2 Makoto Maejima (ii) Non-Gaussian -stable, 0 << 2, if it is not a delta measure, b() does not varnish for any a>0, b() a = b(a 1= )e ic for some c 2

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1 This is page 1 Printer: Opaque this Limit Theorems for Innite Variance Sequences Makoto Maejima ABSTRACT This article discusses limit theorems for some stationary sequences not having nite variances. 1 Independent sequences Although the concern of this volume is \long-range dependence," we start this article by explaining the case of independent rom variables for the sake of completeness. Throughout this article, all rom variables stochastic processes are real-valued. In the following, L() sts for the law of a rom variable, b() 2 R, is the characteristic function of a probability distribution on R. Let f j g be a sequence of independent identically distributed (iid) rom variables suppose that there exist a n > 0 "1 fk n gn k n " 1, c n 2 R such that L 1 k n! j + c n! (1.1) a n for some probability distribution, where! denotes weak convergence of probability distributions. Then is innitely divisible, where is said to be innitely divisible if for any n 1, there exists a probability distribution n such that b() = fb n ()g n. Any innitely dibisible probability distribution occurs as a limiting distribution as in (1.1), (see [4]). Three innitely divisible distributions which are concerned with in this article are the following. (i) Gaussian, if b() = expfim ; g m2 R 2 > 0: AMS Subject classication: Primary 60F05 secondary 60E07, 60G18. Keywords phrases: innite variance stationary sequences, stable semi-stable rom variables, stable semi-stable intergal processes

2 2 Makoto Maejima (ii) Non-Gaussian -stable, 0 << 2, if it is not a delta measure, b() does not varnish for any a>0, b() a = b(a 1= )e ic for some c 2 R: (iii) Non-Gaussian -semi-stable, 0 < <2, if it is not a delta measure, b() does not varnish for some r 2 (0 1) some c 2 R, b() r = b(r 1= )e ic : (1.2) When we want to emphasize r in (1.2), we refer to the distribution as (r )-semi-stable. If k n in (1.1) satises that k n =k n+1! 1asn!1,then in (1.1) is Gaussian or non-gaussian stable, if k n in (1.1) satises that k n =k n+1! r as n! 1 for some r 2 (0 1), then in (1.1) is non- Gaussian (r )-semi-stable, (see [11]). It follows from the dentions that is -stable if only if it is (r )-semi-stable for any r 2 (0 1). The \if" part can be weakened to that if is (r )-semi-stable for r = r 1 r 2 2 (0 1) such that log r 1 = log r 2 is irrational, then is -stable ([11]). Non-Gaussian stability semi-stability are also characterized by the Levy-Khintchine representation of b as follows ([8]): ; b() =exp ic + e is ; 1 ; isi[fjsj 1g] d ; H +(s) (1.3) + Z 0 0 ; e is ; 1 ; isi[fjsj 1g] d H; (jsj) jsj s where H (s) are nonnegative functions on (0 1)such that (i) H (s)=s are nonincreasing in s>0 (ii) H (r 1= s)=h (s), s>0, with r in (1.2). in (1.3) is (r )-semi-stable in general, if H (s) = C (constants), then it is -stable. Non-Gaussian semi-stable distributions do not have nite variances. Actually,if is non-gaussian -semi-stable, then E[jj ] < 1 for any < but E[jj ]=1, ( thus E[jj 2 ]=1, namely its variance is innite). Also if the limiting distribution in (1.1) is non-gaussian semi-stable, then iid rom variables f j g cannot have nite variances. (If f j g are strongly dependent, then partial sums of bounded rom variables may produce an innite variance limit law. See Section 5.) On the other h, if is Gaussian, it, of course, has nite variance, ( moreover all moments). However, this does not necessarily mean that, to get a Gaussian limit in (1.1), we need f j g with nite variance. Namely, some innite variance sequences produce Gaussian limits. We rst explain it. In the following, is the law of j in (1.1). Theorem 1.1. Suppose k n = n in (1:1). Then (1:1) holds for a Gaussian if only if h(z) = Z jxjz x 2 (dx)

