TRANSIENT NEAREST NEIGHBOR RANDOM WALK AND BESSEL PROCESS

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1 TRANSIENT NEAREST NEIGHBOR RANDOM WALK AND BESSEL PROCESS Endre Csáki Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, P.O.B. 27, H-364, Hungary. address: Antónia Földes 2 Department of Mathematics, College of Staten Island, CUNY, 2800 Victory Blvd., Staten Island, New York 034, U.S.A. address: foldes@mail.csi.cuny.edu Pál Révész Institut für Statistik und Wahrscheinlichkeitstheorie, Technische Universität Wien, Wiedner Hauptstrasse 8-0/07 A-040 Vienna, Austria. address: reveszp@renyi.hu Abstract: We prove strong invariance principle between a transient Bessel process and a certain nearest neighbor (NN) random walk that is constructed from the former by using stopping times. We show that their local times are close enough to share the same strong limit theorems. It is also shown, that if the dierence between the distributions of two NN random walks are small, then the walks themselves can be constructed on such a way that they are close enough. Finally, some consequences concerning strong limit theorems are discussed. AMS 2000 Subject Classication: Primary 60F7; Secondary 60F5, 60J0, 60J55, 60J60. Keywords: transient random walk, Bessel process, strong invariance principle, local time, strong theorems. Running head: NN random walk and Bessel process. Research supported by the Hungarian National Foundation for Scientific Research, Grant No. K 6052 and K Research supported by a PSC CUNY Grant, No

2 . Introduction In this paper we consider a nearest neighbor (NN) random walk, dened as follows: let X 0 = 0, X, X 2,... be a Markov chain with E i := P(X n+ = i + X n = i) = P(X n+ = i X n = i) (.) { if i = 0 = /2 + p i if i =, 2,..., where /2 p i /2, i =, 2,.... In case 0 < p i /2 the sequence {X i } describes the motion of a particle which starts at zero, moves over the nonnegative integers and going away from 0 with a larger probability than to the direction of 0. We will be interested in the case when p i B/4i with B > 0 as i. We want to show that in certain sense, this Markov chain is a discrete analogue of a continuous Bessel process and establish a strong invariance principle between these two processes. The properties of the discrete model, often called birth and death chain, its connections with orthogonal polynomials in particular, has been treated extensively in the literature. See e.g. the classical paper by Karlin and McGregor [3], or more recent papers by Coolen- Schrijner and Van Doorn [6] and Dette [9]. In an earlier paper [7] we investigated the local time of this Markov chain in the transient case. There is a well-known result in the literature (cf. e.g. Chung [5]) characterizing those sequences {p i } for which {X i } is transient (resp. recurrent). Theorem A: ([5], page 74) Let X n be a Markov chain with transition probabilities given in (.) with /2 < p i < /2, i =, 2,... Dene Then X n is transient U i := E i E i = /2 p i /2 + p i (.2) k U i <. k= i= As a consequence, the Markov chain (X n ) with p R B/4R, R is transient if B > and recurrent if B <. The Bessel process of order ν, denoted by Y ν (t), t 0 is a diusion process on the line with generator d 2 2 dx + 2ν + d 2 2x dx. 2

3 d = 2ν + 2 is the dimension of the Bessel process. If d is a positive integer, then Y ν ( ) is the absolute value of a d-dimensional Brownian motion. The Bessel process Y ν (t) is transient if and only if ν > 0. The properties of the Bessel process were extensively studied in the literature. Cf. Borodin and Salminen [2], Revuz and Yor [20], Knight [5]. Lamperti [6] determined the limiting distribution of X n and also proved a weak convergence theorem in a more general setting. His result in our case reads as follows. Theorem B: ([6]) Let X n be a Markov chain with transition probabilities given in (.) with /2 < p i < /2, i =, 2,... If lim R Rp R = B/4 > /4, then the following weak convergence holds: X [nt] n = Y (B )/2 (t) in the space D[0,]. In particular, ( ) lim P Xn < x = n n x u B e u2 /2 du. 2 B/2 /2 Γ(B/2 + /2) 0 In Theorems A and B the values of p i can be negative. In the sequel however we deal only with the case when p i are non-negative, and the chain is transient, which will be assumed throughout without mentioning it. Let and dene D(R, ) := + p R := j j= i= U R+i, (.3) 2 + p R D(R, ) = q R (.4) Now let ξ(r, ), R = 0,, 2,... be the total local time at R of the Markov chain {X n }, i.e. Theorem C: ([7]) For a transient NN random walk ξ(r, ) := #{n 0 : X n = R}. (.5) P(ξ(R, ) = k) = p R(q R) k, k =, 2,... (.6) Moreover, η(r, t), R > 0 will denote the local time of the Bessel process, i.e. t η(r, t) := lim I{Y ν (s) (R ε, R + ε)} ds, ε 0 2ε 0 η(r, ) := lim t η(r, t). 3

