Dambis, Dubins Schwarz representation theorem Revuz and Yor, 999, Theorem V..6, p. 8, there exists a Brownian motion bt t such that 2.5 N t = bα t, where α t = αt = N t = [ f f N r ] 2 σ 2 f N r dr t. The process N is manifestly a diffusion; therefore, it has continuous local-time processes L x t N t,x R which satisfy the occupation density formula Revuz and Yor, 999, Corollary VI..6, p. 224 and time-change. By 2., f >, and because σ is bounded away from zero, σ 2 f >. Therefore, the inverse process α t t exists a.s., and is uniquely defined by αα t = t for all t. Let L x t b t,x R denote the local-time processes of the Brownian motion b. It is well known Rogers and Williams, 2, Theorem V.49. that a.s. 2.6 L x t b t,x R = L x αt N t,x R ;

4 4 D. KHOSHNEVISAN, D. A. LEVIN, AND Z. SHI For completeness, we include a brief argument here. Thanks to the occupation density formula, 2.7 hxl x t b dx = hbs ds = hn α s ds, valid for all Borel-measurable functions h : R R. See 2.5 for the last equality. We can change variables [v = α s], and use the definition of α t in 2.5, to note that 2.8 α hxl x t t b dx = hn v dα v = α t hn v d N v = hxl x α N dx. t This establishes 2.6. In particular, with probability one, 2.9 L t N = L αt b t. By 2. and 2.4, d Z t = [ f f N t ] 2 d N t, so if LZ denotes the local times of Z, then almost surely, 2.2 hxl x t Z dx = hz r d Z r = h f N r [ f f N r ] 2 d N r = h f y = hx Lf x t Ly t [ N f f y ] d [ f y ] N f dx x [x = f y]. This proves that a.s., f x L x x t Z = Lf t N. In particular, L t Z = L t N for all t, a.s. Thanks to 2.9, we have proved the following: Almost surely, 2.2 L t Z = L αt b t. Define 2.22 ϕ t = inf s > : L s b > t t. Then thanks to 2.2, 2.2 ϕ t = αθ t t. Thus, 2.24 P Z s λ s [,θ ] Z = = P N s f λ s [,θ ] N = = P b s f λ b =. s [,ϕ ] The last identity follows from 2.5, and the fact that α and α are both continuous and strictly increasing a.s.

5 KHINTCHINE S LIL AND EXTREMES 5 Define N β to be the total number of excursion of the Brownian motion b that exceed β by local-time. Then, P b s f λ b = = P N f λ = b = 2.25 s [,ϕ ] = exp E [ N f λ b = ], because N β is a Poisson random variable Itô, 97. According to Proposition.6 of Revuz and Yor 999, p. 492, E[N β b = ] = 2β for all β >. [See also Revuz and Yor 999, Exercise XII.4..] The result follows. Remark 2.. Also, the following equality holds: 2.26 P Z t λ t [,θ ] Z f = = exp. f λ f This follows as above after noting that f x = f x, and that E[N β b = ] = β, where N denotes the number of excursions of the Brownian motion b that exceed β in β absolute value by local-time. Proof of Theorem.. If we apply the preceding computation to the diffusion X itself, then we find that PM λ = exp /[2Sλ], where S is the scale function of X which satisfies S = and S =. According to 2.2 and 2., Sx = x expy2 /2 d y. Note that Sx x expx 2 /2 as x. Let β n x n= be a sequence which, for x fixed, satisfies β nx as n. We assume, in addition, that α n x := β n x/ln n goes to zero as n. We will press x in the notation and write α n and β n for α n x and β n x, respectively. A little calculus shows that if α n x >, then 2.27 P 2ln j + α n 2 = exp j =n 2S + α n /2 2ln j = exp + α n/2 ln j 2 j =n j +α n/2 2 = exp [ + o] + α n/2 ln x 2 n x dx +α n/2 2 = exp [ + o] + α n/2 ln n 2αn + α n /4α n n α n+α n /4 = exp [ + o] q n x ln n/2 e β n 2βn. Here, 2ln n + βn x 2.28 q n x = exp β2 n x. 2ln n + β n x/2 4ln n If α n x, then the probability on the right-hand side of 2.27 is. Define 2.29 ϕ n x := 2 L 2n L n ln / 2 + x,

6 6 D. KHOSHNEVISAN, D. A. LEVIN, AND Z. SHI and set β n x in 2.27 equal to ϕ n x. This yields 2. P + ϕ nx exp [ + o]cn xe x if ϕ n x >, = 2ln j 2ln n if ϕ n x, where [ 2ln n + ϕn x 2. c n x = exp ϕ2 n + L n ln/ ] 2 + x 2ln n + ϕ n x/2 4ln n 2 L. 2n Note that the little-o in 2. is uniform in x. If x R is fixed, letting n in 2. shows that 2.2 lim n P 2ln n 2ln j ϕ n x = exp e x. This proves 2., whence equation. of Theorem. follows. Using 2.26, we obtain also 2. P + α n = exp [ + o] 2 e β n ln n /2. 2ln j 2 Let β n x = 2 L 2n L n ln 2 + x in 2. to establish.4. 2 β n. PROOF OF THEOREM.2 In light of.6 it suffices to prove that L 2 t Bs. liminf t L t 2sL2 s 4 a.s. We aim to prove that almost surely,.2 2ln j > + c L 2n ln n eventually a.s. if c < 2 Theorem.2 follows from this by the similar reasons that yielded Theorem. from 2.. But.2 follows from 2.27:. P + c L 2n = exp + o 2ln j ln n 2c ln n/2 c. 2 L 2 n Replace n by ρ n where ρ > is fixed. We find that if c < /2 then the probabilities sum in n. Thus, by the Borel Cantelli lemma, for all ρ > and c < /2 fixed,.4 > + c L 2ρ n j ρ n 2ln j lnρ n eventually a.s. Equation.2 follows from this and monotonicity.

