On Functional Versions of the Arc-Sine Law

Transcription

1 J Theor Probab (2) 23: 9 26 DOI.7/s On Functional Versions of the Arc-Sine Law István Berkes Siegfried Hörmann Lajos Horváth Received: 2 April 28 / Revised: July 28 / Published online: 2 August 28 Springer Science+Business Media, LLC 28 Abstract Let X,X 2,...be a sequence of random variables. Let S k = X + +X k and assume that S k /b k converges in distribution for some numerical sequence (b k ). We study the weak convergence of the random processes {Λ n (z), z R}, where Λ n (z) = n I Sk z. b k We consider the same problem when the normalized partial sums S k /b k are replaced by other functionals of the sequence (X n ). In particular, we investigate the case of sample extremes in detail. Keywords Arc-sine law Invariance principles Partial sums Extremes Mathematics Subject Classification (2) 6F7 6G7 6G5 6G52 6G7 Research of I. Berkes supported by NSF grant DMS6467 and OTKA grants K652 and K6796. Research of L. Horváth supported by NSF grant DMS6467. I. Berkes Institute of Statistics, Graz University of Technology, 8 Graz, Austria berkes@stat.tugraz.at S. Hörmann ( ) L. Horváth Department of Mathematics, University of Utah, Salt Lake City, UT 842, USA hormann@math.utah.edu L. Horváth horvath@math.utah.edu

2 J Theor Probab (2) 23: 9 26 Introduction One of the classical results on Brownian motion is the following arc-sine law of Lévy [4]. Let = d denote equality in distribution, and let A ρ,<ρ<, denote the generalized arc-sine distribution with density a ρ (x) = sin πρ π xρ ( x) ρ, <x<. Theorem A (Lévy s first arc-sine law) Let {W(t), t } be a standard Brownian motion. Then I { W(t) } dt d = A /2. If X,X 2,... are independent and identically distributed (i.i.d.) random variables with EX = and EX 2 =, then by Donsker s invariance principle (cf. [3]) we obtain that d I {S k } A /2, (.) n where S k = X + +X k, and d denotes convergence in distribution. Due to the strong dependence between the partial sums S k, the law of large numbers for the averages of I{S k } does not hold. However, Erdős and Hunt [] observed that if the Cesàro averages are replaced with logarithmic averages, then we have log n k I{S k } /2 a.s. (.2) Recently, the following more general result has been proved. Theorem B (Almost sure central limit theorem) Let X,X 2,...be i.i.d. random variables with EX = and EX 2 =. Then sup log n z R k I Sk z Φ(z) k where Φ(z) denotes the standard normal distribution function. a.s., (.3) Under E X 2+δ <, relation (.3) was proved independently by Brosamler [6] and Schatte [7]. Assuming only finite second moments, (.3) is due to Fisher [2] and Lacey and Philipp [3]. Theorem B was extended by Berkes and Csáki [2] to the logarithmic averages Λ n (z) = log n k I{ h k (X,...,X k ) z }, (.4)

3 J Theor Probab (2) 23: 9 26 where h k : R k R are measurable functions such that d h k (X,...,X k ) G (.5) with a nondegenerate distribution G. They showed that Λ n (z) converges a.s. if the dependence of h k (X,...,X k ) on its initial o(k) variables becomes negligible for k. This condition is satisfied for normalized partial sums, extremal statistics, U-statistics, ordered samples, local times, etc. A simple example when the analogue of (.3) fails is the Darling Erdős theorem for the maximum of normalized partial sums of i.i.d. random variables. Since the almost sure result (.2) has a distributional analogue (.), the question arises if the almost sure central limit theorem (.3) has a weak convergence version, i.e., if the sequence of processes n Sk I z k converges weakly. The same question arises for the averages Λ n (z) := n I { h k (X,...,X k ) z }, (.6) where h k : R k R are functionals such that (.5) holds. Since we can write where Λ n (z) = I { R n (t) z } dt, R n (t) = h k (X,...,X k ) if t [ k/n,(k + )/n ), k n, (.7) one might expect that the weak convergence of R n (t) to some limiting process R(t) (i.e., the functional version of (.5)) implies the weak convergence of Λ n (z). However, as we will show in Example 2. below, even the much stronger assumption R n (t) R(t) a.s. for all t [, ] is in general not sufficient to imply the weak convergence of Λ n (z) to I{R(t) z} dt. A weak invariance principle for R n(t) implies only the convergence of the finite-dimensional distributions of Λ n (z), and to obtain the weak convergence of Λ n (z), we have to make additional conditions on the path properties of the limit process R(t), see Theorem 2.2. For example, the weak convergence holds for normalized partial sums h k (X,...,X k ) = S k /b k of i.i.d. random variables, martingale difference sequences, and stationary Gaussian sequences, while Theorems 2.3 and 2.4 yield convergence criteria when the limit process R(t) is a general Gaussian or normalized Lévy process. To give a nonlinear example, we show that Λ n (z) converges weakly in the case of extreme-value statistics of i.i.d. random variables. Our paper is organized as follows. In Sect. 2 we state our main results and consider several examples. We also study the distribution of Λ(z) when z is fixed. We provide

