Chapitre II. A law of iterated logarithm for increasing self similar Markov processes. 1 Introduction. Abstract

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1 Chapitre II A law of iterated logarithm for increasing self similar Markov processes Abstract We consider increasing self similar Markov processes X t, t on ], [. By using the Lamperti s bijection between self similar Markov processes and Lévy processes, we determine the functions f for which there exists a constant c R + \{} such that lim inf t X t /ft = c with probability 1. The determination of such functions depends on the subordinator ξ associated to X through the distribution of the Lévy exponential functional and the Laplace exponent of ξ. We provide an analogous result for the self similar Markov process associated to the opposite of a subordinator. Key Words. Self similar Markov processes, Subordinators, Exponential functional of Lévy process, weak duality of Markov processes. A.M.S Classification. 6G18, 6G17, 6F15. 1 Introduction Let X = X s, s be a strong Markov process with values in ], [ and denote by P x its law starting from X = x >. For α >, we say that X is α self similar α ss, whenever it fulfills the scaling property: for any c > and x > the distribution of cx tc 1/α, t under P x is P cx. 1 Such processes have been introduced by Lamperti [2, 21] under the name of semi stable processes. We refer to Embrechts and Maejima [12] for some account of their properties and applications. Recently, Bertoin and Caballero [3] studied the weak behavior of t α X t as t, in the case when X has increasing sample paths see also Bertoin and Yor [5] for the general case. For any y > fixed, they established the weak convergence P y t α X t P t + X1, 25

2 26 A law of iterated logarithm for increasing self similar Markov processes where P + X1 is the so-called entrance law from +. The problem that we consider here concerns the rate at which an increasing α ss process goes to infinity. More precisely, we should like to determine the functions f : [, [ [, [, for which, for any x > lim inf t X t ft ], [ P x a.s. 2 Fristedt [15] see also Breiman [8] provided an answer to 2 when X has moreover independent and stationary increments, that is X is a stable subordinator. Later, the problem was solved by Watanabe [32] for increasing ss process with independent increments. In this paper we treat the case that does not assume neither stationarity nor independence of the increments. Namely, under a rather natural hypothesis on the entrance laws, we provide an explicit characterization of the functions that satisfies 2. Our approach is based, essentially on the main result of Lamperti [21] about the existence of a bijection between self similar and Lévy processes. Specifically, let ξ = ξ t, t be a Lévy process and F t, t its natural filtration. Denote by P and E the probability and expectation with respect to ξ. Suppose that ξ does not drift to. For α >, define A t = and the time change associated to A by t e ξs/α ds, t, τt = inf{s : A s > t}. For an arbitrary x >, write by P x the law of the process X t = x exp ξ τtx 1/α, t. It is straightforward that under P x, X has the scaling property defined in 1. A classical result on time changes shows that the process X inherits the strong Markov property from ξ. So X is an α ss Markov process. Conversely any α-ss Markov process can be obtained in this way. In our setting X is an increasing process so ξ is a subordinator see Bertoin [1] 3, for background. The law of a subordinator is characterized by its Laplace transform, Ee λξt = exp tφλ λ, t, where φ is the so called Laplace exponent of ξ and can be expressed thanks to the Lévy Khintchine s formula as φλ = dλ + 1 e λx Πdx, ], [ The term d is called the drift coefficient and Π is the Lévy measure associated to the subordinator ξ, that is, a positive measure such that ], [1 xπdx <. We suppose henceforth that the drift coefficient is d =, and we shall exclude the case ξ is arithmetic, that is when Π is supported by k N, for some k >. Bertoin and Caballero [3] showed that if < µ = E ξ 1 = φ + <, then the α ss Markov process X started at x > converges in the sense of finite dimensional distributions when x + cf. Bertoin and Yor [5] for the general case. We then denote by P + the

3 1. Introduction 27 limiting law. Moreover, the law of X 1 under P + is related to the law under P of the Lévy exponential functional associated to the subordinator ξ, i.e. by the formula E + I = e ξs/α ds, 3 fx 1/α 1 = α µ E I 1 f1/i, 4 where f : [, [ [, [ is a measurable and bounded function. Besides, provided that φ + < Carmona, Petit and Yor [1], showed c.f. Proposition 2.1 in [1] that the law of I admits a density ρ which is infinitely differentiable on ], [. Furthermore, Proposition 3.3 op. cit. establishes that the law of I is determined by its integral moments, which in turn are given by the formulae and that E I n = n k=1 k φk/α Ee ri <, n N, for every < r < φ. Let us introduce the following technical hypothesis H The density ρ is decreasing in a neighborhood of, and bounded. Examples which satisfy hypothesis H are given in Section 6. Recall that a Borel function f : R + R +, is regularly varying at infinity resp. at with index β if fxt ft xβ, as t resp. as t for every x >. We refer to Bingham et al. [7] for a complete account of the theory of regular variation. It is well known in the theory of subordinators that the regular variation at infinity resp. at of the Laplace exponent φ, is related to the behavior at resp. at of the subordinator ξ associated to it. So it is natural to expect that the regular variation at of the Laplace exponent should also be related to the local behavior of any α ss process associated to ξ. This is indeed the case, but we first need to recall a result on subordinators in order to give a precise statement: let φ be regularly varying at infinity with index β ], 1[, let ψ be the inverse of φ and then gt = lim inf t log log t ψt 1 log log t, < t < e 1, ξ t = 1 β1 β/β P gt a.s., see e.g. Bertoin [1], section III.4. It follows easily that g is regularly varying at with index 1/β and τt lim = 1, P a.s. t t This being said, it is straightforward that for any x > and X an α ss process associated to ξ we have that X t X lim inf = X αβ 1/αβ t 1 β 1 β/β P x a.s. 5 gt

