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1 Math-Net.Ru All Russian mathematical portal A. Philippe, D. Surgailis, M.-C. Viano, Time-Varying Fractionally Integrated Processes with Nonstationary Long Memory, Teor. Veroyatnost. i Primenen., 2007, Volume 52, Issue 4, DOI: Use of the all-russian mathematical portal Math-Net.Ru implies that you have read and agreed to these terms of use Download details: IP: January 5, 2019, 04:31:20

2 Том 52 ТЕОРИЯ ВЕРОЯТНОСТЕЙ И ЕЕ ПРИМЕНЕНИЯ 2007 Выпуск г. PHILIPPE А.\ SURGAILIS D** VIANO М.-С.*** TIME-VARYING FRACTIONALLY INTEGRATED PROCESSES WITH NONSTATIONARY LONG MEMORY 1 ) В работе вводятся два явных класса A(d), B(d) линейных фильтров, зависящих от времени и определяемых для любой вещественной последовательности d = t Z), таких, что для постоянной последовательности d t = d операторы -4(d) B(d) (I L)~ d совпадают с обычным оператором дробного дифференцирования. Доказано, что эти операторы удовлетворяют соотношениям обратимости В{ d) A(d) A(-d) B(d) = I. В работе исследуется асимптотическое поведение частных сумм фильтрованных процессов белого шума Y t = A(d) Get и Xt B(d) Get в случае, когда последовательность d имеет пределы lim^-too d t d± Е (0, ) в бесконечности, а оператор G образует фильтр с короткой памятью. Доказано, что пределом частных сумм является автомодельный гауссовский процесс, зависящий только от предельных значений d± и суммы коэффициентов оператора G. Кроме того, предельный процесс имеет либо асимптотически стационарные, либо асимптотически стремящиеся к нулю приращения и гладкие траектории. Ключевые слова и фразы; нестационарная дальняя память, зависящее от времени дробное интегрирование, частные суммы, автомодельные процессы, асимптотически стационарные приращения. 1. Introduction, Long memory, also called long-range dependence, is one of the most dynamically developing areas of probability and statistics; see, e.g., the recent volume [8]. Prom the theoretical probability point of view, the most interesting feature of long-range dependence concerns nonclassical limit behavior of the partial sums process. Recall that a stationary time series (X t) t 6 Z) is said to be long-range dependent if its partial sums * Laboratoire de Mathematiques Jean-Leray, UMR CNRS 6629, Universite de Nantes, F Nantes Cedex 3, France; anne.philippe@math.univ-nantes.fr ** Institute of Mathematics and Informatics, Akademijos 4, LT Vilnius, Lithuania; sdonatas@ktl.mii.lt *** Laboratoire Paul Painleve UMR CNRS 8524, UFR de Mathematiques Bat M2, Universite de Lille 1, Villeneuve d'ascq, Cedex, France; marieclaude.viano@univ-lillel.fr ^ This research was supported by the bilateral France-Lithuania scientific project Gilibert and the Lithuanian State Science and Studies Foundation (grant T-15/07).

3 Time-varying fractionally integrated processes 769 process, when suitably normalized, weakly converges to some random process with dependent increments, see, e.g., [5, p. 59], [7, p ]. Under weak additional assumptions, the limit process is self-similar [11]. Probably the most popular stationary long memory processes are the FARIMA(p, d, q) defined by the equations ф(ь) X t = $(L)(I - L)~ d e u where (e u t E Z) is a zero-mean white noise, I is the identity, L is the backward shift operator, ф(ь), t?(l) are polynomials in L of degree p, <j, respectively. Recall that the fractional differencing filter (J L)~ d (0 < d < ~) is defined by the binomial expansion where 4>o(d) ~ 1 and oo (7 - LY d x t = L*x t = X>i(<*)*t-* (1.1) W = j ~ 2 oo j'=0 j=0 dd + 1 d-l+j T(d + j) = -7Щ- 0 > 1), (1-2) see, e.g., [3]. It is well known that, under suitable conditions on the polynomial function ф(-) and (e t, t E Z), the autocovariance function of the FARIMA(p, d, process X t decays as i 2d_1, and the partial sums process jy-d-1/2 ^t^l x t converges in distribution to a fractional Brownian motion (fbm) WH(T) with Hurst parameter H = d +. The convergence of partial sums for general stationary second order linear processes was obtained in [6]. There it is proved that, in the case of i.i.d. noise, the limiting process is always a fbm, a self-similar Gaussian process with dependent increments and nondifferentiable sample paths. In [12] it is proved that the above mentioned convergence result still holds under rather weak assumptions on the noise. Considerable attention in the literature was also given to limit theorems for partial sums processes of nonlinear instantaneous filters of stationary FARIMA and more general stationary linear processes with long memory; see, e.g., [18] for recent developments. Until recently, long-range dependence has been studied almost exclusively in the stationarity setup. One of the reasons for this is the lack of appropriate models with nonstationary long memory. In the present paper we provide a family of such processes presenting long memory and study the limiting behavior of their partial sums. Starting from (1.1), we introduce the time-varying fractional differentiation operators oo oo x A(d) x t = J2 B aj(*) *t-j> ( d x ) t = Y, M*) *-*> (!-3) where d = (d t) t E Z) 3=0 is a given function of t 3=0 E Z, and where a 0 (t) =