4 2 J Theor Probab (2) 23: 9 26 a formula for the first and second moments of Λ(z) when the weak limit of R n (t) is Gaussian. In case of extreme values, we provide explicit expressions for EΛ p (z), p =, 2,...Section 3 contains the proofs. 2 Results Before we state our results, we introduce some basic notation. As usual, D[a,b] denotes the space of càdlàg functions defined on the interval [a,b], i.e., all functions x :[a,b] R such that lim t t x(t) = x(t ) and lim t t x(t) exists. The symbol D[a,b] indicates the weak convergence of random elements in D[a,b] with respect to the Skorokhod topology (cf. [3]). Note that if x D[a,b] and if [c,d] [a,b], then x can be considered as an element of D[c,d] by restricting its domain of definition. If x D[a +, b ] for every <<(a+ b)/2, then we say that x D(a,b). The convergence in D(a,b) means the convergence in D[a +, b ] for all <<(a+ b)/2. Since Λ n (z) and Λ(z) are defined on R, we will also consider the convergence Λ n (z) D[R] Λ(z). This means that there exists a strictly increasing and continuous distribution function T such that Λ n (T (u)) D[,] Λ(T (u)). At the endpoints u = and u =, we set T () = and T () =. Using the fact that Λ n and Λ are (random) distribution functions, we assign the values and when the argument is or. We can assume without loss of generality that all random variables and processes introduced so far (and later on) can be defined on a common probability space (Ω, A,P).If(S, S) is a measurable space and R : Ω S is A S measurable, then P induces a probability measure on (S, S) denoted by PR and defined as usual by PR (A) = P(R A). For some random processes {R n (t), t (, )} in D(, ), let Λ n (z) = I { R n (t) z } dt. (2.) If R n (t) is defined as in (.7), then Λ n (z) reduces to the averages in (.6). The basic assumption throughout this paper will be the existence of some limiting process {R(t), t (, )} in D(, ) such that R n (t) D(,) R(t). (2.2) A typical limit is R(t) = t α W(t), where α /2, and W(t) is standard Brownian motion. Note that this R(t) is not an element in D[, ], and this is why we assume the weak convergence in D(, ). Put Λ(z) = I { R(t) z } dt. (2.3) Our first theorem gives the convergence of the finite-dimensional distributions.

5 J Theor Probab (2) 23: Theorem 2. Let Λ n (z) and Λ(z) be given as in (2.) and (2.3). Assume that (2.2) holds, and let z,...,z M be arbitrary real numbers. Then whenever { Λn (z ), Λ n (z 2 ),...,Λ n (z M ) } d { Λ(z ), Λ(z 2 ),..., Λ(z M ) } (2.4) P ( R(t) = z j ) = for any t (, ) and j M. (2.5) Theorem 2. provides conditions which imply that the finite-dimensional distributions of the processes Λ n (z) converge to Λ(z). If we want to extend this result to Λ n (z) D[R] Λ(z), we need to show that the sequence {PΛ n,n } is tight. As the following example shows, relation (2.2) in general does not imply Λ n (z) D[R] Λ(z), even if the finite-dimensional distributions converge. Example 2. In this example, R n and R have values in [, ], hence it is sufficient to consider Λ n and Λ as elements in D[, ] instead of D[R]. Letξ be uniform in [/3, 2/3] and define for n 3, 2t if t [,(ξ n )/2); ξ n if t [(ξ n R n (t) = )/2,ξ); ξ + n if t [ξ,(ξ + n + )/2); 2t if t [(ξ + n + )/2, ]. Obviously R n (t) converges pointwise to R(t), where Now we have Λ n (z) = 2t if t [,ξ/2); R(t) = ξ if t [ξ/2,(ξ + )/2); 2t if t [(ξ + )/2, ]. z/2 if z [,ξ I { R n (t) z } n ); dt = ξ if z [ξ n,ξ + n ); (z + )/2 if z [ξ + n, ]; and Λ(z) = I { R(t) z } dt = { z/2 if z [,ξ); (z + )/2 if z [ξ,]. It follows that Λ n (z) f.d.d. Λ(z), butλ n (z) does not converge in D[, ].