4 28 A law of iterated logarithm for increasing self similar Markov processes On the other hand, contrary to what we might expect, it is also the regular variation at infinity of the Laplace exponent that gives us the means to determine the behavior at infinity of an increasing self similar Markov process. Indeed, we have the following Theorem 1. Let ξ be a subordinator such that < µ = Eξ 1 < and whose Laplace exponent φ is regularly varying at infinity with index β ], 1[. Suppose that the density ρ, of the Lévy exponential functional I of ξ satisfies hypothesis H. For α >, let X be the α ss process associated to the subordinator ξ. Define φlog log t ft =, t > e. log log t Then for any x > lim inf t This result also holds true under P +. X t tft α = α αβ 1 β α1 β P x a.s. From the equation 5 only the local behavior of X under P + next result we fill this gap. remains to be determined. In the Theorem 2. Under the hypotheses and notations of Theorem 1, we have that lim inf t X t tf1/t α = α αβ 1 β α1 β P + a.s. The rest of this note is organized as follows. In section 2 we state two propositions that enable us to prove Theorem 1. Section 3 is devoted to the proof of these propositions. The proof of Theorem 2 is given in section 4 where we obtain some results on time reversal for a self similar Markov process. There we also obtain a result analog to Theorem 1 for the self similar Markov process associated to the opposite of a subordinator near the first time that it hits. Finally in section 6 we give some examples. 2 Preliminaries Let X be an α-ss Markov process with α >. It is plain that the process Y = X 1/α, is a 1-ss Markov process, in fact it is the 1 ss process associated to 1/αξ. Conversely if Y is a 1-ss Markov process then, for any α >, the process X = Y α is an α ss Markov process. So we can assume henceforth, without loss of generality, that α = 1. We can deduce from equation 4 that the entrance law P +X 1 dx has a density { µx 1 ρx 1 if < x < p 1 x = otherwise, with ρ the density of the law of I. Denote by U = U s, s the Ornstein Uhlenbeck OU process associated to the 1 ss Markov process X, or to the underlying subordinator ξ through Lamperti s transformation if X = x for some x > that is U t = e t X e t 1 t.

5 according whether ρ1/hsds < or =. 2. Preliminaries 29 This process inherits the homogeneity and strong Markov property from X, has transition probabilities P s fx = E x fe s X e s 1 s, for every Borel function f. Moreover, it has a unique invariant probability measure given by the entrance law p 1 xdx. See e.g. Carmona, Petit and Yor [1] for a proof of these facts. The asymptotic behavior of the OU process U, defined above is described in the next proposition. Proposition 1. Let ξ be a subordinator such that < µ = Eξ 1 < and whose Laplace exponent is regularly varying at infinity with index β ], 1[. Suppose that the density, ρ, of the exponential functional I satisfies H. Let U be the Ornstein Uhlenbeck process associated to ξ. If h :], [ ], [, is a decreasing function then for every x > P x Us < hs i.o. s = or = 1, This result also holds true if we suppose that the Lévy measure is finite, Π], [<, instead of the regular variation at infinity of φ. Remark 1. Of course one can derive an integral test from Proposition 1 for the 1 ss Markov process associated to ξ. Indeed, if h is a decreasing function then according whether P x Xs < shs i.o. s = or 1 ρ1/hs ds s < or =. However this result is not really satisfactory unless one has good estimates of ρ. Despite the characterization of the law of the exponential functional I it is not always possible get an explicit representation of its density. But to obtain the result stated at Theorem 1 we will only need estimations of the behavior of log ρ near infinity. That is the purpose of the following Proposition 2. Let I be the exponential functional associated to a subordinator ξ s, s whose Laplace exponent φ, varies regularly at infinity with index β ], 1[. Then log PI > t 1 βϕ t, t, 6 where ϕ t = inf { s >, s } φs > t. If moreover, the density ρ of the law of I, is decreasing on some neighborhood of, then log ρt 1 βϕ t, t. 7 Remark 2. The fact that the tail distribution of I has this asymptotic form implies that the law of I cannot be infinitely divisible see e.g. Steutel [3] or Bingham et al. [7] section If we take for granted Propositions 1 and 2 the proof of Theorem 1 follows by standard arguments.

6 3 A law of iterated logarithm for increasing self similar Markov processes Proof of Theorem 1. By Proposition 2 and the fact that ϕ is regularly varying with index 1 1 β we have that for any constant c > 1 log ρ 1 βc 1 1 β ϕ 1 as t. cft ft Since ϕ is the inverse of s/φs we then have that 1 log ρ 1 βc 1 1 β log log t cft as t. 8 The statement in Theorem 1 is equivalent to the property to be proven that for any ɛ >, and P x Xs < 1 ɛc β sfs i.o. s =, P x Xs < 1 + ɛc β sfs i.o. s = 1, where c β = 1 β 1 β. From the remark after Proposition 1 the former and later equations hold if for any ɛ >, ρ1/f 1,ɛs ds s <, ρ1/f 2,ɛs ds s =, where f 1,ɛ s = 1 ɛc β fs, f 2,ɛ s = 1 + ɛc β fs, respectively. Indeed, let ɛ >, by equation 8 there exists an s ɛ such that for every s > s ɛ, 1 log ρ 1 β1 ɛ1 ɛ 1 f 1,ɛ s Therefore, taking k ɛ = 1 ɛ β 1 β, we have s ɛ = 1 ɛ β 1 β log log s. ρ1/f 1,ɛ s ds s s ɛ log s since k ɛ > 1. Similarly, one shows the divergence of ρ1/f 2,ɛs ds s. 1 β c 1 1 β β kɛ ds s <, log log s We have showed the statement of Theorem 1 for α = 1, to show that the result holds for any α, consider the 1 ss process Y associated to the subordinator α 1 ξ. This subordinator has Laplace exponent φ α λ, such that φ α λ = φα 1 λ α β φλ λ, owed to the regular variation of φ. Then one obtain the result readily by means of the α ss process X = Y α.