4 770 Philippe A., Surgailis D., Viano M.-C. b 0 (t) :=1, a 1 (t) = b 1 (t) :=dt-i, rft-i d t d t d t _j 4- j fltzip^, i > 2 > (L4) C^t-l ^t-j + 1 rff-j+l + 2 <^-2 + j - 1 = ^fl " j k ~ 1+d^~\ j > 2. (1.5) If d t = d is a constant, then A(d) = B(d) = (I L)~ d is the usual fractional differentiation operator. Note that both dj(t) and bj(t) depend on..., d t -j) and that bj(t) is obtained from cij(t) by time reversing dts,..., dt-j) >{dt-j) df-j+i,..., dt-3) dt- 2 ). A remarkable property of these operators is that B(-d) is the inverse of A(d) and vice versa; in other words, B(-d) A(d) = A(-d) B{d) = /, (1. where -d := ( d t, t Z). The invertibility property is very important in regard to applications and statistical inference and was one of the main reasons for studying the operators in (1.3). We introduce the time-varying fractionally integrated processes (X t ) and (Y t ) defined by equations A(-D) X t = Gs u S(-D) Y t = Ge u (1.7) where (e t, t 6 Z) is a stationary martingale difference sequence and G is a short memory filter with absolutely summable coefficients: oo oo Gx t = Y^9jXt-j> with ^ ^ < oo. According to (1.6), the solutions of equations (1.7) are defined by X t = Л(-Ё)- 1 Get = S(D) 0^ = ^(6*^(4)^, (1.8) oo Y t = B(-D)- 1 Ge t = i4(d)ge t = X;(A*(/) i (t)e t. i> (1.9) CO where (6*</Ш :=5>(*)fc-*> (a *$),-(*) == ai(t)fc-i ( 1 Л 0 ) г=0 г=0

5 Time-varying fractionally integrated processes 771 are the impulse responses of the product operators B(d) G, A(d) G, respectively. Note that when d t = d is a constant, the processes (1.8) and (1.9) are stationary and contain the class of (stationary) FARIMA(p, d, q). The asymptotic results below are proved for sequences d = (d t, t Z) such that there exist limits (l.ii) The limit d + = lim^oo d t can be interpreted as the intensity of long memory of Y t and X t in (1.7) in the distant future and d_ = lim^-.oodt as the corresponding intensity in the distant past. We systematically omit the (tedious) estimation of the covariances cov (X S) X t ) and cov (Y s,y t ). The interested reader is referred to the preprint [15], where long memory behavior of these covariances is established. We focus here on the asymptotics of partial sums and prove that for d satisfying (1.11), the partial sums processes of Y t = A(d) Ge t and X t = B(d) Ge t converge to some Gaussian processes which depend on the limits d± in (1.11) and the sum g = YliLi 9% alone. More precisely, according to Theorem 5.1, the partial sums process of Y t = A(d) Ge t weakly converges, under normalization j\t d / 2, to the sum of two independent self-similar processes: a type II fbm and a smooth Gaussian process with asymptotically vanishing increments. On the other hand, as shown in Theorem 5.2, the partial sums process of X t = B(d)Ge t converges to different limits depending on whether d + > d_ or d + < d_ holds: in the first case, the limit process is a type II fbm, while in the second case, it is a Gaussian process with smooth paths and asymptotically vanishing increments. We note that the last fact is very unusual in the context of stationary processes, where the limit of partial sums has nondifferentiable trajectories as a rule. It is also interesting to note that, apart from condition (1.11), we do not assume that the sequence d t belongs to (0, ~) or to any other a priori fixed interval of the real line. When d t has excursions above, the simulated trajectories of processes X t and Y t in (1.8) and (1.9) present sudden «bursts» similar to those encountered in infinite variance processes. The results of the present paper were recently extended into several directions. In [16], we discuss the case of almost periodic sequences d and show that, in the last case, «averaging of nonstationary long memory» occurs and the partial sums processes of time-varying filters Y t and X t converge to a usual fbm with parameter determined by the mean value d of the almost periodic function d. A similar behavior of partial sums was established in the case when d is a random stationary process [9]. The case of time-varying fractionally integrated processes with infinite variance and the convergence of the corresponding partial sums processes was discussed in [4]. In [19] the definition of time-varying fractional operators in (1.3) is extended to