6 4 J Theor Probab (2) 23: 9 26 Due to the fact that the functions Λ n (z) and Λ(z) are monotone and bounded, the tightness solely depends on properties of the limiting process {R(t), t (, )}.This observation is made more precise in the next theorem. Theorem 2.2 Let Λ n (z) and Λ(z) be given as in (2.) and (2.3). Assume that (2.2) holds and that P ( R(t) = z ) = for any t (, ) and z R. (2.6) If for all <</2, the sample paths of Λ (z) = are continuous on R, then Λ n (z) D[R] Λ(z). I { R(t) z } dt (2.7) In order to apply Theorem 2.2, we need criteria to check the continuity of Λ (z). For example, using Kolmogorov s classical continuity criterion (see, e.g., [3]), it suffices to verify sup E Λ (z + h) Λ (z) 2 z R ( = sup E I { R(t) [z, z + h] }) 2 = o ( h +β) for h (β > ). (2.8) z R Using Theorem 2.2 and this criterion, we obtain a simple integral criterion when R(t) is a Gaussian process. Theorem 2.3 Let Λ n (z) and Λ(z) be given as in (2.) and (2.3). Assume that (2.2) holds. Further assume that the limiting process {R(t), t (, )} in (2.2) is Gaussian with σ s = ER 2 (s) and ρ(s,t) = Corr(R(s), R(t)). If for every >, then Λ n (z) D[R] Λ(z). t ds dt <, (2.9) σ s σ t ρ 2 (s, t) Theorem 2.3 contains many important examples. The following corollary highlights the most important special case, which motivated this paper. Corollary 2. Assume that X,X 2,...are i.i.d. with EX = and EX 2 =. Then Sk D[R] W(t) I z I z dt, n k t where {W(t), t } is a standard Brownian motion.

7 J Theor Probab (2) 23: Corollary 2. can be extended to a large class of dependent random variables. We illustrate this fact by the following: Corollary 2.2 Assume that X,X 2,... is a martingale-difference sequence with finite second moments. Let F n = σ(x k,k n) and define σn 2 = E[X2 n F n ]. Further set Bn 2 = Var(S n) and sn 2 = σ 2 + +σ n 2. Assume that s2 n /B2 P n and that Bn 2 = nα L(n), where L is a slowly varying function at infinity, and α is a positive constant. Finally let {W(t), t } be a standard Brownian motion. If the Lindeberg condition EXk 2 I{ Xk 2 } B2 k for all > holds, then we have B 2 n n Sk I z B k D[R] { W(t α } ) I t α/2 z dt. As another application of Theorem 2.3, we consider partial-sum processes of strongly dependent sequences converging to a fractional Brownian motion. Recall that a fractional Brownian motion is a Gaussian process {B H (t), t } with EB H (t) = and Cov ( B H (s), B H (t) ) = 2( t 2H + s 2H (t s) 2H ), s t. The constant H (, ) is called the Hurst index. We note that B /2 (t) is a standard Brownian motion. Corollary 2.3 Let X,X 2,...be a stationary Gaussian sequence with EX = and assume that for some <H <, Var(S n ) = n 2H L 2 (n) for n, where L is a slowly varying function at infinity. Then Sk I n k H L(k) z D[R] BH (t) I t H z dt, where B H (t) is a fractional Brownian motion with Hurst index H. In all our examples so far, the limit process R(t) was a Gaussian process. To consider different situations, let X,X 2,... be i.i.d. random variables belonging to the domain of attraction of a stable distribution G α, i.e., assume that there are constants d a n and b n such that S n /b n a n G α. As is known, the characteristic function ϕ of G α has the form ( ϕ(t) = exp {iγt c t α + iβ t t ω(t,α) )},