7 3. Proofs 31 3 Proofs This section contains two parts. In the first one, we give the proof of the Proposition 1, which is rather technical so that we decompose it in to several Lemmas. The second part contains the proof of the Proposition Proof of Proposition 1 Let Ũ be process {Ũs = e s X e s, s R}. Under P + the process Ũ is a stationary strong Markov process, whose transition probabilities are those of the OU process U defined in the preceding section. In fact, the law of the process Ũs, s under P + is the same as that of the OU process U s, s with initial measure the entrance law P +X 1 dx = p 1 xdx. This process will enable us to describe the local behavior of the OU process U and in section 4 prove the Theorem 2. The first ingredient in the proof of Proposition 1 is the following Lemma 1. For any x > P + lim h Ũ h Ũ h = Ũ Ũ = x = 1. Remark 3. In Lemma 1 we do not impose any constraint in the way we make h tend to. That is why we postpone its proof until section 4. We suppose in the sequel that the starting point of the OU process U is fixed, U = x >, unless otherwise stated. The main argument in the proof of Proposition 1 is that of Breiman s [8] proof of a law of iterated logarithm for stable subordinators, which in turn is an adaptation of Motoo s [24] proof of Kolmogorov s test for diffusions. Here is an outline of such a method, see e.g. Ito and McKean [18] for Motoo s proof of Kolmogorov s test. Let {R n, n } be the successive return times of the OU process U to its starting point, i.e., R n+1 = inf{t > R n : U t = U }, with R =. Denote by R = R 1 and T y the first hitting time of a level y > by the OU process U, i.e., T y = inf{t > : U t = y}. Define the function gx, y by gx, y = P x T y < R = P x inf U t < y, y < x. t [,R] By the homogeneity and strong Markov property of U the random variables {R n+1 R n, n } are independent and identically distributed with the same law as R. The fact that the OU process U has a unique invariant probability implies that E x R <.

8 32 A law of iterated logarithm for increasing self similar Markov processes Then, by the strong law of large numbers R n n n E xr, P x a.s. Besides, we can deduce from the Lemma 1, using the homogeneity and the strong Markov properties of the OU process Ũ that the OU process U hits points from above and it leave it from below and more importantly the range of the excursion process {U s, s [, R]} is a compact interval with U in its interior. Thanks to these facts Motoo s arguments apply to show that for any decreasing function h :], [ ], [ we have the P x a.s. inclusion of sets { Us < hs i.o. s } { inf U s < hc 1 n i.o. n } s [R n,r n+1 ] { inf U s < hc 2 n i.o. n } { U s < hs i.o. s } s [R n,r n+1 ] with c 1, c 2 > constants that depend only on E x R. Therefore, by a standard application of the Borel Cantelli Lemma we get that if the integral gx, hsds 9 converges then P x inf U t < hc 1 n i.o. n =, t [R n,r n+1 ] whereas if 9 diverges then P x inf U t < hc 2 n i.o. n = 1. t [R n,r n+1 ] The proof reduces then to estimate the function gx, y, that is, estimate the distribution of the depth of the excursion and to show that the criterion does not depend of x. Namely that that is, there exists two positive constants b 1, b 2 such that gx, y ρ1/y as y, 1 b 1 ρ1/y gx, y b 2 ρ1/y as y. In [2] Bertoin gets an estimate for the function g when the underlying self similar process is a stable subordinator. His proof provides the key steps for our estimation of the function g. Lemma 1 enable us to follow the arguments of section 3 in [2] and this yields Lemma 2. Assume ρ is bounded. For every x, y > and q > we have i ii 1 lim ɛ ɛ both P x a.s. and in L 1 P x. R 1 {Us [y ɛ,y]}ds = 1 y Card{ t [, R[: U t = y } lim ɛ E y 1 ɛ R e qs 1 {Us [y ɛ,y]}ds = 1 y.

9 3. Proofs 33 iii iv E x E x R = µ ρ1/x t [,R[ 1 {Ut=y} = ρ1/y ρ1/x. Proof. First note that an application of Dynkin s formula shows that the measure νf = E x R fu sds, E x R is an invariant law for the OU process. Moreover, by the uniqueness of the invariant law we have that νf = fzµz 1 ρ1/zdz, 11 for every function f non negative and measurable. Next, if we take for granted Lemma 1, then we may simply repeat the arguments of [2] to prove i iii. The statement in iv follows from i, iii and the identity in equation 11. A standard application of the strong Markov property at time R shows that for every y > E x = P x T y < R 1 + P y R < T x + P y R < T x 2 + t [,R[ 1 {Ut=y} = P xt y < R P y T x R. 12 Therefore, by comparing iv in Lemma 2 and equation 12 we get that Since by hypothesis H we have that P x T y < R = P y T x R ρ1/y ρ1/x. lim ρ1/y =, y then we may conclude that the statement in 1 is equivalent to lim inf y P yt x R >. 13 We next focus in the proof of 13. To that end, we will obtain more precise information on the duration R of the excursion as the starting point tends to using the well known fact that the distribution of R can be characterized in terms of the resolvent density. We introduce some notation. Define by {L y t, t > } the local time at y > of the OU process U, that is L y t = 1 1 y {Us=y}. <s t Let x, y >, and u 1 x, y the 1-potential of L y t under P x, for x > and P + u 1 x, y = E x ], [ e s dl y s. for x = i.e.,

10 34 A law of iterated logarithm for increasing self similar Markov processes We have by the strong Markov property that Lemma 3. For every y >, we have u 1 x, y = y 1 E x e Ty 1 E y e R. 14 u 1, y = 1 µ 1/y dzρz. In particular u 1, y is a bounded and continuous function. Proof. Let R 1 denote the 1-resolvent operator of the OU process, that is, R 1 fx = E x e s fu s ds = R 1 x, dyfy, for any Borel positive function f, and x. Our first aim is to show that the measure R 1, dy has a density that coincides with u 1, y. Indeed, by a change of variables, an application of Fubini s Theorem and the self similarity of X, we get R 1 f = E + = E + Straightforward calculations shows that = = 1 1 E + 1 du R 1 f = e s fu s ds fux 1 u/u du fux1 du. dxµx 1 ρ1/xfxu. dyfyvy, with vy = µ 1 1/y dxρx. This shows that R 1, dy has a density vy, that is continuous and bounded. In particular y vy = lim ɛ 1 vxdx. ɛ y ɛ On the other hand, y y ɛ vxdx = E + T y = I ɛ + II ɛ. By the strong Markov property e s 1 {Us [y ɛ,y]}ds + E + II ɛ = E +e Ty 1 E y e R E y R T y e s 1 {Us [y ɛ,y]}ds. e s 1 {Us [y ɛ,y]}ds