3 Limit Theorems for Innite Variance Sequences 3 is slowly varying. For the proof, see [4] Section 2.6. If h(z) is a slowly varying function with h(z)!1, then, of course, E[j j j 2 ]=1, but we get a Gaussian limit. In the following, whenever we say \semi-stable" without any further comments, we include \stable" case as its special case. A stochastic process f(t) t 0g is said to be a Levy process if it has independent stationary increments, it is stochastically continuous, its sample paths are rightcontinuous have left limits, (0) = 0 almost surely. Iff(t) t 0g is a Levy process L((1)) is -semi-stable, then it is called an -semi-stable Levy process denoted by fz (t) t 0g. \Process" version of the limit theorem (1.1) is the following. See [15]. Theorem 1.2. Suppose (1:1) holds = L(Z (1)). Then 1 a n [knt] j + c n w! Z (t) where w! means the convergence in the space D([0 1)). For later use, we extend the denition of fz (t) t 0g to the case for t<0 in the following way. Let fz (;) (t) t 0g be an independent copy of fz (t) t 0g dene for t<0, Z (t) =;Z (;) (;t). 2 Semi-stable integrals Let f : R! R be a nonrom function. We consider integrals of f with respect to fz (t) t2 Rg. From now on, for simplicity, weassumethatthey are symmetric in the sense that L(Z (t)) = L(;Z (t)) for every t. Denote the characteristic function of L(Z (1)) by '() 2 R. In the rest of this article, we always assume that 6= 2, because we are only interested in innite variance cases. Theorem 2.1. If f 2 L (R), then = f(u)dz (u) can be dened in the sense of convergence in probability, is also symmetric -semi-stable such that E e i = exp log '(f(u))du : (2.1) For the stable case, see, e.g. [14], for the semi-stable case, see [13]. We call a semi-stable integral.

4 4 Makoto Maejima We dene semi-stable integral processes by (t) := f t (u)dz (u) where f t : R! R, f t 2 L (R) for each t 0. t 0 3 Limit theorems converging to semi-stable integral processes Let H > 0. A stochastic process f(t)g is said to be H-semi-selfsimilar if for some a 2 (0 1) f(at)g d = fa H (t)g (3.1) to be H-selfsimilar if (3.1) holds for any a>0, where = d denotes equality of all nite dimensional distributions. For more about semi-selfsimilarlity, see [10]. Here we consider two semi-stable integral processes of moving average type with innite variances represented by ; 1 (t) := jt ; uj H= ;juj H= dz (u) t 0 0 <H< 1 H 6= 1 2 (t) := (3.2) log t ; u u dz (u) t 0 1 <<2: (3.3) Both integrals are well dened because the integrs are L -integrable in the respective cases. f 1 (t) t 0g is H-semi-selfsimilar ([10]), is H-selfsimilar if fz (t)g is stable ([9]). f 2 (t)g is (1=)-semi-selfsimilar, is (1=)-selfsimilar if fz (t)g is -stable ([5]). Both processes have - semi-stable joint distributions ( thus do not have nite variances). The dependence structure of the increments of f 1 (t)g f 2 (t)g in the stable case is studied in [2]. Here we give limit theorems converging to f k (t)g k = 1 2: In the following,! denotes convergence in law. Suppose f j g are iid symmetric d rom variables satisfying when fz (t)g is -stable, or n = r n= n [r ;n ] j d! Z (1) (3.4) j d! Z (1)

5 Limit Theorems for Innite Variance Sequences 5 when fz (t)g is (r )-semi-stable. Take such that 1 ; 1 < < 1, dene a stationary sequence where Y k = j2z c j k;j k =1 2 ::: c j = 8 >< >: 0 if j =0 j ; if j > 0 ;jjj ; if j < 0: (3.5) We can easily see that the innite series Y k is well dened for each k Y k does not have nite variance. Dene further for H = 1 ;, W n (t) =n ;H V n (t) =r nh [nt] [r ;n t] Y k (3.6) Theorem 3.1. Let f 1 (t)g f 2 (t)g be the processes dened in (3:2) (3:3), respectively. Y k : (i) Suppose that fz (t)g is -stable. Then ( W n (t) ) d jj 1 (t) when 6= 0 2 (t) when =0 where d ) denotes convergence of all nite dimensional distributions. (ii) Suppose that fz (t)g is (r )-semi-stable. Then ( V n (t) ) d jj 1 (t) when 6= 0 2 (t) when =0: As to the proofs, for the case of stable f 1 (t)g, see [9], for the case of stable f 2 (t)g, see [5]. For the semi-stable case, the proof can be carried out in almost the same way as for the stable case. Remark 3.1. If <0 (necessarily >1), then H =1= ; >1=. Thus the normalization n H in (3.6) grows much faster than n 1= in (3.4), the case of partial sums of independent rom variables. This explains that fy k g exhibits long range dependence.