8 6 J Theor Probab (2) 23: 9 26 where tan( πα 2 ) if α, ω(t,α) = 2 log t if α =, π and c,α,β, and γ are constants (γ is a real, c, <α 2, β ). We refer to α as the characteristic exponent of the stable law. Note that if α, we can assume without loss of generality that a n = (cf. Feller [], Theorem 3, p. 58), and hence P(S n ) G α (). Letting ρ = G α (), it follows from a classical result of Spitzer [9] (see Doney [8] for an improvement) that n n d I{S k } A ρ, and thus the occupation time distribution for the sequence (X n ) is the generalized arc-sine D[,] distribution. By Skorohod [8] wehaves [nt] /b n L α (t), where {L α (t), t } is a stable process with L α () having distribution G α. (For the properties of stable processes, we refer to Borodin and Ibragimov [5].) As a simple calculation shows, in the case of the partial-sum functional h k (X,...,X k ) = S k /b k, the limiting process in (2.2) is the normalized stable process L α (t)/t /α. The following theorem extends Theorem 2.3 to the case where, instead of a Gaussian process, the limiting process R(t) in (2.2) is a normalized Lévy process. Theorem 2.4 Let Λ n (z) and Λ(z) be given as in (2.) and (2.3) and assume that (2.2) holds. Further assume that R(t) = L(t)/b t, where L(t) is a Lévy process, and b t is a normalizing function such that the distribution of R(t) does not depend on t. Finally assume that l(z) = dp(r(t) z)/dz exists and is uniformly bounded. Then if for every >, ( t ) b t ds dt <, (2.) b t s we have Λ n (z) D[R] Λ(z). Corollary 2.4 Assume that X,X 2,... are i.i.d. random variables. Assume that S n /b n converges weakly to some stable distribution with characteristic exponent α. Then n Sk I z b k D[R] Lα (t) I t /α z dt, where L α (t) is a stable process with characteristic exponent α. Motivated by the arc-sine law and the almost sure central limit theorem, all our examples so far were related to partial-sum processes. In what follows, we give an example involving nonlinear functionals of the sequence {X n }.LetX,X 2,... be i.i.d. random variables with common distribution function F(x)and set M n = max{x,x 2,...,X n }.

9 J Theor Probab (2) 23: If P(M n /b n a n x) converges to some nondegenerate limiting distribution G(x) ( is attracted to G ), then by the classical theory, G belongs to one of three different types of limiting distributions. As an illustration, we consider here the case where {, x ; G(x) = exp( λx α ), x > (λ, α > ). If M n is attracted to the G above, then we can choose the constants as a n = and b n = inf F(t) /n t (2.) (2.2) (see, e.g., Bingham et al. [4], Sect. 8.3). Dwass [9] and Resnick [5] proved that M nt /b n D[,] E(t), (2.3) where {E(t),t [, ]} is the extremal process of type G. The finite-dimensional distributions of E(t) can be characterized as follows: If = t <t <t 2 < < t k, then ( E(t ),...,E(t k ) ) d ( = U, max{u,u 2 },...,max{u,u 2...,U k } ), (2.4) where U,...,U k are independent with P(U i x) = G t i t i (x). Theorem 2.5 Assume that X,X 2,... are i.i.d. random variables and define M n = d max{x,...,x n }. Assume further that M n /b n G, where G is given in (2.), and b n is given in (2.2). Then Mk D[R] E(t) I z I n b k t /α z dt, where E(t) is an extremal process of type G. 2. Distribution of the Limiting Process The purpose of this section is to investigate the distribution of the random variables Λ(z) defined in (2.3). In the case of partial sums of i.i.d. random variables with mean and variance, R(t) in (2.2) equals W(t)/ t, and Λ(z) = I { W(t)/ t z } dt. (2.5) By Lévy s arc-sine law we have Λ() d = A /2, but computing the distribution of Λ(z) for z seems to be a much harder problem. The combinatorial approaches used by Andersen [] and Spitzer [9] seem not to carry over to this case. In this section we will compute EΛ p (z) for some special functionals h k in (.7). Our first observation is that if the distribution of R(t) does not depend on t, then EΛ(z) = P(R(/2) z).