4 It is well-known that η(r, ) has exponential distribution (see e.g. [2]). P(η(R, ) < x) = exp ( ν ) R x. (.7) For 0 < a < b let τ := τ(a, b) = min{t 0 : Y ν (t) / (a, b)}. (.8) Then we have (cf. Borodin and Salminen [2], Section 6, 3.0. and 3.0.4). Theorem D: For 0 < a < x < b we have where P x (Y ν (τ) = a) = P x (Y ν (τ) = b) = x 2ν b 2ν, (.9) a 2ν b 2ν E x e ατ = S ν(b 2α, x 2α) + S ν (x 2α, a 2α) S ν (b 2α, a, (.0) 2α) S ν (u, v) = (uv) ν (I ν (u)k ν (v) K ν (u)i ν (v)), (.) I ν and K ν being the modied Bessel functions of the rst and second kind, resp. Here and in what follows P x and E x denote conditional probability, resp. expectation under Y ν (0) = x. For simplicity we will use P 0 = P, and E 0 = E. Now consider Y ν (t), t 0, a Bessel process of order ν, Y ν (0) = 0, and let X n, n = 0,, 2,... be an NN random walk with p 0 = p = /2, p R = (R ) 2ν R 2ν (R ) 2ν (R + ), R = 2, 3,... (.2) 2ν 2 Our main results are strong invariance principles concerning Bessel process, NN random walk and their local times. Theorem.. On a suitable probability space we can construct a Bessel process {Y ν (t), t 0}, ν > 0 and an NN random walk {X n, n = 0,, 2,...} with p R as in (.2) such that for any ε > 0, as n we have Y ν (n) X n = O(n /4+ε ) a.s. (.3) Our strong invariance principle for local times reads as follows. 4

5 Theorem.2. Let Y ν (t) and X n as in Theorem. and let η and ξ their respective local times. As R, we have ξ(r, ) η(r, ) = O(R /2 log R) a.s. (.4) We prove the following strong invariance principle between two NN random walks. Theorem.3. Let {X n () resp. Assume that and } n=0 and {X(2) n } n=0 p() j p(2) j be two NN random walk with p() j and p (2) j, B 4j C (.5) j γ B 4j C (.6) j γ j =, 2,... with B >, < γ 2 and some non-negative constant C. Then on a suitable probability space one can construct {X n () } and {X n (2) } such that as n X () n X (2) n = O((X () n + X (2) n ) 2 γ ) = O((n log log n) γ/2 ) a.s. The organization of the paper is as follows. In Section 2 we will present some well-known facts and prove some preliminary results. Sections 3-5 contain the proofs of Theorems.-.3, respectively. In Section 6 we prove strong theorems (most of them are integral tests) which easily follow from Theorems. and.2 and the corresponding results for Bessel process. In Section 7, using our Theorem.3 in both directions, we prove an integral test for the local time of the NN-walk, and a strong theorem for the speed of escape of the Bessel process. 2. Preliminaries Lemma 2.. Let Y ν ( ) be a Bessel process starting from x = R and let τ be the stopping time dened by (.8) with a = R and b = R +. Let p R be dened by (.2). Then as R p R = 2ν + ), (2.) 4R + O ( R 2 E R (τ) = + O ( R ), (2.2) V ar R (τ) = O(). (2.3) 5