11 3. Proofs 35 Using ii in Lemma 2 and equation 14 we get that lim ɛ ɛ 1 II ɛ = y 1 E +e Ty 1 E y e R = u 1, y. Thus the proof will be completed if we show that, ɛ 1 I ɛ as ɛ. Let H y be the first time that the OU process jumps above the level y >, H y = inf{s > : U s > y}. Indeed, by the Markov property applied at the first passage time of the OU process above the level y ɛ we get I ɛ E + 1{Hy ɛ<h y}e H H y y ɛ sup E z e s 1 {Us [y ɛ,y]}ds. z [y ɛ,y] Applying repeatedly the Markov property at the stopping time R we get for every z >, H y E z e s 1 {Us [y ɛ,y]}ds E R z e s 1 {Us [y ɛ,y]}ds 1 E z e R. 1 {R<Hy} The claimed result now follows from an application of ii in Lemma 2 and the fact that H y ɛ H y as ɛ a.s. We assume throughout the rest of this section that either φ is regularly varying at with index β ], 1[ or the Lévy measure is finite, Π], [<. Lemma 4. One has lim inf y y 1 E + + e T y >. Before proving this Lemma let us define a function that will be used in the sequel. function φλ/λ is decreasing, there exists a function β y such that Since the φβ y /β y = y, we denote δ y = e 1/βy 1. Proof. The statement in Lemma 4 means that lim inf y + y 1 P +T y < e >, with e an exponential random variable independent of the OU process. To show this fact we will need to introduce some notation and recall some results. Let Hy X be the first passage time above the level y by the 1-ss process X, that is Hy X = inf { s > : X s > y }. Bertoin and Caballero [3] showed that under the entrance law P +, the law of the pair is the same as that of H X y, X H X y yi exp{ V Z}, y exp{1 V Z}, where V, Z and I are independent and V is uniformly distributed on [, 1] and the law of Z is given by PZ dz = µ 1 zπdz z >.

12 36 A law of iterated logarithm for increasing self similar Markov processes So by taking S y = log1 + H X y we get that where K is a random variable with law USy /y, S y D y ek,, PK dk = µ 1 Πkdk, and Πk = Πk,. Recall that H y is the first time that the OU process U jumps above the level y. It is plain that the OU process hits a level [y, [ only if the ss process X is already at this level, i.e. log1 + H X y H y, for every y >. Moreover, the weak convergence of U Sy /y implies that P + log1 + H X y < H y P +U Sy [, y[ as y. So we can suppose henceforth that log1 + H X y = H y, for all y small enough. Let t > fixed and ɛ y an arbitrary function vanishing at. For every y > such that t > ɛ y we have by the strong Markov property applied at time S y, that P +T y < t = t y y1+δy t ɛy y P +U Sy dz, S y drp z s [, t r], U s = y P +U Sy dz, S y drp z s [, ɛ y ], U s y, the inequality in the former equation is owed to the fact that the OU process does not have negative jumps and hits the points from above. Using the Lamperti s transformation, it is straightforward that for every z ]y, y1 + δ y [ P z s [, ɛ y ], U s y = P s [, ɛ y ], ze s exp{ξ τe s 1/z} y P s [, ɛ y ], y1 + δ y e s exp{ξ τe s 1/y} y = P y s [, ɛ y ], 1 + δ y U s y. The weak convergence of U Sy /y, S y as y, implies that P +U Sy [y, y1 + δ y ], S y [, t ɛ y ] P e K ]1, 1 + δ y [, as y. Putting together equations 15 & 16 and the later fact we get the estimation P +T y < t P +U Sy [y, y1 + δ y ], S y [, t ɛ y ]P y s [, ɛ y ], 1 + δ y U s y P e K ]1, 1 + δ y [ P y s [, ɛ y ], 1 + δ y U s y, as y. Furthermore, by the definition of δ y we have that Pe K [1, 1 + δ y ] = P K [, βy 1 ] = 1 µ β 1 y Πrdr, 17 18

13 3. Proofs 37 and the last term in the former equation can be estimated in terms of the Laplace exponent. Specifically, there exist two constant c 1, c 2 depending only on φ such that φβ y c 1 1 β 1 y φβ y Πrdr c 2, β y µ β y see e.g. [1] Proposition III.1. Since β y is the inverse of φz/z we have by equations 17 & 18 that P +T y < t yc 1 P y s [, ɛ y ], 1 + δ y U s y, for every y small enough. Now, we shall show in Lemma 5 below that the function ɛ y can be chosen such that lim inf P y s [, ɛ y ], 1 + δ y U s y = ϑ >. 19 y Taking for granted this statement we end the proof since we have showed that for all t >, P +T y < e e t P +T y < t e t Cy as y, where e is an exponential random variable independent of U and C = c 1 ϑ. Lemma 5. We may choose ɛ y such that 19 holds true. Proof. Recall that β y is determined by φβ y /β y = y and that δ y = e 1/βy 1. The regular variation at infinity of φ will enable us to show that the functions ɛ y = ed/βy 1 y and a y = d 1/β y, with d > 1 arbitrary, are such that i ii ɛ y, a y,, as y, lim Pξ ɛ y a y = ϑ 1 >. y The reason why we require the functions ɛ y and a y to have this behavior is the following. Let s y = log1 + yɛ y, y > and note that τs s, for every s, since A s s for every s. Then on the event ξ ɛy a y, we have the inequalities exp{ξ τe sy 1/y} exp{ξ ɛy } e ay, due to the fact that ξ is an increasing process and τ e sy 1/y e sy 1/y = ɛ y.

14 38 A law of iterated logarithm for increasing self similar Markov processes So, on this event, we have also the inequalities y1 + δ y e sy exp{ξ } y1 + δy e sy+ay y, τ e sy 1/y from the definition of the functions s y and a y. Since s y ɛ y for every y small enough and the OU process does not have negative jumps, we can conclude by ii that lim inf y P s [, ɛ y ], y1 + δ y e s exp{ξ τe s 1/y} y lim y Pξ ɛy a y >. Then the proof reduces to show that the functions ɛ y and a y so defined satisfies i,ii. Let φ be the derivative of φ, and λ y the function determined by the relation Λu = φu uφ u, u >, φ λ y = a y ɛ y. Since φλ is concave and regularly varying with index β ], 1[ then φλ/λ is regularly varying with index β 1 and φ λ βφλ/λ. This implies in turn that β y and yβ y as y. Thus it is straightforward that ɛ y, and a y satisfy i, and moreover a y = Oyɛ y. This and the regular variation of φ imply that λ y = Oβ y. According to Jain and Pruitt [19] Theorem 5.1 the statement in ii is equivalent to The former is indeed true in our construction, Therefore lim ɛ yλλ y <. y φλy Λλ y = λ y φ λ y λ y 1 β λ y φ λ y β a y 1 β = λ y. ɛ y β ɛ y Λλ y λ y 1 β d 1 = O1. β y β This ends the proof in the case φ is regularly varying at infinity. When the Lévy measure is a finite measure, that is ξ is a compound Poisson process, we can take a y and ɛ y = e 1/βy 1/y y >. This choice of the functions a y, ɛ y is due to the fact that a compound Poisson process remains at zero during an exponential time and a fortiori lim P ξ ɛy a y >. y + The rest of the proof follows as in the case φ is regularly varying at.

15 3. Proofs 39 The last ingredient in the proof of Proposition 1 is the following result. Lemma 6. One has i ii lim sup y + E y e R < 1, lim inf y P yt x R >. Proof. i We know from equation 14 that Moreover, by Lemma 3 one has Thus Lemma 4 implies u 1, y = y 1 E + e T y 1 E y e R E + e T y u1 y, y. lim u 1 1, y = lim y y µ In particular, using equation 14 one gets 1/y dzρz = 1 µ. lim sup y + yu 1 y, y = θ <. lim sup E y e R = y ii The statement in i shows that for every t > lim sup P y R t y θ 1 + θ. θ 1 + θ. Since the OU process U hits the points continuously from above, it is plain that for every y < x P y T x < R = P y H x < R. Thus, for every t > and as a consequence P y H x < R P y H x < t P y R t, lim inf y + P yh x < R P +H x < t lim sup P y R t y P +H x < t θ 1 + θ. Since the OU process U is recurrent and without negative jumps we can ensure that P +H x < = 1. Then there exists a t > such that the right hand term in the former inequality is strictly positive. Lemma 6 ends the proof of Proposition 1 since we have noted that 1 is equivalent to 13.

16 4 A law of iterated logarithm for increasing self similar Markov processes 3.2 Proof of Proposition 2 This proof is based on the fact that one can relate the behavior of PI > t to that of the Laplace exponent φ of ξ by using connections between the behavior of Ee λi as λ and that of PI > t as t. This result can be proved using the results in Geluk [16]. However, for ease of reference we provide a complete proof based on a result due to Kasahara. We note that a similar result has been obtained in Haas [17] Proposition 11. Proof of Proposition 2. Since the moment generating function of I, is well defined, that is, ρs = Ee si < s >, we have the conditions to use Kasahara s Tauberian Theorem Bingham et al. [7] Theorem , it links the regular variation of log ρs as s with that of log PI > t as t. On the other hand the characteristic function of I, say f, is an entire function, admits a Taylor series fz = n a n z n, with a n = i n EIn n! = i n n k=1 φk n N, and its maximum modulus, coincides with ρs, that is e.g. Lukacs [23] Theorem Ms, f = sup { fz : z s}, Ms, f = ρs, s >, In order to apply Kasahara s Theorem we must check that log ρs, i.e. log Ms, f, is asymptotically regularly varying. To this end, we recall that we can estimate the behavior of log Ms, f in terms of the coefficients of the Taylor expansion of f. More precisely, suppose that lim sup n n log n log1/ a n = 1 β. 2 By Levin [22] section 1.13, if there exists a regularly varying function with index β, say ψ, such that then with ψ the asymptotic inverse of ψ. A version of ψ is lim a n 1/n ψn = e β, 21 n log Ms, f lim s ψ = β, 22 s ψ s = inf{r > ψr > s}. With the aim of obtaining the asymptotic behavior of log PI > t, let ϕs = s/ψs. Then ϕ is a regularly varying function with index 1 β and its asymptotic inverse, ϕ, varies regularly with index 1 β 1. Using equation 22, a straightforward application to Theorem in Bingham et al. [7] leads to log PI > t 1 βϕ t, t, and, provided that ρ decreases in some neighborhood of, we can apply Theorem op. cit. to get log ρt 1 βϕ t, as t.

17 3. Proofs 41 The rest of the proof is devoted to the proof of 2 and the fact that φ satisfies the equation 21. With this aim, recall that n 1 a n = EI n /n! = φk. As φ is regularly varying with index β, it can be expressed as φs = s β ls, with l a slowly varying function. Moreover, there exist two functions ε and c and a positive constant a, such that { t lt = exp ct + εs ds } s and εt and ct c with c R, as t. Therefore n n log 1/ a n = log φk = β log k + k=1 k=1 a k=1 Since n k=1 lim log k n n log n and for every slowly varying function l we have = 1, log lt lim =, t log t n log lk. it is straightforward that the lim sup in 2 is in fact a limit and equals 1/β. Next, we show that lim φn a n 1/n = e β. n To do this, observe that due to the fact that n! 1/n ne 1 we get a n 1/n ne 1 β exp{ 1 n log lk}. n Moreover, 1 n = 1 n = n log lk k=1 n n a n a 1 a εs ds n 1 s + n k εs ds s 1 n 1 n εs ds s + c, k=1 k=1 k+1 k k k+1 k k=1 εs ds s + 1 n k=1 εs ds + 1 s n n ck the last line is a consequence of Cesaro s theorem since ck c, and as k. Therefore k k+1 k εs ds s, a n 1/n e β φn 1. k=1 n ck k=1