7 KHINTCHINE S LIL AND EXTREMES 7 4. AN EXPECTATION BOUND We can use our results to improve on the bounds of Dobric and Marano 2 for the rate of convergence of E[ B s 2sL 2 s /2 ] to. Proposition 4.. As t, [ ] Bs 4. E = + L t 2sL2 s 4 L 2 t L 4 t 2 L 2 t + γ ln / 2 2 L 2 t where γ.5772 denotes Euler s constant. Proof. Define 4.2 U n := 2ln n 2ln j + o, L 2 t 2 L 2n + L n + ln / 2. We have shown that U n converges weakly to Λ. We now establish that n E [ U 2 n] <. This implies uniform integrability, whence we can deduce that E[U n ] x dλx. Let ϕ n x be as defined in 2.29, and note that x n := /2L 2n L n ln/ 2 solves ϕ n x n =. Recalling the definition of c n x in 2. and rewriting 2. shows that for x > x n 4. P U n x = exp [ + o]c n xe x. Consequently, for n large enough, 4.4 xpu n x x n x exp 2 c n xe x dx. For n sufficiently large and x < xn, c n x e 2 //2 /2. Thus, for n sufficiently large, x n 4.5 xpu n x xe 24 ex dx xe 24 ex dx <. Also for n sufficiently large, 4.6 xpu n > x x e 2 c nxe x dx 2 xc nxe x dx. We can get the easy bound for x >, c n x /2 + x//4 = 6 + 4x, yielding 4.7 xpu n > x dx Now we can write 4.8 E [ Un 2 ] = 2 xp U n > x dx = 2 2 x 6 + 4x e x dx <. xpu n > x dx + 2 xpu n < x dx. We have just show that the two terms on the right-hand side are bounded uniformly in n, which establishes uniform integrability. From Lemma 2., it follows that 4.9 2L 2 t E [ B s 2sL2 s ] 2 L t + L 4 t + ln 2 x dλx. It suffices to prove that x dλx = γ. But this follows because [ ] 4. x d e e x = lnte t dt = d t z e t dt dz = Γ z z= Γz. z=

8 8 D. KHOSHNEVISAN, D. A. LEVIN, AND Z. SHI Cf. Davis 965, Equation AN APPLICATION TO RANDOM WALKS Let X,X 2,... be i.i.d. random variables with 5. E[X ] =, VarX =, and E [ X 2 L 2 X e e] <. Let S n = X + + X n n denote the corresponding random walk. Then, according to Theorem 2 of Einmahl 987, there exists a probability space on which one can construct S n n= together with a Brownian motion B such that S n Bn 2 = on/l 2 n a.s. On the other hand, by the Borel Cantelli lemma, Bt Bn 2 = on/l 2 n uniformly for all t [n, n+] a.s. These remarks, and a few more lines of elementary computations, together yield the following. Proposition 5.. If 5. holds, then for all x R, lim P n lim P n 2L 2 n 2L 2 n k n k n S k 2kL2 k 2 L n + L 4 n + ln S k 2kL2 k 2 L n + L 4 n + ln x = Λx, x = Λx. Remark 5.2. It would be interesting to know if the preceding remains valid if only E[X ] = and VarX =. We believe the answer to be, No. REFERENCES Davis, Philip J Gamma Function and Related Functions, Handbook of Mathematical Functions M. Abramowitz and I. Stegun, eds., Dover, New York, pp Dobric, Vladimir and Lisa Marano. 2. Rates of convergence for Lévy s modulus of continuity and Hinchin s law of the iterated logarithm, High Dimensional Probability III J. Hoffmann-Jørgensen, M. Marcus, and J. Wellner, eds., Progress in Probability, vol. 55, Birkhäuser, Basel, pp Einmahl, Uwe Strong invariance principles for partial sums of independent random vectors, Ann. Probab. 5, Erdős, Paul On the law of the iterated logarithm, Ann. Math. 4, Horváth, Lajos and Davar Khoshnevisan Weight functions and pathwise local central limit theorems, Stoch. Proc. Their Appl. 59, 5 2. Itô, Kyosi. 97. Poisson point processes attached to Markov processes, Proc. Sixth. Berkeley Symp. Math. Statis. Probab., vol., University of California, Berkeley, pp Khintchine, A. Ya. 9. Asymptotische Gesetz der Wahrscheinlichkeitsrechnung, Springer, Berlin. Motoo, Minoru Proof of the law of iterated logarithm through diffusion equation, Ann. Inst. Stat. Math., Resnick, Sidney I Extreme Values, Regular Variation, and Point Processes, Springer Verlag, New York. Revuz, Daniel and Marc Yor Continuous Martingales and Brownian Motion, Third Edition, Springer, Berlin. Rogers, L.C.G. and David Williams. 2. Diffusions, Markov Processes and Martingales, Second Edition, vol. 2, Cambridge University Press, Cambridge. DEPARTMENT OF MATHEMATICS, THE UNIVERSITY OF UTAH, 55 S. 4 E. SALT LAKE CITY, UT address: davar@math.utah.edu URL: davar

9 KHINTCHINE S LIL AND EXTREMES 9 DEPARTMENT OF MATHEMATICS, THE UNIVERSITY OF UTAH, 55 S. 4 E. SALT LAKE CITY, UT address: levin@math.utah.edu URL: levin LABORATOIRE DE PROBABILITEÉS, UNIVERSITÉ PARIS VI 4 PLACE JUSSIEU F PARIS CEDEX 5 FRANCE address: zhan@proba.jussieu.fr URL: