ERGODIC PATH PROPERTIES OF PROCESSES WITH STATIONARY INCREMENTS

Transcription

1 J. Austral. Math. Soc. 72 (2002, ERGODIC ATH ROERTIES OF ROCESSES WITH STATIONARY INCREMENTS OFFER KELLA and WOLFGANG STADJE (Received 14 July 1999; revised 7 January 2001 Communicated by V. Stefanov Abstract For a real-valued ergodic process X with strictly stationary increments satisfying some measurability and continuity assumptions it is proved that the long-run average behaviour of all its increments over finite intervals replicates the distribution of the corresponding increments of X in a strong sense. Moreover, every Lévy process has a version that possesses this ergodic path property Mathematics subect classification: primary 60G17, 60G51, 60F Introduction Let X =.X.t// t 0 be a real-valued process with strictly stationary increments, that is, the distribution of.x.s + t/ X.s// t 0 is the same for every s 0. All strictly stationary processes and all Lévy processes have this property. We assume that the underlying probability space is complete and that X is separable, measurable, and ergodic. For example, every separable centered Lévy process satisfies these conditions [3, pages 422 and ]. We will show that under certain regularity conditions almost all sample paths are connected to the distribution of X in the following strong sense: Call a function x :[0; / R an X-function if for every n N, disoint finite intervals I 1 ;::: ;I n [0; / that are open from the left and closed from the right, and real numbers u 1 ;::: ;u n the following asymptotic relation holds: (1.1 lim T 1 ½{t [0; T ] 1x.I + t/ u for = 1;::: ;n} = ( 1X.I / u for = 1;::: ;n ; c 2002 Australian Mathematical Society /2000 $A2:00 + 0:00 199

2 200 Offer Kella and Wolfgang Stade [2] where ½ denotes Lebesgue measure, I + t ={s + t s I} is the interval I shifted by t, and1x.i/ = x.b/ x.a/ is the increment of x. / in I =.a; b]. THEOREM 1. Assume that (1.2 X.I/ = sup{ 1X.I / I I} 0; as ½.I/ 0 and that (1.3 ( 1X.I/ = u = 0 for all intervals I [0; / and u R: Then almost every sample path of X is an X-function. This theorem is proved in Section 2. In Section 3, X is taken to be an arbitrary Lévy process. In this case we show that there is always a version of X for which almost all sample paths are X-functions (even without the conditions (1.2 and (1.3. It is not difficult to prove that for any fixed n N, any prespecified u 1 ;::: ;u n N and any intervals I 1 ;::: ;I n as above the limiting relation (1.4 lim T 1 ½{t [0; T] 1X.I + t/ u for = 1;::: ;n} = ( 1X.I / u for = 1;::: ;n holds almost surely; but the exceptional null set on which (1.4 is not valid depends on n, u 1 ;::: ;u n and I 1 ;::: ;I n. In order to show that there is a universal null set on whose complement (1.4 holds for all n, u i and I i, we need that the increments of X are locally small uniformly in probability (that is, assumption (1.2 and pointwise convergence of the distribution function of the random vector.1x.j 1 /;::: ;1X.J n // to that of.1x.i 1 /;::: ;1X.I n //, if the intervals J i increase or decrease to the bounded intervals I i, i = 1;::: ;n, which requires assumption (1.3. For Lévy processes, there is always, after suitably centering, a version satisfying (1.2. Moreover, by a classical theorem due to Hartman and Wintner [5], X.t/ can have an atom for some t > 0 only if X is a compound oisson process with drift. Hence, (1.3 holds except for this special case. If X is compound oisson with drift zero, the set of discontinuity points of the distribution function (d.f. of X.t/ is the same for every t > 0. Using this observation and the explicit form of the d.f. of 1X.I/ as a oisson sum of convolutions, we will show in Section 3 that the assertion of Theorem 1 remains true for compound oisson processes (also with nonzero drift and thus for Lévy processes in general. We conclude the paper with a new short and elementary proof of the Hartman-Wintner theorem, which is seen to be an immediate consequence of a neat inequality, of independent interest, for sums of i.i.d. symmetric random variables.

