J. Austral. Math. Soc. 72 (2002, 199 208 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. 2000 Mathematics subect classification: primary 60G17, 60G51, 60F15. 1. 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 511 512]. 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 1446-8107/2000 $A2:00 + 0:00 199
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] 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 ;
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 315 316]. 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] 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/ ] because @ 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
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 (2.5. 3. 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] 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
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 (3.6 2 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 144 145], (3.8 ( U 1 + +U 2 = 0 = lim.2t / 1 ².Þ/ 2 dþ = lim.2t / 1 [E.cos ÞU 1 /] 2 dþ
[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
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, 1996. [2] K. L. Chung, A course in probability theory (Harcourt, Brace & World, New York, 1968. [3] J. L. Doob, Stochastic processes (Wiley, New York, 1953. [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. 241 396. [5]. Hartman and A. Wintner, On the infinitesimal generator of integral convolutions, Amer. J. Math. 64 (1942, 273 298. [6] W. Stade, Two ergodic sample-path properties of the oisson process, J. Theor. robab. 11 (1998, 197 208. [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. 119 166. [8] D. W. Stroock, robability theory: an analytic view (Cambridge University ress, Cambridge, 1992. Department of Statistics Department of Mathematics The Hebrew University of Jerusalem and Computer Science Mount Scopus, Jerusalem 91905 University of Osnabrück Israel 49069 Osnabrück e-mail: offer.kella@hui.ac.il Germany e-mail: wolfgang@mathematik.uni-osnabrueck.de