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Some Simple Properties of Sums of Randoms Variable Having Long-Range Dependence Author(s): A. C. Davison and D. R. Cox Source: Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 424, No. 1867 (Aug. 8, 1989), pp. 255-262 Published by: Royal Society Stable URL: http://www.jstor.org/stable/2398367 Accessed: 27-06-2017 15:15 UTC JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://about.jstor.org/terms Royal Society is collaborating with JSTOR to digitize, preserve and extend access to Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences

Proc. R. Soc. Lond. A 424, 255-262 (1989) Printed in Great Britain Some simple properties of sums of random variables having long-range dependence BY A. C. DAVISON1 AND D. R. COX2, F.R.S. 'Department of Mathematics, Imperial College, London SW7 2BZ, U.K. 2Nuffield College, Oxford OXI 1NF, U.K. (Received 17 November 1988) The higher-order moments and cumulants of sums of a special case of random variables having long-range dependence are investigated. Tensor methods are used to simplify the calculations. The limiting form of the second and third cumulants as the number of variables added becomes large is studied by analytical and numerical methods. The implications are discussed for the existence of non-gaussian limits of sums of random quantities of finite variance and long-range dependence. 1. INTRODUCTION Phenomena that can be represented by a stationary random process in time or space occur quite widely in the natural sciences and in technology. Provided such a process {Xt} has finite variance, key properties in discrete time are the mean E(Xt), and the second-order properties, namely the autocovariance and autocorrelation yt = COy (XO,Xt), pt = yt/yo, where y0 is the variance of the process. We use index notation for random variables and their cumulants, including also pt and yt above, rather than the more familiar power notation. Thus, X2 is the second element of X1,..., x8, and the square of X1 is X1Xl. An equivalent specification of the second-order properties of the process is by the power spectral density function gt( -itw* t=-oo Such processes can be classified in various ways. One important distinction is between short-range and long-range dependence. Essentially in the former E yt < oo and the power spectrum is finite at the origin. For long-range dependence, E yt is divergent and the power spectrum has a singularity at origin. This is broadly equivalent to power law decay of yt with small index; yt flt-a for 0 < a < 1. Processes satisfying this last condition have a property of asymptotic self-similarity. That is, sums of n consecutive values X1 +... +Xn,xn+l +... +x2n, (1.1) have exactly or approximately the same autocorrelation function as the original [ 255 ]

256 A. C. Davison and D. R. Cox process. If the autocorrelation functions are exactly the same, it can be shown that t= 182(t2-a), (1.2) where 82 denotes a central second difference operator, and we shall for simplicity use this form below. If {Xt} are independently and identically distributed with finite variance, the limiting distribution of the sums in (1.1) as n --oo is gaussian. This conclusion-carries over with minor conditions to processes with short-range dependence. In these situations non-gaussian limits for sums can only arise through components of infinite variance. Rosenblatt (i96i) showed that when long-range dependence is present non-gaussian limits can result, even though the process has finite variance. The book edited by Elberlein & Taqqu (I986) gives a survey of related mathematical results, and includes a bibliography on long-range dependence. A standard and in principle elementary method of investigating the speed of convergence to gaussian form and possible non-gaussian limits is by the calculation of moments, or more commonly cumulants, of order three or higher. McCullagh (I987) has recently shown that such calculations can be systematized in an elegant way by use of tensor notation. The present paper applies such methods to obtain some simple results about a special class of processes with long-range dependence and non-gaussian limits. We use a combination of exact calculation by tensor methods, elementary asymptotic analysis of the resulting expressions, exact numerical evaluation and computer simulation. 2. NON-G*AUSSIAN LIMITS FOR SUMS We consider the following special case. First we suppose that {Yt} is a stationary gaussian process of zero mean having the exactly self-similar autocorrelation function (1.2). The exactly self-similar character is not needed for our qualitative conclusions. Let Xt = h(yt) for some function h( ). We suppose without loss of generality that E{h(Yt)} = 0. Now suppose that h( ) can be expanded in a Taylor series. Our arguments are general but it will be enough for our purposes to consider the first two terms of the expansion h(yt) = a, Yt+a2(Yt Yt 1)..., (2.1) where in general the rth term is conveniently written in the form ar Hr(Yt); here Hr is the rth Hermite polynomial. It is known that the limiting distribution of the sums (1.1) is determined by the relation between the rate of decay of yt for large t and the Hermite rank of h( ), defined as the smallest non-zero coefficient in expansion (2.1). Thus if we take just two terms, we expect a different behaviour when al = 0, a2 = 0, Hermite rank 2, from that when a, = 0, Hermite rank Taqqu (I975) shows that if yt flt-a and the Hermite rank of h(o) is 1 then th sums in (1. 1) have a limiting gaussian distribution if 0 < a < 1, but that if h( ) has Hermite rank 2, a non-gaussian limit results if 0 < a < 2. We concentrate therefore on the study for large n of S = al(y? +... + Yn) + a2(y1 Y1 + + Yn yn-n). (2.2)