3 [3] Ergodic path properties 201 An interesting consequence of our results is that there exist càdlàg functions x :[0; / R for which (1.5 lim T 1 ½{t [0; T ] 1x.I + t/ u ; = 1;:::n} n = lim T 1 ½{t [0; T] 1x.I + t/ u } =1 for all n N and all I 1 ;::: ;I n ; u 1 ;::: ;u n as above, and these limits are strictly between 0 and 1. Indeed, the set of these functions has probability 1 under any distribution on the space of càdlàg functions generated by some non-deterministic Lévy process. Constructing such a function seems to be quite difficult; we know no explicit example of a function with this property. Note that the limits on the right-hand side of (1.5 can be specified to be given by lim T 1 ½{t [0; T ] 1x.I + t/ u} = ( 1X.I/ u = ( X.b a/ u for all I =.a; b] and u R,whereX is an arbitrary Lévy process. By suitably choosing the underlying Lévy process we can also achieve various additional properties of x. / besides (1.5, such as continuity, monotonicity, having only positive umps, etc. It is one of the fundamental ideas in probability theory that the average behaviour of a single realization of a stochastic process over a long time horizon should replicate the underlying distribution of the process. roperty (1.1 is a strong version of this principle. For sample properties of Lévy processes see Fristedt [4] and Bertoin [1]. Recently, a sample path approach has been frequently used to analyze stochastic systems by studying a fixed typical realization (see for example Stidham and El- Taha [7]. For the oisson process, relation (1.1 and related questions were studied in [6]. 2. roof of Theorem 1 All intervals below are open from the left and closed from the right. Fix the intervals I; I 1 ;::: ;I n and the real numbers u; u 1 ;::: ;u n. Define the auxiliary processes Y t = 1 {1X.I +t/ u ; =1;::: ;n}; Z t = 1 { X.I+t/ u}: Obviously, Y t and Z t can be written in the form Y t = f (.X.s + t/ X.t// s 0 ; Z t = g (.X.s + t/ X.t// s 0 ;

4 202 Offer Kella and Wolfgang Stade [4] so that the processes.y t / t 0 and.z t / t 0 are stationary and ergodic. They are also measurable and bounded. Thus, the ergodic theorem yields (2.1 (2.2 T 1 Y t.!/ dt 0 T 1 Z t.!/ dt 0 a:s: E.Y 0 / = ( 1X.I / u ; = 1;::: ;n ; a:s: E.Z 0 / = ( X.I/ u (see for example [8, pages ]. The exceptional null sets on which convergence in (2.1 orin(2.2 does not hold depend on n; I 1 ;::: ;I n, u 1 ;::: ;u n or on I; u, respectively. Taking the (denumerable union of all these null sets over n N, intervals I; I 1 ;::: ;I n [0; / with rational endpoints and u; u 1 ;::: ;u n Q we get a set of probability 0. On its complement C (a set of probability 1 relations (2.1 and (2.2 hold for all n N, u; u 1 ;::: ;u n Q and I; I 1 ;::: ;I n [0; / with rational endpoints. From now on we only consider sample paths corresponding to points in C. Now let I 1 ;::: ;I n, u 1 ;::: ;u n be arbitrary. Let.u.m/ sequences of rational numbers such that u.m/ where " m is rational and " m 0. Furthermore, choose intervals J rational endpoints such that J cover the left (right endpoint of I and satisfy I J / m N, = 1;::: ;n, be u,asm,andu u.m/ 2" m > 0,, L, R with I approximates I from inside and the L (R L R ; = 1;::: ;n; ½.L / 0and½.R / 0; as k : Clearly, the following inclusion holds for every, m, k and t: { } { 1X.J + t/ u.m/ 1X.I + t/ u or X.L + t/ ( u u.m/ =2 or X.R + t/ ( } u u.m/ =2 : Hence, for every sample path of X and for every T > 0wehave { t [0; T] 1X.J + t/ u.m/ ; = 1;::: ;n } { t [0; T ] 1X.I + t/ u ; = 1;::: ;n } n =1 n =1 { t [0; T ] X.L { t [0; T ] X.R + t/ ( } u u.m/ =2; = 1;::: ;n + t/ ( } u u.m/ =2; = 1;:::;n :