6 Proof: For ν = /2, i.e. for d = 3-dimensional Bessel process, in case x = R, a = R, b = R + we have E R (e λτ ) = cos( 2λ) which does not depend on R. We prove that this holds asymptotically in general, when ν > 0. Using the identity (cf. [2], page 449 and [22], page 78) and the series expansion π 2 sin(νπ) (I ν(x) I ν (x)) if ν is not an integer K ν (x) = lim µ ν K µ (x) if ν is an integer I ν (x) = k=0 (x/2) ν+2k k!γ(ν + k + ), one can see that the coecient of α in the Taylor series expansion of the Laplace transform (.0) is (b 2 x 2 )a 2ν + (x 2 a 2 )b 2ν (b 2 a 2 )x 2ν E x (τ) = 2(ν + ) a 2ν b 2ν from which by putting x = R, a = R, b = R +, we obtain E R (τ) = (2R + )(R ) 2ν + (2R )(R + ) 2ν 4R 2ν 2(ν + ) (R ) 2ν (R + ) 2ν giving (2.2) after some calculations. (2.3) can also be obtained similarly, but it seems quite complicated. A simpler argument is to use moment generating function and expansion of the Bessel functions for imaginary arguments near innity. Put α = λ into (.0) to obtain E x (e λτ ) = S ν(ib 2λ, ix 2λ) + S ν (ix 2λ, ia 2λ) S ν (ib 2λ, ia, (2.4) 2λ) where i =. We use the following asymptotic expansions (cf. Erdélyi et al. [], page 86, or Watson [22], pages 202, 29) I ν (z) = (2πz) /2 ( e z + ie z+iνπ + O( z ) ), 6

7 ( ) π /2 (e K ν (z) = z + O( z ) ). 2z Hence one obtains for λ > 0 xed, and x < b, S ν (ib 2λ, ix 2λ) = ( 2λbx) ν (I ν (ib 2λ)K ν (ix 2λ) I ν (ix 2λ)K ν (ib 2λ)) = ( (e 2 ( 2λbx) ν /2 i(b x) 2λ e i(b x) 2λ + O, x. x)) One can obtain asymptotic expansions similarly for S ν (ix 2λ, ia 2λ), S ν (ib 2λ, ia 2λ). Putting these into (2.4), with x = R, a = R, b = R +, we get as R E R (e λτ ) = (R2 + R) ν /2 + (R 2 R) ν /2 (R 2 ) ν /2 e i = ( ) cos( 2λ) + O. R 2λ e i 2λ + O ( ) R e 2i 2λ e 2i 2λ + O ( R Hence putting λ =, there exists a constant C such that E R (e τ ) C for all R =, 2,... By Markov's inequality we have P R (τ > t) = P R (e τ > e t ) Ce t, from which E R (τ 2 ) 2C, implying (2.3). Here and throughout C, C, C 2,... denotes unimportant positive (possibly random) constants whose values may change from line to line. Recall the denition of the upper and lower classes for a stochastic process Z(t), t 0 dened on a probability space (Ω, F, P ) (cf. Révész [9], p. 33). The function a (t) belongs to the upper-upper class of Z(t) (a (t) UUC(Z(t)) if for almost all ω Ω there exists a t 0 (ω) > 0 such that Z(t) < a (t) if t > t 0 (ω). The function a 2 (t) belongs to the upper-lower class of Z(t) (a (t) ULC(Z(t)) if for almost all ω Ω there exists a sequence of positive numbers 0 < t = t (ω) < t 2 = t 2 (ω) <... with lim i t i = such that Z(t i ) a 2 (t i ), (i =, 2,...). The function a 3 (t) belongs to the lower-upper class of Z(t) (a 3 (t) LUC(Z(t)) if for almost all ω Ω there exists a sequence of positive numbers 0 < t = t (ω) < t 2 = t 2 (ω) <... with lim i t i = such that Z(t i ) a 3 (t i ), (i =, 2,...). ) 7

8 The function a 4 (t) belongs to the lower-lower class of Z(t) (a 4 (t) LLC(Z(t)) if for almost all ω Ω there exists a t 0 (ω) > 0 such that Z(t) > a 4 (t) if t > t 0 (ω). The following lower class results are due to Dvoretzky and Erd s [0] for integer d = 2ν+2. In the general case when ν > 0, the proof is similar (cf. also Knight [5] and Chaumont and Pardo [4] in the case of positive self-similar Markov processes). Theorem E: Let ν > 0 and let b(t) be a non-increasing, non-negative function. t /2 b(t) LLC(Y ν (t)) (b(2 t )) 2ν dt <. It follows e.g. that in case ν > 0, for any ε > 0 we have Y ν (t) t /2 ε (2.5) almost surely for all suciently large t. In fact, from our invariance principle it will follow that the integral test in Theorem E holds also for our Markov chain (X n ). In the proof however we need an analogue of (2.5) for X n. One can easily calculate the exact distribution of ξ(r, ), the total local time of X n of Theorem. according to Theorem C. Lemma A: If p R is given by (.2), then ξ(r, ) has geometric distribution (.6) with p R = Lemma 2.2. For any δ > 0 we have almost surely for all large enough n. + p 2 R D(R, ) = ( + p 2 R)((R + ) 2ν R 2ν ) = ν ( ) (R + ) 2ν R + O. (2.6) R X n n /2 δ Proof: From Lemma A it is easy to conclude that almost surely for some R 0 > 0 ξ(r, ) CR log R if R R 0, with some random positive constant C. Hence the time particle spent up to in [, S] is less than S R= ξ(r, ) which the R 0 R= S ξ(r, ) + C R log R C S 2+δ R=R 0 8

9 with some (random) C > 0. Consequently, after C S 2+δ steps the particle will be farther away from the origin than S. Let n = [C S 2+δ ], then ( ) n /(2+δ) S C and hence ( ) n /(2+δ) X n n /2 δ C for n large enough. This proves the Lemma. 3. Proof of Theorem. Define the sequences (τ n ), t 0 = 0, t n := τ τ n as follows: τ := min{t : t > 0, Y ν (t) = }, τ 2 := min{t : t > 0, Y ν (t + t ) = 2}, τ n := min{t : t > 0, Y ν (t + t n ) Y ν (t n ) = } for n = 3, 4,... Let X n = Y ν (t n ). Then (cf. (.2)) it is an NN random walk with p 0 = p = /2, p R = (R ) 2ν R 2ν (R ) 2ν (R + ), R = 2, 3,... 2ν 2 Let F n be the σ-algebra generated by (τ k, Y ν (τ k )) n k= and consider n M n := (τ i E(τ i F i )). i= Then the sequence (M n ) n is a martingale with respect to (F n ) n. It follows from (2.2) of Lemma 2. that for i = 2, 3,... we have ( ) E(τ i F i ) = E(τ i Y ν (t i )) = + O. Y ν (t i ) Hence t n n M n + τ + C n i=2 Y ν (t i ) = M n + τ + C n i=2 X i 9

10 with some (random) constant C. By (2.3) of Lemma 2. we have EM 2 n Cn. Let ε > 0 be arbitrary and dene n k = [k /ε ]. From the martingale inequality we get ( ) P max M n C n /2+ε n k n n k k hence we obtain by the Borel-Cantelli lemma almost surely for large k. Hence we also have By Lemma 2.2 consequently max M n C n /2+ε n k n n k k n i=2 M n = O(n /2+ε ) a.s. X i = O(n /2+ε ) C 2, a.s., n 2ε k t n n = O(n /2+ε ) a.s. (3.) It is well-known (cf. [2], p. 69) that Y ν (t) satises the stochastic dierential equation where W (t) is a standard Wiener process. Hence consequently, dy ν (t) = dw (t) + 2ν + dt, (3.2) 2Y ν (t) X n Y ν (n) = Y ν (t n ) Y ν (n) = W (t n ) W (n) + X n Y ν (n) W (t n ) W (n) + (2ν + ) t n n 2 n 2ν + t n 2Y ν (s) ds, max min(n,t n) t max(n,t n) Y ν (t). Now by (3.) and (2.5) the last term is O(n 2ε ) almost surely and since for the increments of the Wiener process (cf. [8], page 30) as n, we have (.3) of Theorem.. W (t n ) W (n) = O(n /4+ε ) a.s. 0