18 42 A law of iterated logarithm for increasing self similar Markov processes 4 On time reversal of X. The aim of this section is to obtain a result on time reversal for a self similar process and then use it to prove Lemma 1 and Theorem 2. Let z > and P z the law of the process X defined by with the time change X t = z exp ξ τt/z, t τt = inf{s >, s e ξr dr > t}, and the convention that X t = if τt/z =. Define P the law of the process identical to. Then, under the family P z, z the process X is Markovian and has the scaling property defined in equation 1 with α = 1. We will say that X is the dual 1 self similar Markov process, cf. Bertoin and Yor [5] and the reference therein. Observe that is an absorbing state for X and let J be its lifetime, i.e., J = inf{t : X t = }. It should be clear that the distribution of J under P z is that of zi, where I is the Lévy exponential functional defined in 3. Last, denote F t, t the natural filtration and P t z, dy the semigroup of the dual 1 ss Markov process. Lemma 2 in [5], states that the q resolvents R q and R q of the processes X and X, respectively, are in weak duality with respect to the Lebesgue measure cf. Vuolle-Apiala and Graversen [31] for a related discussion. Thus duality also holds for the respective semigroups. We will refer to this result as the duality Lemma and we will use it to show, roughly speaking, that the law of the process Xr t, t < r X r = x, under P + is the same as that Xt, t < r J = r, under P x, with r, x > fixed. A rigorous statement will be done by using the method of h transform of Doob, see e.g. Sharpe [29] section 62, Fitzsimmons et al. [13],... To this end, note that by the self similarity of X, for any s > the law of the random variable X s under E + density p s z = µz 1 ρs/z, z >, has a for every s, t > and z > P t p s z = p t+s z. 23 The identity 23 follows from the duality Lemma and the fact that P +X s dz, s > is a family of entrance laws for the semigroup P t, of the 1 ss Markov process X. Equation 23 and the Markov property of X implies that for any r, x > the process h r t = p r t X t p r X 1 {t<r}, t >,

19 4. On time reversal of X. 43 is a P x martingale. Let Ω be the space of càdlàg maps from [, [ to [, [ killed at the first hitting time of. After Sharpe [29] Theorem 62.19, there exists a unique probability measure Q r x on Ω equipped with its natural filtration, rendering the process X an inhomogeneous Markov process with semigroup Q r t,t+sz, dy = P s z, dyp r t s y, 24 p r t z and such that Q r x X = x = 1. The measure Q r x has the property that for any s > for every F in F s. Q r xf 1 {s<j} = P x F h r s, 25 Lemma 7. i If F is F J measurable and g is a Borel function, then Ê x F gj = µ drp r xgrq r x F. 26 Thus, Q r x r> is a regular version of the family of conditional probabilities E + P x J = r, r >. ii Let r > fixed and G a bounded functional then G Xr t, t < r = Q r p r G X t, t < r, 27 where Q r p r denotes the law of the process X under Q r x with initial measure P +X r dx. It is implicit in the statement in ii of Lemma 7 that Q r x is the image under time reversal of a measure Q r x on Ω corresponding to X t, t < r under the conditional law P + X r = x. So the support of Q r x is the set Ω r of càdlàg paths that start at x and are absorbed at at time r. Proof. i By the Monotone class Theorem, to prove 26, it suffices to check that for any s the formula holds for every element of the form F = F {J > s} with F in F s. Indeed, note that on the set {s < xi} we have that xi = x τs/x e ξt dt + xe ξ τs/x I = s + xe ξ τs/x I, with I independent of ξ τu/x, u s and equal in law to I, owed to the strong Markov property of ξ. Using the fact that under P x the law of J is that of xi, the former equality and the strong Markov property of X we get that Ê x gj1 {s<j} F s = W s, X s,

20 44 A law of iterated logarithm for increasing self similar Markov processes where W s, z = Êz 1{<J} gs + J = E gs + zi = = µ s s drz 1 ρ r s/z gr drp r s zgr. Note that P z J dr = µp r z. dr Thereby an application of formula 25 gives Ê x F gj = Êx F W s, X s = µêx F dr p r s X s gr = µ = s drp r xgrêx F h r s drµp r xgrq r xf. ii We first verify that under P + the process Y t = X r t, < t < r, admits the semigroup defined in equation 24. Let a, b : [, [ ], [ be Borel functions and t, t + s [, r[. Indeed, by the duality lemma for ss Markov processes, we have that E + ayt by t+s = dz p r t s zbz E z ax s = dzazêzp r t s X s b X s = with Q r t,t+s the semigroup defined in equation 24. dzazp r t zêzp r t s X t b X t p r t z = E +ay t Q r t,t+sby t, By the Monotone class theorem, to prove ii it suffices to check that equation 27 holds for every G of the form f 1 X r t1 f n X r tn with f 1,..., f n positive bounded Borel functions and t 1 < < t n < r. Using the fact that the ss process X does not have fixed jumps we get that for n = 2 E + f1 X r t1 f 2 X r t2 = E + f1 X r t1 f 2 X r t2 Q r,t1 fq r t 1,t 2 f 2 X r the general case follows by iteration. Now we have all the elements to provide a = E + = dxp r xq r x f 1 X t1 f 2 X t2,

21 4. On time reversal of X. 45 Proof of Lemma 1. When h +, thanks to the Markov property of X, applied at time 1, our problem reduces to show that for every x > U h U P x lim = U = 1. h + h To this end, we recall that since ξ is a subordinator we have i ii ξ at time τ1/x is right continuous and ξ s lim s s =, iii τs lim = 1 P a.s. s s Using these facts and Lamperti s transformation it is straightforward that X ɛ X lim ɛ + ɛ =, P x a.s. The rest of the proof, in the case h +, follows by standard arguments. Next we use Lemma 7 to study the case h. By equation 27 we know that Ũ h P + lim Ũ = Ũ Ũ = x = Q 1 e h X1 e h X h x lim = h h + h X. Since for any x > and ɛ > the measure Q 1 x is absolutely continuous with respect to P x on the trace of {ɛ < J} in F ɛ, the result follows as in the case h + but this time for the dual self similar process X. Other interesting results on time reversal can be deduced from the duality Lemma by using the classical Theorem on time reversal of Nagasawa or its generalized version in Theorem 47 chapter XVIII Dellacherie et al. [11]. We will content ourselves with the following result and refer to Bertoin and Yor [5] and the reference therein for a related discussion. Proposition 3. Let x > fixed. Under Q 1 x the dual Ornstein Uhlenbeck process Û = {e t X1 e t, t > }, is an homogeneous strong Markov process with semigroup where H t is the dilatation H t fz = ftz. Q 1,1 e s H e sf, Proof. The homogeneity is obtained from the expression of the semigroup in 24 using the self similarity enjoyed by X under P x. Indeed, let f, g positive Borel functions then Q 1 x fe t X1 e tge t+s X1 e t+s = Q 1 x fe t X1 e tq 1 H 1 e t,1 e t+s e t+sg X 1 e t.