Long-range dependence 257 3. SOME CUMULANTS In this section we use the tensor notation developed by McCullagh (I987). Arrays of joint cumulants are denoted by Ki'j'k, for example, which denotes the joint cumulant cum (Y', y47 yk), and Kt)jk, which denotes the generalized cumulant cum (yi, Y yk). These and other cumulant arrays are invariant to permutations of indices. In the tvensor notation (2.2) becomes S=b+b, Y`+bi1 Y1 YY, where summation is implied over indices appearing both as subscripts and superscripts, b = - na2, bi = a,, b1, = a2 8,j, and &j is the Kronecker de Our primary interest is in the second and third cumulants K2(S) and K3(S) of S. The second cumulant is bibj K'j+?b bik K%,ik + b bk Ki, k +b b, Kii,kl which reduces to b1biki'j+2b1jbklk jkkl upon expressing the generalized lants in terms of ordinary cumulants and noting that cumulants of order three or more are zero for gaussian random variables. The third and fourth cumulants are likewise expressed as K3(S) = 6b, bi bkl K ' k Kj 1 + 8b1i bkl bmn Ki' k Kj,m K', n, K4(S) = 48b, b3 bkl bmn K ' k Kj)m K', n + 48b13 bkl bmn bop K ' k Kj,m Kl,o Kn, p, from which the expressions for the rth cumulant may be deduced. The coefficients of its first and second terms are respectively 2r-3r! and 2r-1(r-1)! (Taqqu I975). If S has a limiting normal -distribution, the rate at which its distribution approaches normality as n -- 0o is governed by the third standardized cumulant K3(S)/K2(S)2. If without loss of generality we let yo = 1, the second and third cumulants of S can be reexpressed as n n p-+22 pi-ip i-i K2(S) = al2 E p1-?2a2 E p p i,j=l ij>=1 (3.1) n n K3(S) = 6a2 a2 E pl j pjk + 8a32 pl j pkk pk-. i,j,k=1 i, j, k=1 4. ASYMPTOTIC ANALYSIS E p3- = n+2 E pi-i, p? lijpl-j = n+2 E p'-j p -i The behaviour of the cumulants (3.1) for large n depends on the rate of increase of the summations on the right-hand side, which can be expressed as n n i,j=l i>j i,j=1 i>j n,p i pfk = n + 4 E pi- + 2? pi pp-? + 4 E pi-kp-k + 2 E pi-j p-k i, j,k=1 i>j i>j i>j>k i>j>k n ~ p)- p-k pk-t = n+6 E p -p + 6 E p i pkpk. i,j,k=1 i>j i>j>k (4.1)