10 8 J Theor Probab (2) 23: 9 26 Fig. The graph of EΛ 2 (z) EA 2 Φ(z) shows that for z, Λ(z) has not the same distribution as A Φ(z), although the differences are very small Lemma 2. Assume that {R(t), t (, )} is a Gaussian process with ER(t) =, R 2 (t) =, and Corr(R(s), R(t)) = ρ(s,t). Then EΛ 2 (z) = n= γ n ( Φ (n) (z) ) 2, n! where γ n = ρn (s, t) ds dt, and Φ (n) (z) is the nth derivative of the standard normal distribution function. Returning to the special case (2.5), a reasonable conjecture is that Λ(z) is generalized arc-sine. Since in this case we have EΛ(z) = Φ(z), this would mean that Λ(z) = d A Φ(z). In order to get evidence for this conjecture, we used Lemma 2. and computed EΛ 2 (z) and compared it with EA 2 Φ(z) = Φ(z)( + Φ(z))/2. If the conjecture were true, the functions have to coincide. However, the computations (see Fig. ) disproves this conjecture, although the difference is very small. It remains open whether it is possible to extend Lemma 2. and get explicit representations for higher moments. The following lemma gives such a formula for the moments of Λ(z) = {E(t)/t/α z} dt, when E(t) is the extremal process of type G with G given in (2.). In this case the moments uniquely determine the distribution of Λ(z) (cf. Feller [], Sect. VII.3]). Lemma 2.2 Let G be given as in (2.), and let E(t) be an extremal process of type G. Let Λ(z) = I{E(t)/t/α z} dt. Then if p 2, p 2 EΛ p (z) = G p (z)(p )! k= u k e uλz α du. 3 Proofs 3. Convergence of the Finite-Dimensional Distributions and Tightness Lemma 3. Assume that (.7) and (2.2) hold. For <</2, define Λ (z) as in (2.7) and set Λ n (z) = I{R n (t) z} dt.

11 J Theor Probab (2) 23: (A) In order to prove (2.4), it is enough to show that, for all <</2, { Λ n (z ), Λ n (z 2),...,Λ n (z M) } d { Λ (z ), Λ (z 2 ),..., Λ (z M ) }. (3.) (B) In order to prove Λ n (z) D[R] Λ(z), it is enough to show that, for all <</2, Λ D[R] n (z) Λ (z). Proof We will show part (B); the proof of part (A) is similar but easier. Recall that Λ D[R] n (z) Λ (z) means Λ n (T (u)) D[R] Λ (T (u)) for some strictly increasing distribution function T. Obviously, ( sup Λ n T (u) ) ( Λ n T (u) ) 2 and u [,] sup u [,] Λ ( T (u) ) Λ ( T (u) ) 2. Hence, for every bounded and uniformly continuous function f : D[, ] R and every η>, there is some > such that Ef (Λ n T ) η + Ef (Λ n T ). Thus by the weak convergence of Λ n and the Portmanteau theorem we have lim sup n Ef (Λ n T ) η + Ef (Λ T ) 2η + Ef (ΛT ). Similarly we obtain lim inf n Ef (Λ n T ) 2η + Ef (ΛT ). Since these estimates hold for every η>, the statement is proven. Proof of Theorem 2. Let (z,...,z M ) R M be such that (2.5) holds, and let (λ,...,λ M ) R M.Let<</2 and consider the mapping τ : D[, ] R with M τ (x) = λ i I x(t) z i dt. i= If x n and x are elements of D[, ] with x n x in the topology of D[, ], we know that x n (t) x(t) for almost all t [, ]. Hence I{x n (t) z} I{x(t) z} for {t [, ]:t N c x(t) z} (N is a set of Lebesgue measure depending on x n and x, and N c is its complement). This shows that x n x in the topology of D[, ] implies τ (x n ) τ (x), provided that the time x(t) spends in {z,...,z M } has Lebesgue measure, i.e., I { x(t) {z,...,z M } } dt =. Let ρ denote the Skorokhod metric on D[, ]. Since R D(, ), we can consider R as an element of D[, ] by restricting its domain of definition. We showed that τ is PR a.s. ρ -continuous if PR {x D[, ] I { x(t) {z,...,z M } } } dt> =,