22 according whether ρ1/hsds < or =. 46 A law of iterated logarithm for increasing self similar Markov processes The expression of the semigroup can be reduced to Q 1 H 1 e t,1 e t+s e t+sgz = p e tz 1 Ê z g e t+s Xe t 1 e s p e t+s Xe t 1 e s = p 1 e t z 1 Ê e t z gûsp e s X 1 e s, = Q 1,1 e s H e sge t z where the second equality is owed to the self similarity and the obvious identity cp rc u = p r c 1 u. The strong Markov property follows from 25 by the optional stopping theorem using standard arguments. We have now the elements to prove the Theorem 2. Proof of Theorem 2. The statement in ii in Lemma 7 shows that for every positive and bounded functional F, F Ût, t R = E + F e t X e t, t R X 1 = x, Q 1 x with R resp. R the first return time of the process {e t X e t, t } resp. of Û, to its starting point. Moreover, by the stationarity of the OU process Ũ defined at the beginning of the subsection 3.1, one gets that E +R X 1 = x = E x R, and P + inf <t<r et X e t > y X 1 = x = P x inf <t<r e t X e t 1 > y. Recall that our proof of Proposition 1 is based on the fact that the OU process U is homogeneous and strong Markov and the probabilities that we considered there depend only on the excursion away its starting point. It should be then clear that thanks to Proposition 3 one can repeat the arguments in the proof of Proposition 1 to show that for any decreasing Borel function h we have Q 1 x Ût < ht i.o. t = or 1, We deduce from this criterion, the equation 27 and a time change that for any increasing Borel function l such that l = we have P + X t < tlt i.o. t = or 1, according whether ρ1/ls ds < or =. + s Rewriting the arguments in the proof of Theorem 1 we obtain the result. The former proof provides further information on the behavior of the dual 1 ss Markov process near its lifetime.

23 according whether 1/hs b e ashs b ds < or =. 5. Examples 47 Corollary 1. Let ξ be a subordinator such that its Laplace exponent φ is regularly varying at infinity with index β ], 1[ and < φ + <. Suppose that the density of the Lévy exponential functional associated to ξ satisfies hypothesis H. If X is the dual 1 ss process associated to ξ with lifetime J and f is the function defined in Theorem 1 then for any x > lim inf s X r1 s sf1/s = r1 β1 β Q r x a.s. Proof. In the previous proof we showed that for any x > and l an increasing Borel function we have Q 1 x X1 s < sls i.o. s = or = 1 according whether ρ1/ls ds < or =. + s Moreover, a straightforward verification of the finite dimensional distributions shows that the scaling property of X under P is translated for the dual OU process in the form: under Q r x the law of the process 1 r et Xr1 e t, t > is that of the dual OU under Q 1 x/r. The result follows as in the proof of Theorem 1. 5 Examples Example Watanabe process Let ξ be a subordinator with zero drift and Lévy measure νdx = abe bx dx, with a, b >. That is, ξ a compound Poisson process with jumps having an exponential distribution. Carmona et al. [1] 2 showed that in this case the density of the law of I = e bξs ds is given by ρx = a 2 xe ax, x >. So ρx satisfies the hypothesis H. The 1/b ss Markov process associated to ξ by Lamperti transformation is a process that arises in the study of extremes. More precisely, the 1/b ss Markov process associated to ξ is a Q Extremal process with { x, Qx = ax b x >. See Resnick [25]. This family of process is usually called generalized Watanabe process in honor to Watanabe S. who studied them, when b = 1, using the theory of Brownian excursions, see e.g. Revuz et Yor [26] pp. 54. We refer also to Carmona et al. [9] and the reference therein for the study of this process as a ss Markov process and its generalizations. Hence, thanks to Proposition 1 we obtain Corollary 2. Let X be a generalized Watanabe process and h an increasing function such that hs b /s is a decreasing function. Then P x Xs < hs i.o. s = or 1

24 48 A law of iterated logarithm for increasing self similar Markov processes This result appears in Yimin Xiao [33] Corollary 4.1 in the case b = a = 1. With the aim of providing a larger class of examples, in the following construction we make some assumptions on the subordinators that ensure that the density of I satisfies hypothesis H. It uses the recent results of Bertoin and Yor [6, 4]. Let Udx, be the renewal measure of ξ, i.e. E fξ s ds = [, fxudx. If the renewal measure is absolutely continuous with respect to Lebesgue measure, the function ux = Udx/dx, is usually called the renewal density. Proposition 4. Let ξ be a subordinator. Suppose that its renewal measure is absolutely continuous with respect to Lebesgue measure and that its renewal density ux, is a decreasing and convex function such that lim ut = 1 ], [, t µ i.e., Eξ 1 = µ. Then the density ρ, of the exponential functional associated to ξ satisfies the hypothesis H. Examples of such subordinators are those arising in Mandelbrot s construction of regenerative sets see e.g. Fitzsimmons et al. [14]. Proof. It is well known, that the renewal measure and the Laplace exponent of ξ are related by the formula 1 φλ = e λx uxdx. 28 An integration by parts in the former equation leads κλ = λ φλ = 1 µ + 1 e λx gxdx, where gx is the left hand derivative of ux. That is, κ is the Laplace exponent of a subordinator with killing term 1 µ, zero drift and Lévy measure with density gx. Integrating by parts, once more, we obtain that ψλ = λκλ = λ µ + e λx 1 λxν dx,, with νdx = dgx a Stieltjes measure. Specifically, ψλ is the Laplace exponent of a Lévy process, say ζ s, s, with no-positive jumps, drift term 1/µ and no Gaussian component. We have furthermore, that Eζ 1 = ψ + = 1 ], [, µ then ζ drifts to. This implies that the law of the exponential functional, I ψ, associated to ζ, is self decomposable, i.e., for every < a < 1, there exists an independent random variable J a such that J a + ai ψ has the same law as I ψ, we refer to Sato [27] for background on self decomposable laws. To see this, consider the first passage time above the level log a, that is ϱ a = inf{s > : ζ s > log a}. By the strong Markov property of ζ we have that ζ s = ζ ϱa+s ζ ϱa