258 A. C. Davison and D. R. Cox Suppose that {Yt} is a self-similar process with autocorrelation (1.2) and unit variance: y0 = 1. Then pt = yt - flt-a for 0 < a < 1, and 1 1 n-i n+i>j 2 +2 LI1 Epi? l du dv(v-u)a, which is,8(1 -a)-' [(n2-a 1)/(2-a)- (n-1) 2a-1]. Similarly we see that ^, 1-2a)-1 [ (n 2-2a _1)/ (I-a) -(n- 1) 2 2a-1] a0l E pj p-k j i>j>k fl2{n lnn+ (n-1) (In 2-1)}, a = More complicated calculations are needed to find approximations to the remaining sums on the right-hand side of (4.1). Note firstly that which is roughly n t-1 r-1 E pi-k pj-k = E E E prpr-s i>j>k t=3 r=2s=1 n+- r2 rr2 A l32 f dt f drra f ds (r- The inner integral is (1 -a)-1 {(r-)1-a - 2a-1}, and we write I =,82(Iii-12)1(12-a). The second integral here is I =2 a-1 (I -a)%"1 { (n2-a -22-a)/ (2 -a) - (n-2 ) (3)1-a}. To approximate to Il,, we let Rn+21 Ill = I(t, e) dt, J22 where I(t, e) = f (r-.)1-a r-a dr, and make a Taylor series expansion of I(t, e) to first order in e: I(t, 0) = 1(t2-2a_1)/(1-a) 1+C Thus, di,o (-1(2-3a) t1-2a -al/(1-2a) a0l ac =22+(I1-a)lnt a=-1 =- (- (n + 1)3-2a _ (5)3-2a )i (n-l2) 21a3-2a ' 2/\2/, 2(1-2a) [2(1_) {(n+ )2 2a ()2 2a} (n-2)al provided that a : A with appropriate modification otherwise.

Long-range dependence 259 Second, n+1-1 1s- E pj pfki 12J' dt dr i>j>k J2 12 J say. Interchanging the order of integration for the inner two integrals gives.a2 n+i %-1 3 12 = _ [?2 dt ds s-a {(t-s-2)1-1-a 2 -l('1 'I22)1 1-aj22ji -a 22 say. Another change of the order of integration gives I21 /-%. (2 -a)-' [B(I 1-a, 3 -a) I1-1,2n ( 1-a, 3-a)-I1j2nf(1-a) 3-a)}- (n- 1) 2 a-] I22 = 2a-1( -a)-' [1/(2-a) {(n-2)2-a _ (3)2-a}- (n-2) 2a-1] where I,(p, q) is the cumulative distribution function of the beta distribu parameters p and q, and B(p, q) is the beta function, 1 Juv-'(I 1-u)2-1 du. The final summation is,,+2 1 1 1S22 pi-j -kpi-k ; /J3 J dt ds s-a dr r-a(s,r)-a. i>j>k J 2 1 1 Call this I3. The inner integral can be written I(s,8) drr=-a(,-)-a 8c and we expand this by Taylor series to first order in d. We shall choose c so that &J(s, O)/I is finite and non-zero, as is not the case if c = 1. We find that I(s, O) = sl-2ab( 1-a, 1-a), and that U(s, 8)/& = - 2c(r -6c)-aa 6c-l-ac We choose c = 1/(1 -a) and 8 = 2a-1, so that U(s, O)/IM and &c = 2. Thus the inner integral is roughly s B(1-a,~ ~ a1-12a (I -a I- ) (I a)2 t_ 2) }'aand we write I3 = /3{B(l-a, 1-a)I31-2aI32}/(l-a). Then ((2-3a)-1{1(n 33a-23-3a)/(l-a)-(n-2) (3)2-3a} a:a 2 32 nlnn-21n2-(n-2){1 +In3} a = 2 (I - -2a)-lf 1(n2-2a -22-2a)/(l -a) -(n-2) ( and I 2 V 32 ni t.n.- 2n 2 - (n - 2) I + In 12 a = 1