11 4. Proof of Theorem.2 For R > 0 integer dene κ := min{t 0 : Y ν (t) = R}, δ := min{t κ : Y ν (t) / (R, R + )}, κ i := min{t δ i : Y ν (t) = R}, δ i := min{t κ i : Y ν (t) / (R, R + )}, κ := max{t 0 : Y ν (t) = R}, i = 2, 3,... Consider the local times at R of the Bessel process during excursions around R, i.e. let ζ i := η(r, δ i ) η(r, κ i ), i =, 2,..., ζ := η(r, ) η(r, κ ). We have Lemma 4.. where and p R = η(r, ) = E ( e λη(r, )) = ξ(r, ) i= ζ i + ζ. p R ϕ(λ) q R ϕ(λ), (4.) A R A R + B R (R + ) 2ν R 2ν (R + ) 2ν, q R = p R, (4.2) ϕ(λ) = ν(a R + B R ) ν(a R + B R ) λr 2ν+ A R B R, (4.3) A R = (R ) 2ν R 2ν, B R = R 2ν (R + ) 2ν. (4.4) Proof: By ([2], p. 395, 3.3.2) ζ i are i.i.d. random variables having exponential distribution with moment generating function ϕ(λ) given in (4.3). Moreover, it is obvious that ζ is independent from ξ(r, ) i= ζ i. Furthermore, ζ is the local time of R under the condition that starting from R, Y ν (t) will reach R + before R. Hence its distribution can be calculated from formula 3.3.5(b) of [2], and its moment generating function happens to be equal to ϕ(λ) of (4.3).

12 We can see θ := E(ζ i ) = E( ζ) = ν(a ( ) R + B R ) = + O, R. R 2ν+ A R B R R ξ(r, ) P( η(r, ) ξ(r, ) u) = P (ζ i θ) + ζ θ u i= ( ) k P(ξ(R. ) > N) + P max (ζ k N i θ) u i= ( ) (qr) N + e λu e λθ N ( ) e λθ N +. + λθ λθ In the above calculation we used the common moment generating function (4.3) of ζ i and ζ, the exact distribution of ξ(r, ) (see (.6)) and the exponential Kolmogorov inequality. Estimating the above expression with standard methods and selecting we conclude that N = CR log R, u = CR /2 log R, λ = u θ 2 N P( η(r, ) ξ(r, ) CR /2 log R) C exp ( C log R 2θ With a big enough C the right hand side of the above inequality is summable in R, hence Theorem.2 follows by the Borel-Cantelli lemma. 5. Proof of Theorem.3 Let p () j and p (2) j as in Theorem.3. Dene the two-dimensional Markov chain (X n (), X (2) as follows. If p () j p (2), then let k P ( (X () n+, X (2) n+) = (j +, k + ) (X () n, X n (2) ) ) = (j, k) ) = 2 + p(2) k P ( (X () n+, X (2) n+) = (j +, k ) (X () n, X (2) n ) = (j, k) ) = p () j p (2) k P ( (X n+, () X n+) (2) = (j, k ) (X n (), X n (2) ) = (j, k) ) = 2 p() 2. j. n )

13 If, however p () j p (2), then let k P ( (X () n+, X (2) n+) = (j +, k + ) (X () n, X n (2) ) = (j, k) ) = 2 + p() j P ( (X n+, () X n+) (2) = (j, k + ) (X n (), X n (2) ) = (j, k) ) = p (2) k p () j P ( (X () n+, X (2) n+) = (j, k ) (X () Then it can be easily seen that X n () and X n (2) Consider the following 4 cases. (i) p () j p (2) k, j k, (ii) p () j p (2) k, j k, (iii) p () j p (2) k, j k, (iv) p () j p (2) k, j k. In case (i) from (.5) and (.6) we obtain implying n, X n (2) B 4j C j B γ 4k + C k B γ 4k + C kj, γ k j 2Cj 2 γ B/4 Cj γ = O(j2 γ ) if j. So in this case if X n () = j and X n (2) = k, then we have 0 X (2) n X () n = O((X () n ) 2 γ ) ) = (j, k) ) = 2 p(2) k. are two NN random walks as desired. if n. In case (ii) either X () n+ X (2) n+ = X n () X n (2), or X () n+ X (2) n+ = X n () X n (2) 2, so that we have 2 X () n+ X (2) n+ X n () X n (2). Similar procedure shows that in case (iii) 2 X (2) n+ X () n+ X n (2) X n () and in case (iv) 0 X n () X n (2) = O((X n (2) ) 2 γ ). Hence Theorem.3 follows from the law of the iterated logarithm for X n (i) (cf. [3]). 3