12 2 J Theor Probab (2) 23: 9 26 which is { P I { R(t) {z,...,z M } } } dt> =. Since the last relation is implied by (2.5), by the mapping theorem (see Billingsley [3], Theorem 2.7), we have τ (R n ) τ (R). Since this is true for every d (λ,...,λ M ) R M, we get (3.) by the Cramér Wold theorem. Hence the result follows from part (A) of Lemma 3.. Proof of Theorem 2.2 Theorem 2., (2.6), and Lemma 3. yield that Λ n Λ (z). Hence, according to part (B) of Lemma 3., it is enough to show the tightness of Λ n (Φ (u)), u, where Φ(x) is the standard normal distribution function. (Instead of Φ(x), we can use any continuous and strictly increasing distribution function.) The tightness will follow if we show that for every δ>, lim K { lim sup P n (z) f.d.d. sup Λ ( n Φ (u) ) Λ ( n Φ (v) ) } >δ =. (3.2) u,v u v /K Since Λ n (Φ (u)) Λ n (Φ ()) = a.s.(u ) for all η>, there is a (, ) such that P { Λ n( Φ (a) ) >δ/6 } η. (3.3) Similarly, Λ n (Φ (u)) Λ n (Φ ()) = 2 a.s. (u ), and therefore, for all η>, there is some b (, ) such that P { 2 Λ n( Φ (b) ) >δ/6 } η. (3.4) We note that ( sup Λ n Φ (u) ) Λ n( Φ (v) ) u,v u v /K 2Λ n( Φ (a) ) + 2 ( 2 Λ ( n Φ (u) )) + 2 sup Λ ( n Φ (u) ) Λ ( n Φ (v) ). a u,v b u v /K Let u i = a + i(b a)/k, i K. Now sup Λ n( Φ (u) ) Λ n (Φ (v)) a u,v b u v /K ( 2 max Λ n Φ (u i+ ) ) Λ n( Φ (u i ) ). i K

13 J Theor Probab (2) 23: By the convergence of the finite-dimensional distributions we have { Λ n( Φ (a) ),Λ n( Φ (b) ) (, max Φ (u i+ ) ) Λ n( Φ (u i ) ) } { d Λ n i K Λ ( Φ (a) ),Λ ( Φ (b) ), max i K Λ ( Φ (u i+ ) ) Λ ( Φ (u i ) ) }. By the almost sure continuity of Λ (z) on [Φ (a), Φ (b)] we get that, for all δ>, { lim P max Λ ( Φ (u i+ ) ) Λ ( Φ (u i ) ) } >δ/6 =. K i K Hence, combining the last relation with (3.3) and (3.4), we have lim K { lim sup P n sup u,v u v /K ( Λ n Φ (u) ) Λ ( n Φ (v) ) } >δ 2η. Since this is true for any η>, relation (3.2) is proven. 3.2 Proof of the Main Results Proof of Theorem 2.3 Since R(t) is Gaussian, P(R(t) z) is a continuous function of z for every t (, ). By Theorem 2.2 and the continuity condition (2.8) itis enough to show that ( E = ) 2 I{R(t) [z, z + δ]} dt P (( R(s),R(t) ) [z, z + δ] 2) ds dt = o ( δ +η), (3.5) uniformly in z and with some η>. The density f s,t (x, y) of the vector (R(s), R(t)) is bivariate normal, and hence f s,t (x, y) 2πσ s σ t ρ 2 (s, t) for all (x, y) R 2. This obviously implies that P ( (R(s), R(t)) [z, z + δ] 2) δ 2 ( 2πσ s σ t ρ 2 (s, t)). Hence, if (2.9) holds, then (3.5) follows if we choose η</2. Thus we have Λ (z). Since the argument is true for all <</2, Theorem 2.3 follows from Lemma 3.. Λ D[R] n (z)