5 [5] Ergodic path properties 203 It follows that ½ { t [0; T] 1X.I + t/ u ; = 1;::: ;n } ½ { t [0; T ] 1X.J =1 ½ { t [0; T] X.L + t/ u.m/ ; = 1;::: ;n } +t/>" m } From (2.1 and (2.2 we can now conclude that =1 ½ { t [0; T ] X.R +t/>" m } : (2.3 lim inf T 1 ½{t [0; T ] 1X.I + t/ u ; = 1;::: ;n} ( 1X.J =1 / u.m/ ; = 1;::: ;n ( X.R />" m : =1 ( X.L />" m Now let k in (2.3. Condition (1.2 clearly implies that X is stochastically continuous so that.1x.j 1 /;::: ;1X.J n //.1X.I 1 /;::: ;1X.I n // with respect to the Euclidean metric. It follows that (.1X.J 1 /;::: ;1X.J n // B (.1X.I 1 /;::: ;1X.I n // B for all Borel sets in R n satisfying (.1X.I 1 /;::: ;1X.I n B = 0. We can take B = n =1. ; u.m/ ] B {y R n y = u.m/ for some } and the increments 1X.I / have continuous distributions (by condition (1.3. Therefore, we obtain lim ( 1X.J / u.m/ ; = 1;::: ;n = ( 1X.I / u.m/ ; = 1;::: ;n : k The other probabilities on the right-hand side of (2.3 all tend to zero because " m > 0 and m is still fixed. Thus, letting first k and then m, inequality (2.3 yields (2.4 lim inf T 1 ½{t [0; T ] 1X.I + t/ u ; = 1;::: ;n} ( 1X.I // u ; = 1;::: ;n : For the reverse direction, take sequences v.m/ u as m, v.m/ v.m/ v > 2" m > 0; " m Q. Then { } { 1X.I + t/ u 1X.J + t/ v.m/ or X.L + t/ >" m or X.R + t/ >" m } : Q and

6 204 Offer Kella and Wolfgang Stade [6] As above, (2.1 and (2.2 imply that lim sup T 1 ½{t [0; T ] 1X.I + t/ u ; = 1;::: ;n} ( 1X.J / v.m/ + =1 ( X.L />" m + ( X.R />" m : =1 Letting first k, then m and reasoning as above we obtain (2.5 lim sup T 1 ½{t [0; T] 1X.I + t/ u ; = 1;::: ;n} ( 1X.I / u ; = 1;::: ;n : The assertion follows from (2.4 and ( Lévy processes We now consider the result for Lévy processes. Every Lévy process has, after a suitable deterministic centering, a version with càdlàg paths, and we will take such a version X from now on. Furthermore, we assume that X.0/ = 0. Note that for this X we have X.I/ = D a:s: sup 0 t ½.I/ X.t/ 0. Thus, Theorem 1 implies that almost every sample path of X is an X-function if the distribution of X.t/ is continuous for every t > 0. What happens if (1.3 does not hold, that is, if ( X.t/ = u > 0forsome t > 0andsomeu R? Then a classical result by Hartman and Wintner [5] states that X must be a compound oisson process with drift (see also [1, pages 30 31]. The next theorem covers this case. THEOREM 2. If X is a compound oisson process with drift, then almost all its paths are X-functions. ROOF. If (1.1 holds for the function x and the process X, it is also valid for.x.t/ + þt/ t 0 and.x.t/ + þt/ t 0. Thus, we can assume that the drift of X is zero, so that X is piecewise constant between umps distributed according to some d.f. F,and (3.1 ( X.I/ >0 = e b½.i/ ; where b > 0 is the intensity of the underlying oisson process. We have ( 1X.I/ u = e b½.i/ i=0.b½.i// i F i.u/; i!

7 [7] Ergodic path properties 205 where F i is the i-fold convolution of F. LetD be the union of the sets of discontinuity points of F i, i Z +. Then D is denumerable and it is the set of atoms of 1X.I/ for any I. Now we repeat the construction of Theorem 1 but take C to be the set of all points! for which (2.1 and (2.2 hold for all n N, all intervals I; I 1 ;::: ;I n [0; / with rational endpoints and all u; u 1 ;::: ;u n Q D. Then.C/ = 1 and it suffices to consider paths corresponding to points in C. as in Theo- For arbitrary I 1 ;::: ;I n and u 1 ;::: ;u n take sequences J rem 1, but set Then for all, k, m, t { 1X.J + t/ ũ.m/ so that we obtain, for all k, m, ũ.m/ = { u.m/ if u = Q D; u if u Q D:, L, R } { 1X.I + t/ u or X.L />0or X.R />0 } (3.2 lim inf T 1 ½{t [0; T] 1X.I + t/ u ; = 1;::: ;n} ( 1X.J / ũ.m/ ; = 1;::: ;n =1 ( X.L />0 =1 ( X.R />0 : Let k. Then, by (3.1, the two sums in (3.2 converge to 0. The first term on the right-hand side is equal to n ( =1 1X.J / ũ.m/ and (3.3 ( 1X.J / ũ.m/ = e b½.j / b½.i / e i=0 [b½.j /] i F i.ũ.m/ / i! [b½.i /] i i=0 i! F i.ũ.m/ /; as k by bounded convergence, since ½.J / ½.I / as k, and the renewal function F i i=0.t/ is finite for all t 0. Now let m. If u = Q D, then F i.ũ.m/ / F i.u / because every F i is continuous in u.butifu Q D, then F i.ũ.m/ / = F i.u / for all i Z +. Hence, the limit in (3.3 tends to b½.i / e [b½.i /] i F i.u / = ( 1X.I / u : i! i=0

8 206 Offer Kella and Wolfgang Stade [8] We have proved that (3.4 lim inf T 1 ½{t [0; T ] 1X.I + t/ u ; = 1;::: ;n} n ( 1X.I / u : =1 For the other direction we follow the proof of Theorem 1 and obtain, for every k and m, (3.5 lim sup T 1 ½{t [0; T] 1X.I + t/ u ; = 1;::: ;n} n =1 ( 1X.J / v.m/ + ( X.L />0 + =1 =1 ( X.R />0 : As k ; the two sums in (3.5 converge to zero by (3.1, while the product tends to n ( =1 1X.I / v.m/. By right-continuity, this latter product converges to n ( =1 1X.I / u as m ; since v.m/ u, = 1;::: ;n. We have shown that for every centered Lévy process there is a version X for which almost all paths are X-functions. But the centering function, say f, can be chosen to be additive, that is, to satisfy the equation f.t + s/ = f.t/ + f.s/ for all t; s 0. Now note that if a function x :[0; / R satisfies (1.1 for some process X, then x + f satisfies (1.1 for the process X + f. Hence, for every (not necessarily centered Lévy process there is a version with almost all paths being X-functions. Finally, we remark that the Hartman-Wintner theorem on which Theorem 2 relies is a straightforward consequence of the following interesting inequality. THEOREM 3. Let U 1 ; U 2 ;::: be a sequence of i.i.d. symmetric random variables satisfying ( U 1 = 0 = 0. Then for every N, ( U 1 + +U 2 = 0 ( 2 ( and (3.7 ( U 1 + +U 2 +1 = 0 ( U 1 + +U 2 = 0 : ROOF. Let ².Þ/ = E.e iþu1 /. By Fourier inversion ([2, pages ], (3.8 ( U 1 + +U 2 = 0 = lim.2t / 1 ².Þ/ 2 dþ = lim.2t / 1 [E.cos ÞU 1 /] 2 dþ