14 6. Strong theorems As usual, applying Theorem. and Theorem.3, we can give limit results valid for one of the processes to the other process involved. In this section we denote Y ν (t) by Y (t) and dene the following related processes. M(t) := max 0 s t Y (s), The future inmums are dened as Escape processes are dened by I(t) := inf s t Y (s), A(t) := sup{s : Y (s) t}, Q n := max k n X k. H n := inf k n X k. G n := sup{k : X k n}. Laws of the iterated logarithm are known for Bessel processes (cf. [2]) and NN random walks (cf. [3]) as well. Upper class results for Bessel process read as follows (cf. Orey and Pruitt [7] for integral d, and Pardo [8] for the case of positive self-similar Markov processes). Theorem F: Let a(t) be a non-decreasing non-negative continuous function. Then for ν 0 t /2 a(t) UUC(Y (t)) (a(x)) 2ν+2 e a2 (x)/2 dx <. x Now Theorems.,.3 and Theorems E and F together imply the following result. Theorem 6.. Let {X n } be an NN random walk with p R satisfying p R = B ( ) 4R + O, R R +δ with B > and for some δ > 0. Let furthermore a(t) be a non-decreasing non-negative function. Then n /2 a(n) UUC(X n ) If b(t) is a non-increasing non-negative function, then n /2 b(n) LLC(X n ) 4 (a(k)) B+ e a2 (k)/2 <. k= k (b(2 k )) B <. k=

15 Next we prove the following invariance principles for the processes dened above. Theorem 6.2. Let Y (t) and X n as in Theorem.. Then for any ε > 0 we have M(n) Q n = O(n /4+ε ) a.s. (6.) and I(n) H n = O(n /4+ε ) a.s. (6.2) Proof: Dene s, s, k, k by Y ( s) = M(n), Y (s ) = I(n), X k = Q n, X k = H n. Then as n, we have almost surely and Q n M(n) = X k Y ( s) X k Y ( k) = O(n /4+ε ) M(n) Q n = Y ( s) X k = Y ( s) Y ([ s]) (X [ s] Y ([ s])) + X [ s] X k Y ( s) Y ([ s]) (X [ s] Y ([ s]) = Y ( s) Y ([ s]) + O(n /4+ε ) By (3.2) and recalling the results on the increments of the Wiener process (see [8] page 30) we get s 2ν + Y ( s) Y ([ s]) = W ( s) W ([ s]) + [ s] 2Y (s) ds sup sup 0 t n 0 s W (t + s) W (t) + 2ν + 2 max [ s] t s Y (t) = O(log n) a.s., since Y (t) in the interval ([ s], s) is bounded away from zero. Hence (6.) follows. To show (6.2), note that n s n +α and n k n +α for any α > 0 almost surely for all large n. Then as n I(n) H n Y (k ) X k = O((k ) /4+ε ) = O(n (+α)(/4+ε) ) a.s. On the other hand, H n I(n) X k Y ([s ]) + Y ([s ]) Y (s ) = O(n (+α)(/4+ε) ) + Y ([s ]) Y (s ). By (3.2), taking into account that when applying this formula the integral contribution is negative, and recalling again the results on the increments of the Wiener process, we get Y ([s ]) Y (s ) W ([s ]) W (s ) sup sup 0 t n +α 0 s 5 W (t + s) W (t) = O(log n) a.s.