14 22 J Theor Probab (2) 23: 9 26 Proof of Theorem 2.4 We proceed as in the proof of Theorem 2.3. Assume that s<t. From our assumptions we get for f s,t (x, y), the joint density of (R(s), R(t)), that f s,t (x, y) = l(x) b t l(b t y/b t s b s x/b t s ) sup l(x) 2 b t. b t s b t s x R Now the result follows from (2.). Proof of Theorem 2.5 First we note that b n = n /α L(n), where L(n) is slowly varying as n (cf. Bingham et al. [4], Theorem 8.3.2]). Hence M nt /b nt D(,) E(t)/t /α. Note that the distribution function of the limiting process does not depend on t and is continuous. By Theorem 2. we get I{M k /b k z}= I { M nt /b nt z } f.d.d. I { E(t)/t /α z } dt. n Our goal is again to establish (2.8) with R(t) = E(t)/t /α. It can be easily seen that it is sufficient to show that, for every h> and for sufficiently small η>, t sup z h P ( E(s)/s /α [z, z + δ], E(t)/t /α [z, z + δ] ) ds dt = o ( δ +η) (3.6) as δ. If h =, then (2.8) trivially follows from (3.6) because of R(t). Since the distribution of R(t) (which does not depend on t) is continuous, it is sufficient to show (3.6) for every h>. Relation (2.4) shows that, for s<t, we have P ( E(s) x,e(t) y ) { G s (x)g t s (y) if x y; = G s (y)g t s (y) if x>y. We assume throughout that s t. Since λ in (2.) has no influence on the following arguments, we assume for simplicity that λ =. Now we have to distinguish between the cases (i) (z + δ)s /α >zt /α and (ii) (z + δ)s /α zt /α. Under assumption (i), we get by some algebra that P ( E(s)/s /α [z, z + δ], E(t)/t /α [z, z + δ] ) = [ G t s( (z + δ)t /α) G t s( zt /α)] [ G s( (z + δ)s /α) G s( zs /α)] + [ G s( (z + δ)s /α) G s( zt /α)] G t s( zt /α) =: H (s,t,z,δ)+ H 2 (s,t,z,δ). Under (ii), we have the simpler relation P ( E(s)/s /α [z, z + δ], E(t)/t /α [z, z + δ] ) = H (s,t,z,δ).

15 J Theor Probab (2) 23: Thus (3.6) will follow if, for every h>, ( t H i (s,t,z,δ)ds r i sup z h ) dt = o ( δ +η) (i =, 2), (3.7) where r = and r 2 = t( z+δ z )α. Using the mean-value theorem and the explicit form of G, we obtain t t H (s,t,z,δ)dt δ e ( s/t)(z+δ) α e ( s/t)(z+δ) α dt = δt ( L(z + δ) L(z) ) with L(z) = z α ( G(z)). Observe that sup z L(z + δ) L(z) =O(δ α ). Thus, if η is chosen sufficiently small, we get (3.7) when i =. Next we consider H 2. A straightforward calculation shows that H 2 (s,t,z,δ)ds {(z/(z+δ)) α s/t } [ = t z α( G(z + δ) G(z) ) ( ( ) z α ) ] G(z). z + δ First we assume that α. By the mean-value theorem we get that ( ) z α αδ(z + δ) z + δ and z α( G(z + δ) G(z) ) δz α G (z) = αδz G(z). Since G(z) and z h, we obtain by combining our previous results that H 2 (s,t,z,δ)ds dt 2 αδ( z (z + δ) ) = O ( δ 2). sup z h {(z/(z+δ)) α s/t } The case α> can be treated similarly. Proof of Corollaries 2. and 2.2 We only have to prove Corollary 2.2, which is the more general one. We set B = and define ξ n (t) = S k B n if t [ Bk 2 ) /B2 n,b2 k+ /B2 n, k n. Brown [7] showed that under the assumptions of Corollary 2.2, the polygonal line process obtained by connecting the points of discontinuity of ξ n (t) with straight lines converges to a standard Brownian motion. The argument there can be easily transposed to ξ n (t) D[,] W(t).Let η n (t) = S k B n if t [ k/n,(k + )/n ), k n.