260 A. C. Davison and D. R. Cox One use of expressions like those above is to study the limiting behaviour of the standardized third cumulant Yl(S) = K3(S)/K2(S)2 of S as n increases. Thus if h(q) has Hermite rank 1, then it follows that the standardized third cumulant y1(s) - n-a/2 for large n, consistent with slow approach of S to a limiting gaussian distribution. However, if the Hermite rank of h(o) is 2, then y1(s) is asymptotically constant when 0 < a < 1, and the limit distribution of S is non-gaussian. If the Hermite rank of h( ) is 2 and 2 < a < 1, the position is more complicated. Examination of the various asymptotic expressions above shows that (ln n)-i a =2, Yi ) fn2lnn a = 2 3, n- 2 < a <1. Thus, the standardized third cumulant is asymptotically zero provided that 2 < a < 1. If a > A, then K2(S) - n, and the fact that y (S)--0 as n -oo that n-2s may have a limiting gaussian distribution. This in fact follows from the results of Breuer & Major (I983); see Sun & Ho (I986). The fact that K2(S) - n ln n when a = 2 but that yl(s) --0 leads us to conjecture that (n ln then has a limiting gaussian distribution. Qualitative results such as these depend only on the leading terms of the TABLE 1. EXACT AND APPROXIMATE VALUES OF THE STANDARDIZED THIRD CUMULANT Yi(S) OF THE SUM (2.2) WHEN THE UNDERLYING RANDOM VARIABLES Yt ARE GAUSSIAN WITH AUTOCORRELATION FUNCTION (1.2) Hermite rank 1 Hermite rank 2 a,=a2= 1 a,=0,a2= I a n exact approximate exact approximate 0.2 10 2.48 2.46 2.63 2.63 30 2.39 2.39 2.59 2.60 100 2.30 2.30 2.57 2.57 103 2.10 2.55 106 1.35 2.55 0.4 10 2.04 2.02 2.04 2.03 30 1.83 1.83 1.82 1.82 100 1.63 1.63 1.65 1.65 103 1.24 1.46 106 0.37 1.24 0.6 10 1.51 1.50 1.35 1.33 30 1.21 1.20 0.96 0.95 100 0.96 0.95 0.66 0.65 103 0.59 0.32 106 0.09 0.04 0.8 10 1.09 1.08 0.98 0.97 30 0.73 0.72 0.58 0.57 100 0.48 0.47 0.32 0.32 103 0.22 0.10 106 0.02 0.00

Long-range dependence 261 expansions, and do not require the continuity corrections that complicate the approximations derived above. The continuity corrections are useful, however, when the asymptotic expansions are used for approximate numerical calculation of cumulants, for values of n for which exact evaluation of the sums in (3.1) is prohibitively expensive. Table 1 shows exact and approximate values of the standardized third cumulant y1(s) when the underlying random variables are gaussian with autocorrelation (1.2). The approximations are excellent for n as small as 10. The limiting non-gaussian distribution for processes of Hermite rank 2 when 1 < a < 1 is clearly illustrated, but the standardized third cumulants for processes of Hermite rank 1 are large, even for values of n as large as one million. The limiting gaussian distribution is then approached exceedingly slowly. Interestingly, when the limiting distribution is normal for cases of Hermite ranks both I and 2, i.e. 2 < a < 1, the approach to normality - as judged by the third standardized cumulant - is quicker for processes of Hermite rank 2. 5. RESULTS OF MONTE CARLO SIMULATIONS To gain some insight into the shape of the distribution of S when Carlo experiment was performed. For each value of a from 0.1, 0.2,..., 0.9, and n = 100, one million independent replicates of S based on normal variables with covariance (1.2) were generated using NAG routines G05CBF, G05EAF and G05EZF. -4-9 -1 0 1 2 3 x FIGURE 1. Plot of log-density for one million simulations of (2.2) with covariance (1.2), a = 0.1, n = 100. -4 A-9-141 -1 0 1 2 3 x FIGURE 2. Plot of log-density for one million simulations of (2.2) with covariance (1.2), a = 0.3, n = 100.

262 A. C. Davison and D. R. Cox Figures 1 and 2 show plots of the logarithms of the resulting estimates of the density of S, when a takes values 0.1 and 0.3. The behaviour of the functions in the figures is evidently non-gaussian: the upper tail seems to be exponential, with lnf(x) - - g(a, n) x. Both distributions are positively skewed. Similar plots for the data with a > 0.5 are consistent with the parabolic shape associated with the gaussian distribution. This work was supported in part by a Science and Engineering Research Council Senior Research Fellowship, and a Nuffield Foundation Newly Appointed Lecturer Award. REFERENCES Breuer, P. & Major, P. I983 J. multivariate Analysis 13, 425-441. Eberlein, E. & Taqqu, M. S. (eds) I 986 Dependence in probability and statistics: a survey of recent results. Boston: Birkhauser. McCullagh, P. I987 Tensor methods in statistics. London: Chapman & Hall. Rosenblatt, M. I96I Proc. 4th Berkeley Symp. Mathematical Statistics and Probability, pp. 431-443. University of California. Sun, T. C. & Ho, H. C. I986 In Dependence in probability and statistics: a survey of recent results (ed. E. Eberlein & M. S. Taqqu), pp. 3-19. Boston: Birkhauser. Taqqu, M. S. 1975 Z. Wahrscheinlichkeitstheorie verw. Geb. 31, 287-302.