16 as n. Hence I(n) H n = O(n (+α)(/4+ε) ) Since α > 0 and ε > 0 are arbitrary, (6.2) follows. This completes the proof of Theorem 6.2. Theorem 6.3. Let X n () and X n (2) as in Theorem.3 and let Q () n and Q (2) n be the corresponding maximums, while let H n () and H n (2) be the corresponding future inmum processes. Then for any ε > 0, as n we have a.s. Q () n Q (2) n = O(n γ/2+ε ) a.s. (6.3) and H () n H (2) n = O(n γ/2+ε ) a.s. (6.4) Proof: Dene k i, k i, i =, 2 by Then X (i) k i = Q (i) n, X (i) k i = H (i) n. Q () n Q (2) n max(x () k X (2) k, X () k 2 X (2) k 2 ) = O((n log log n) γ/2 ) a.s., proving (6.3). Moreover, for any α > 0, n k i n +α almost surely for large n, hence we have H n () H n (2) max(x () k X(2) k, X() k 2 X(2) k ) = O((n log log 2 n)(+α)( γ/2) ) a.s. Since α is arbitrary, (6.4) follows. This completes the proof of Theorem 6.3. Khoshnevisan et al. [4] (for I(t) and A(t)), Shi [2] (for Y (t) I(t)), Gruet and Shi [2] (for M(t)), Adelman and Shi [] (for M(t) I(t)), proved the following upper and lower class results. Theorem G: Let ϕ(t) be a non-increasing, and ψ(t) be a non-decreasing function, both non-negative. Then for ν > 0 t /2 ψ(t) UUC(I(t)) t 2 ϕ(t) LLC(A(t)) 6 (ψ(x)) 2ν e ψ2 (x)/2 dx <, x xϕ ν (x) e /2ϕ(x) dx <.

17 t /2 ψ(t) UUC(Y (t) I(t)) xψ 2ν 2 (x) e ψ2 (x)/2 dx <, Theorem H: Let h(t) > 0 be a non-decreasing function and let ρ(t) > 0 be such that (log ρ(t))/ log t is non-decreasing. Then for ν 0 t /2 h 2 (x) /h(t) LLC(M(t)) x e j2 ν h2 (x)/2 dx <, where j ν is the smallest positive zero of the Bessel function J ν (x), /ρ(t) LLC(M(t) I(t)) dx x log ρ(x) <. Taking into account that H n and G n are inverses of each other, immediate consequences of Theorems F, G, H, Theorems 6.2 and 6.3 are the following upper and lower class results. Theorem 6.4. Let X n be as in Theorem 6. and let ϕ(t) be a non-increasing and ψ(t) be a non-decresing function, both non-negative. Then n /2 ψ(n) UUC(H n ) n 2 ϕ(n) LLC(G n ) n /2 ψ(n) UUC(X n H n ) k= (ψ(k)) B e ψ2 (k)/2 <, k= k kϕ (B )/2 (k) e /2ϕ(k) <, k= kψ B 3 (k) e ψ2 (k)/2 <. Theorem 6.5. Let X n be as in Theorem 6. with B. Assume that h(t) > 0 is a nondecreasing function and let ρ(t) > 0 be such that (log ρ(t))/ log t is non-decreasing. Then n /2 /h(n) LLC(Q n ) k= where ν = (B )/2, and j ν is as in Theorem H, h 2 (k) k e j2 ν h2 (k)/2 <, /ρ(n) LLC(Q n H n ) k=2 k log ρ(k) <. 7

18 7. Local time We will need the following result from Yor [23], page 52. Theorem J: For the local time of a Bessel process of order ν we have η(r, ) D = (2ν) R 2ν Y 2 0 (R 2ν ), where Y 0 is a two-dimensional Bessel process and D = means equality in distribution. Hence applying Theorem F for ν = 0, we get Theorem K: If f(x) is non-decreasing, non-negative function, then Rf(R) UUC(η(R, )) From this and Theorem.2 we get the following result. f(x) x e νf(x) dx <. Theorem 7.. If f(x) is non-decreasing, non-negative function, then Rf(R) UUC(ξ(R, )) k= f(k) k e νf(k) <. In [7] we proved the following result. Theorem L: Let p R = B 4R + O ( R γ ) with B >, and γ >. Then with probability there exist innitely many R for which ξ(r + j, ) = for each j = 0,, 2,..., [log log R/ log 2]. Moreover, with probability for each R large enough and ε > 0 there exists an ( + ε) log log R R S log 2 such that ξ(s, ) >. Remark : In fact in [7] we proved this result in the case when p R = B/4R but the same proof works also in the case of Theorem L. This theorem applies e.g. for the case when p R is given by (.2), which in turn, gives the following result for the Bessel process. Let 8