9 [9] Ergodic path properties 207 lim sup.2t / 1 = lim sup.2t / 1 E = lim sup E.cos 2 ÞU 1 / dþ ( cos 2.ÞU 1 / dþ U1 E (.2TU 1 / 1 cos 2 xdx U 1 ( 2 = 2 2 : The inequality in (3.8 follows from.e.cos ÞU 1 // 2 E.cos 2.ÞU 1 //, which is a consequence of Jensen s inequality, and for the second-last equality we have used the substitution x = ÞU 1, which is possible as ( U 1 = 0 = 0. Finally, since ².Þ/ [ 1; 1] by symmetry, (3.7 follows from ( U 1 + +U 2 +1 = 0 = lim ².Þ/ 2 +1 dþ lim ².Þ/ 2 dþ = ( U 1 + +U 2 = 0 : REMARK. Inequality (3.6 states that in the considered class of random walks the probability ( U 1 + +U 2 = 0 is maximal for the simple ±1-walk for which ( U 1 = 1 = ( U 1 = 1 = 1=2. Now assume that Y is a Lévy process which is not a compound oisson process with drift. Let Y be an independent copy of Y. Then X = Y Y is a symmetric Lévy process, and since ( Y.t/ = a 2 ( X.t/ = 0 for all a R and t > 0, we have to prove ( X.t/ = 0 = 0. For arbitrary Ž>0letX Ž 1 be the process obtained from X by deleting all umps that are smaller than Ž in absolute value and set X Ž = X 2 XŽ. Then 1 X Ž is a symmetric compound oisson process of intensity ¹ 1 Ž, say, and lim Ž 0 ¹ Ž =. Let Ž ( Ž be the characteristic function of X Ž.t/ t;1 t;1 1 (XŽ 2.t/. By Fourier inversion, (3.9 ( X.t/ = 0 = lim.2t / 1 lim.2t / 1 = ( X Ž 1.t/ = 0 = e ¹Žt =0 ( 2 Ž.Þ/ Ž t;1 t;2.þ/ dþ Ž t;1.þ/ dþ =0 2 2 ( [¹Ž t] 2 e [¹ Žt] ¹Žt ( U Ž 1! + +U Ž = 0 + [¹ Žt] 2 +1 ;.2 /!.2 + 1/! where the U Ž i are the ump sizes of X Ž 1, which are certain i.i.d. symmetric random variables satisfying ( U Ž i = 0 = 0. The first inequality in (3.9 follows from

10 208 Offer Kella and Wolfgang Stade [10] t;1.þ/ 0and t;2.þ/ 1 for all Þ R and the second from the Lemma. But as Ž 0, we have ¹ Ž and thus the right-hand side of (3.9 tends to zero. Hence, ( X.t/ = 0 = 0. References [1] J. Bertoin, Lévy processes (Cambridge University ress, Cambridge, [2] K. L. Chung, A course in probability theory (Harcourt, Brace & World, New York, [3] J. L. Doob, Stochastic processes (Wiley, New York, [4] B. Fristedt, Sample functions of stochastic processes with stationary independent increments, in: Advances in probability and related topics, Vol. 3 (eds.. Ney and S. ort (Dekker, New York, 1974 pp [5]. Hartman and A. Wintner, On the infinitesimal generator of integral convolutions, Amer. J. Math. 64 (1942, [6] W. Stade, Two ergodic sample-path properties of the oisson process, J. Theor. robab. 11 (1998, [7] S. Stidham and M. El-Taha, Sample-path techniques in queueing theory, in: Advances in queueing (ed. J. W. Dshalalow (CRC ress, Boca Raton, 1998 pp [8] D. W. Stroock, robability theory: an analytic view (Cambridge University ress, Cambridge, Department of Statistics Department of Mathematics The Hebrew University of Jerusalem and Computer Science Mount Scopus, Jerusalem University of Osnabrück Israel Osnabrück offer.kella@hui.ac.il Germany wolfgang@mathematik.uni-osnabrueck.de