16 24 J Theor Probab (2) 23: 9 26 Hence η n (t) = ξ n (g n (t)), where g n (t) = B tn 2 /B2 n = ( nt /n)α L( nt )/L(n). By the uniform convergence theorem for regularly varying functions (cf. Bingham et al. [4], p. 6) sup t [,] g n (t) t α (n ). In order to prove that the measures generated by the processes {ξ n (t), t [, ]} are tight, Brown [7] showed that, for every >, we have lim δ ( lim sup P n sup s<t s t <δ ξ n (t) ξ n (s) ) >. (The argument remains valid for the broken line process.) This immediately implies that sup t [,] ξ n (t α P ) η n (t), respectively η n (t) D[,] W(t α ).Nowweset R n (t) = η n (t) B nt B n = S k B k if t [ k/n,(k + )/n ), k n. It is clear that R n (t) D(,) W(t α )/t α/2. Hence the limiting process R(t) = W(t α )/t α/2 is Gaussian with σ s = and Corr 2 (R(s), R(t)) = (s/t) α,fors t. Since condition (2.9) holds for every α>, the result follows from Theorem 2.3. Proof of Corollaries The proofs are the same as the proof of Corollary 2.2 with a corresponding invariance principle for the partial-sum process. In order to get the weak convergence of the partial sums in Corollary 2.3, we use Taqqu [2, Lemma 5.]. It remains to show (2.9). After a change of variables s = t h condition (2.9)is t dh dt <, (3.8) ρ 2 (t h, t) where ρ(t h, t) = [( h ) H ( + h ) H h 2H ] 2 t t (t h) H t H. It can be easily seen that the integrand in (3.8) is only large when h. By some routine analysis we obtain uniformly in t [, ] that ρ 2 (t h, t) ( ) h 2H as h. t Hence (3.8) holds. Finally, for Corollary 2.4 we use Theorems 2.4 and 2.4 in Borodin and Ibragimov [5]. The verification of (2.) is elementary and therefore omitted. 3.3 Distribution of the Limits Proof of Lemma 2. Assume that (X, Y ) are jointly Gaussian with Corr(X, Y ) = ρ. A convenient way to compute Ef (X)f 2 (Y ) is to use the Hermite expansions of f

17 J Theor Probab (2) 23: and f 2 (cf. Rozanov [6, p. 82]). We obtain where α (i) n with Ef (X)f 2 (Y ) = ρ n n! α() n α(2) n, n= are the Hermite coefficients of the corresponding functions f i, i.e., α (i) n = f i (x)φ(x)h n (x) dx (i =, 2) H n (x) = ( ) n e x2 /2 dn dx n e x2 /2, and φ(x) is the standard normal density function. (Of course, we need to assume that the Hermite expansions of f i exist.) Hence EI { R(s) z } I { R(t) z } = n Proof of Lemma 2.2 By (2.4)wehave EΛ p (z) = p! = p! tp... dt...dt p tp... dt...dt p = p!g(z) tp t t... ρ n (s, t) ( Φ (n) (z) ) 2. n! P ( E t zt /α,e t zt /α,...,e tp zt /α ) p G t ( zt /α t ) ( G t t /α) ( zt...g t p t p /α ) zt p e (t /t )λz α...e (t p 2/t p )λz α dt...dt p. The result can be easily obtained by changing variables s i = t i /t i. References. Andersen, E.: On the fluctuations of sums of random variables. II. Math. Scand. 2, (954) 2. Berkes, I., Csáki, E.: A universal result in almost sure central limit theory. Stoch. Process. Appl. 94, 5 34 (2) 3. Billingsley, P.: Convergence of Probability Measures. Wiley, New York (999) 4. Bingham, N., Goldie, C., Teugels, J.: Regular Variation. Encyclopedia of Mathematics and its Applications. Cambridge (987) 5. Borodin, A., Ibragimov, I.: Limit Theorems for Functionals of Random Walks. Proc. Steklov Inst. Math. (995) 6. Brosamler, G.A.: An almost everywhere central limit theorem. Math. Proc. Camb. Philos. Soc. 4, (988) 7. Brown, B.: Martingale central limit theorems. Ann. Math. Stat. 42, (97)

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