6 6 Makoto Maejima 4 Rom walks in rom scenery Consider two independent Levy processes fz (t)g fz (t)g. Suppose that fz (t)g is -semi-stable with 0 <<2 fz (t)g is -stable with 1 < 2. Then the local time of fz (t)g, L t (x), exists (see [3]) we can dene (t) = L t (x)dz (u) t 0 since fz (t)g is a semimartingale (see [12]). f(t)g is H-selfsimilar if fz (t)g is stable ([6]), is H-semi-selfsimilar if fz (t)g is semi-stable ([1]), where H = 1 ; 1= +1=() (> 1=2). Although (t) itself is not distributed as to be semi-stable, the variance of (t) is innite, because of the innite variance property of fz g. Let fs n n 0g be an integer-valued rom walk with mean zero f(j) j 2 Zg be a sequence of independent identically distributed symmetric rom variables, independent of fs n g.further assume that n = S n d! Z (1) r r= n = [r ;n ] n (j) d! Z (1) (j) d! Z (1) when fz (t)g is stable when fz (t)g is semi-stable. Consider a stationary innite variance sequence f(s k ) k 0g. This is the problem of rom walks in rom scenery. Theorem 4.1. Let H =1; 1= +1=(). Under the above assumptions, we have r nh [nt] n ;H [r ;n t] (S k ) d ) (t) when fz (t)g is stable (4.1) (S k ) d ) (t) when fz (t)g is semi-stable. (4.2) Kesten-Spitzer [6] proved (4.1) Arai [1] extended (4.1) to the semistable case (4.2). Remark 4.1. If > 1, then H = 1 ; 1= +1=() > 1=. By the same reason as in Remark 3.1, f(s k ) k 0g exhibits long range dependence.

7 Limit Theorems for Innite Variance Sequences 7 5 An innite variance limit law may arise from sums of bounded rom variables Recently, Koul Surgailis [7] gave an example that shows the title of this section above. Consider a stationary sequence fy k g in Section 3, assume that (3.4) holds for -stable Z (1) with 1 < < 2 in (3.6) is negative. They proved the following. Theorem 5.1. Let h be a real-valued measurable function of bounded variation such that E[h(Y0)] = 0. Then n = n h(y k )! Ce d hz (1) where C > 0, e h = ; RR f(x)dh(x) f is the density function of L(Y 0). Their comment on this theorem is \This theorem is surprising in the sense it shows that an -stable (1 < < 2) limit law may arise from sums of bounded rom variables fh(y k )g. This is unlike the case of independent identically distributed or weakly dependent summs." References [1] T. Arai (2001) : A class of semi-selfsimilar processes related to rom walks in rom scenery. To appear in Tokyo J. Math. [2] A. Astraukas, J.R. Levy M.S. Taqqu (1991) : The asymptotic dependence structure of the linear fractional Levy motion. Lithuanian Math. J. 31 (1), 1{28. [3] E. Boylan (1964) : Local times for a class of Marko processes. Illinois J. Math. 8, 19{39. [4] B.V. Gnedenko A.N. Kolmogorov (1968): Limit Distributions for Sums of Independent Rom Variables. Addison-Wesley. [5] Y. Kasahara, M. Maejima W. Vervaat (1988) : Log-fractional stable processes. Stoch. Proc. Appl. 30, 329{229. [6] H. Kesten F. Spitzer (1979) : A limit theorem related to a new class of self similar processes. Z. Wahrscheinlichkeitstheorie verw. Gebiete 50, 5{25. [7] H.L. Koul D. Surgailis (2001) : Asymptotics of weighted emprical processes of long memory moving averages with innite variance. To appear in Stoch. Proc. Appl.

8 8 Makoto Maejima [8] P. Levy (1937) : Theorie de l'addition des Variables Aleatoires. Genthier-Villars. [9] M. Maejima (1983) : On a class of self-similar processes. Z. Wahrscheinlichkeitstheorie verw. Gebiete 62, 235{245. [10] M. Maejima K. Sato (1999) : Semi-selfsimilar processes. J. Theoret. Probab. 12, 347{383. [11] D. Mejzler (1973) : On a certain class of innitely divisible distributions. Israel J. Math. 16, 1{19. [12] P.A. Meyer (1976) : Un cours sur les integrales stochastiques. Seminaire de Probabilites. Univ. de Strasbourg. Lecture Notes in Math. 511, Springer. [13] B. Rajput K. Rama-Murthy (1987) : Spectral representation of semistable preocesses, semistable laws on Banach spaces. J. Multivariate Anal. 21, 139{157. [14] G. Samorodnitsky M.S. Taqqu (1994) : Stable Non-Gaussian Rom Processes. Chapman Hall. [15] A.V. Skorohod (1957) : Limit theorems for stochastic processes with independent increments. Theory Probab. Appl. 2, 138{171. Makoto Maejima Department of Mathematics, Keio University, , Hiyoshi, Kohoku-ku, Yokohama , Japan maejima@math.keio.ac.jp