19 (i) κ(r) := inf{t : Y ν (t) = R}, (ii) κ (R) := sup{t : Y ν (t) = R}, (iii) Ψ(R) be the largest integer for which the event occurs. A(R) = Ψ(R) j= {κ (R + j) < κ(r + j + )} A(R) means that Y ν (t) moves from R to R + before returning to R, it goes from R + to R + 2 before returning to R,... and also from R + Ψ(R) to R + Ψ(R) + and it never returns to R + Ψ(R). We say that the process Y ν (t) escapes through (R, R + Ψ(R)) with large velocity. Theorem 7.2. For Ψ( ) dened above, we have for all ν > 0 lim sup R Ψ(R) log log R = log 2 a.s. Remark 2: The statement of Theorem 7.2 (for integral d = 2ν + 2) was formulated in [9], p. 29 as a Conjecture. References [] ADELMAN, O. and SHI, Z.: The measure of the overlap of past and future under a transient Bessel process. Stochastics Stochastics Rep. 57 (996), [2] BORODIN, A.N. and SALMINEN, P.: Handbook of Brownian Motion Facts and Formulae. Birkhäuser, Basel, 996. [3] BRÉZIS, H., ROSENKRANTZ, W. and SINGER, B.: An extension of Khintchine's estimate for large deviations to a class of Markov chains converging to a singular diusion. Comm. Pure Appl. Math. 24 (97), [4] CHAUMONT, L. and PARDO, J.C.: The lower envelope of positive self-similar Markov Processes. Electr. J. Probab. (2006),

20 [5] CHUNG, K.L.: Markov Chains with Stationary Transition Probabilities. 2nd ed. Springer-Verlag, New York, 967. [6] COOLIN-SCHRIJNER, P. and VAN DOORN, E.A.: Analysis of random walks using orthogonal polynomials. J. Comput. Appl. Math. 99 (998), [7] CSÁKI, E., FÖLDES, A. and RÉVÉSZ, P.: Transient nearest neighbor random walk on the line. J.Theor. Probab., to appear. [8] CSÖRGŽ, M. and RÉVÉSZ, P.: Strong Approximations in Probability and Statistics. Academic Press, New York, 98. [9] DETTE, H.: First return probabilities of birth and death chains and associated orthogonal polynomials. Proc. Amer. Math. Soc. 29 (200), [0] DVORETZKY, A. and ERDŽS, P.: Some problems on random walk in space. Proc. Second Berkeley Symposium on Mathematical Statistics and Probability, 950. pp University of California Press, Berkeley and Los Angeles, 95. [] ERDÉLYI, A. MAGNUS, W. OBERHETTIGER, F. and TRICOMI, F.G.: Higher Transcendental Functions. Vol. 2, McGraw - Hill, New York, 953. [2] GRUET, J.C. and SHI, Z.: On the Spitzer and Chung laws of the iterated logarithm for Brownian motion. Séminaire de Probabilités, XXIX, Lecture Notes in Math., 63, Springer, Berlin, 995, , [3] KARLIN, S. and McGREGOR, J.: Random walks. Illinois J. Math. 3 (959), 668. [4] KHOSHNEVISAN, D., LEWIS, M.L. and SHI, Z.: On a problem of Erd s and Taylor. Ann. Probab. 24 (996), [5] KNIGHT, F.B.: Essentials of Brownian Motion and Diusion. Am. Math. Soc., Providence, R.I., 98. [6] LAMPERTI, J.: A new class of probability limit theorems. J. Math. Mech. (962), [7] OREY, S. and PRUITT, W.E.: Sample functions of the N-parameter Wiener process. Ann. Probab. (973), [8] PARDO, J.C.: The upper envelope of positive self-similar Markov processes. J.Theor. Probab., to appear. arxiv:math.pr/

21 [9] RÉVÉSZ, P.: Random Walk in Random and Non-Random Environments. 2nd ed. World Scientic, Singapore, [20] REVUZ, D. and YOR, M.: Continuous Martingales and Brownian Motion. 3rd ed. Springer-Verlag, Berlin, 999. [2] SHI, Z.: On transient Bessel processes and planar Brownian motion reected at their future inma. Stoch. Process. Appl. 60 (995), [22] WATSON, G.N.: A Treatise on the Theory of Bessel Functions. 2nd ed. Cambridge University Press, Cambridge, 944. [23] YOR, M.: Some Aspects of Brownian Motion. Part I: Some Special Functionals. Birkhäuser, Basel,