25 5. Examples 49 is a Lévy process independent of {ζ r, r < ϱ a } and the same law as ζ. Moreover, by the absence of positive jumps and the fact that Eζ 1 ], [, we have that ζ ϱa = log a a.s. Therefore, e ζs ds = ϱa e ζs ds + e ζϱa e ζ sds = J a + ai ψ. As a consequence the density ρ ψ, of the law of I ψ, is unimodal, i.e., there exists a b > such that ρ ψ x is increasing on ], b[ and decreasing on ]b, [, see e.g. Sato [27] Theorem Besides, Bertoin and Yor [4] section 3, showed that 1 µ E fi φ = E I 1 ψ fi 1 ψ, for every positive measurable function f, in the obvious notation. In particular, the densities of I ψ and I φ are related by 1 µ ρ φx = 1 1 ψ x ρ, for every x >. 29 x We derive from this that ρ φ is a bounded and decreasing function on some neighborhood of. Remark 4. Equation 29 and the uniqueness of the invariant law for the OU process show that the law of I ψ is the invariant law of the OU process associated to the subordinator with Laplace exponent φ. Remark 5. Since every self decomposable law is infinitely divisible, then the law of I ψ is infinitely divisible. According to Steutel [3] its tail distribution is of the form log PI ψ > x = Ox log x, and since its density is decreasing on a set ]b, [ it follows by Theorem in Bingham et al. [7] that its density has the same behavior at infinity, i.e. log ρ ψ x = Ox log x x. This provides a complementary result to Proposition 2, log xρ φ x = Ox 1 log1/x, x. We take the following examples from Fitzsimmons et al. [14] and Bertoin and Yor [4], respectively. Example Let ξ be a subordinator without killing term, with zero drift and Lévy measure Πdx = βe x Γ1 βe x dx, 1 1+β with β ], 1[. An integration by parts in the Lévy Khintchine formula and a use of the beta integral show that the Laplace exponent of ξ is given by φλ = Γλ + β. Γλ Using equation 28 we get that the potential measure of ξ is absolutely continuous with respect to Lebesgue measure and that the renewal density is given by ux = 1 e x 1 β Γβ e x x >. 1

26 5 A law of iterated logarithm for increasing self similar Markov processes Therefore u is a convex decreasing function. Moreover, φλ λ β as λ. According to Lamperti [21], the increasing 1/β ss Markov process X, associated to ξ is a β stable subordinator. Then by Theorem 1 one gets lim inf t X t t 1/β log log t β 1/β = β1 β 1 β β. That is we recover the law of iterated logarithm for stable subordinators of Fristedt [15]. Furthermore, since under P + the law of X1 is that of an β stable random variable one can use Proposition 1 and the estimations of the stable density, see e.g. Zolotarev [34], to recover the Breiman s [8] test for stable subordinators. Example Let β ], 1[ and ξ be a subordinator with zero drift and Lévy measure Πdx = e x/β Γ1 β1 e x/β dx. 1+β By straightforward calculations we get that its Laplace exponent, say φ, can be expressed as and by the Stirling formula φλ = Γβλ + 1 Γβλ 1 + 1, φλ β β λ β as λ. Proceeding as in the former example we get that the renewal density of ξ is given by ux = 1 Γ1 + β ex/β 1 1 β, and is a convex decreasing function. Besides, since the law of the exponential functional I associated to this subordinator is characterized by its entire moments it is immediate that its Laplace transform is given by Ee si s n = E β s = Γnβ + 1. The function E β x is the so called Mittag-Leffler function. Hence, I follows the Mittag Leffler distribution, that is, I follows the same distribution as γ β β with γ β a β stable random variable. Furthermore, it can be showed without the use of Proposition 2 that log PI > x 1 ββ n= β 1 β x 1 1 β, x, see e.g. Bingham et al. [7] Theorem or Sato [27] solution to exercise This fact can be considered as a motivation for our proof of Proposition 2. More generally, one can consider the subordinator with Laplace exponent φ θ λ = Γβλ + θ Γβλ 1 + θ, 3

27 Bibliography 51 for β ], 1[ and θ β. See Bertoin and Yor [4] for a description of the Lévy measure corresponding to this Laplace exponent. The renewal density associated to this Laplace exponent admits the expression u θ x = 1 Γθ + 1 e xθ 1/β e x/β 1 1 β, x. Which is easily seen to be a decreasing and convex function. The entire moments of the exponential functional I θ associated to this subordinator are given by EI n θ = n!γθ Γβn + θ, n 1. We recognize in this formula the entire moments of a generalized Mittag Leffler distribution see e.g. Schneider [28]. Schneider showed that this distribution admits a density ρ β,θ x, whose behavior at infinity is ρ β,θ x Bx δ exp{c β x σ }, x, with σ = 1/1 β, δ = β θ + 1/2, c β = 1 ββ β 1 β, 31 1 β and B = 2π 1/2 Γθσ 1/2 β δ. This fact enables us to state the sharper result Corollary 3. Let X be the 1 ss process associated to a subordinator ξ with Laplace exponent defined by 3. If h : [, [ [, [ is a decreasing function then according whether with σ, c β and δ as in 31. P x X s < shs i.o s = or 1, hs δ exp { c β hs σ} ds s < or = Acknowledgement. I should like to express my gratitude to J. Bertoin who carefully read the manuscript and gave helpful advice. I am also grateful to M. E. Caballero for her permanent support and valuable comments. Research supported by a grant from CONACYT National Council of science and technology Mexico. Bibliography [1] J. Bertoin. Lévy processes, volume 121 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, [2] J. Bertoin. Darling-Erdős theorems for normalized sums of i.i.d. variables close to a stable law. Ann. Probab., 262: , [3] J. Bertoin and M.-E. Caballero. Entrance from + for increasing semi-stable Markov processes. Bernoulli, 82:195 25, 22. [4] J. Bertoin and M. Yor. On subordinators, self-similar Markov processes and some factorizations of the exponential variable. Electron. Comm. Probab., 6:95 16 electronic, 21.

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