6 772 Philippe A., Surgailis D., Viano M.-C. the continuous time setup, as a limit of discrete operators corresponding to slowly varying sequences of the form d N j = d(t/n), t E Z, where d(x), x E R, is a given function on the real line. This extension is useful for modeling multifractality. See [14] and [1] for an approach to multifractality involving time substitution of a fbm. The paper is organized as follows. In Section 2 we study the general properties of time-varying filters (1.3), (1.4), and (1.5). Section 3 is devoted to the proof of invertibility (1.6). Section 4 discusses some properties of self-similar processes later obtained in Section 5 as limits of the partial sums. Section 5 formulates and proves the main convergence theorems (Theorems 5.1 and 5.2). Technical lemmas are relegated to Section General properties of time-varying fractional Alters. Let S v (p ^ 1) denote the class of all real-valued random processes (x u t E Z) such that sup t E x p < oo. Let 3 С 3 2 denote the subclass of all orthogonal sequences (x t, t E Z) with zero mean, i.e., such that Ex t = 0, T&x t x s = 0 for any t, s E Z, t ф s. Everywhere in what follows we assume that the sequence d = (d u t 6 Z) is bounded, i.e., sup \d t \ < oo. t Z Definition 2.1. Let D be a real number. We say that a sequence d = (d t, t E Z) satisfies condition srf(d) if there exists К < oo such that for all integers s < t such that t s > K, t-u s<u<t Similarly, we say that d = (d t, t E Z) satisfies condition 33(D) if there exists К < oo such that for all integers s < t such that \t s\ > K, jri, + и s s<u<t Clearly, if (d t, t E Z) satisfies condition srf(d) (respectively, condition 38(D)), then the sequence (d t+r, t E Z) also satisfies condition srf(d) (respectively, condition 38(D)), for any r E Z; in other words, conditions srf(d) and 38(D) are invariant with respect to translations. It is also obvious that if (d t, t E Z) satisfies condition srf(d) (respectively, 38(D)), then the time-reversed sequence (d_ t, t E Z) satisfies condition 38(D) (respectively, condition &/(D)). The following example shows that conditions stf(d) and 38(D) are not equivalent. Example 2.1. Let d t = t = 0, * = 1, t = 2, 4^0,1,2.

7 Time-varying fractionally integrated processes 773 The sequence (d t, t E Z) satisfies condition srf(q) and does not satisfy condition 38(0). Indeed, s<u<t t u _2 _1 1 ~t + t-1~t < 0, t-1 t-2 ' 1 t-2 0, 1-2, and ^(0) holds. On the other hand, ^2o <u<t d u /u that 38(0) is not satisfied. s < 0, t > 2, s = 0, t > 2, 3 = 1, t > 2, s < 0, t = 2, 3 < 0, t = l, = > 0 for t > 2 so Proposition 2.1. Let D $ {x еъ: x ^ 0}. (i) Le (d t, t 6 Z) satisfy condition srf(d). Then there exists a constant С < oo siic/i that for all t E Z and j ^ 1 a,(t) <C ( ). (ii) Let (d, t E Z) satisfy condition 38(D). Then there exists a constant С < oo such that for all t E Z and j ^ 1 М<) ^сшл). Proof, (i) Let j 0 be large enough so that \ < (d u + к 1)/ (D + к - 1) < holds for all и Z, fc > j 0 - Then A(t).-hM--c(ti) n d *-* +fc ~ 1 where J0 d t - k + k-l C(t,j 0 ) П fc=i D + k-1 is bounded by a constant C(j 0 ) uniformly in t E Z, since d is bounded. Using the inequality: 1 я < е ж (x ^ 1), this implies Л,(*)^ Ыехр{ Write,,, D + k- 1 ~ ^ + Д,

8 774 Philippe A., Surgailis D., Viano M.-C. where Yli=i(dt~k D)/k ^ 0 by condition Next, (D), provided j is large enough. where \R'\ and \R"\ are bounded uniformly in t E Z and j > j 0. This proves Aj(t) < С uniformly in (ii) We omit this proof which is similar to the proof of (i). Corollary 2.1. Let d satisfy condition srf(d) (respectively, condition 88(D)). Moreover, let there exist C,8 > 0 such that for any t E Z, Ы4СГ 1 -'. (2.1) (i) If D < 0, T/ien j4(d) G (respectively, B(d) G) is a bounded operator (ii) If D < 1, T/ien -A(d) G (respectively, B(d)G) is a bounded operator Proof, (i) It suffices to show that there exist G', 5' > 0 such that for all t E Z, j 3* 0 (a*3)j(*)l ^GT^', (6* 5 ) J (T)UG / (l + i)- 1 - y. (2.2) In view of Proposition 2.1 and well-known properties of FARIMA coefficients in (1.2), inequalities (2.2) follow from the elementary bound: (1 + i) D ~\l + j - i)- 1-6 < C(l + j)" 1 " 6 ', (2.3) г=0 where 5' = min(5, D) > 0. (ii) Follows by the same argument and (2.3), where 8 f = D >. Proposition 2.2. (i) A sequence d satisfies conditions srf(d) and 38(D) for any D such that D > limsup (7-sup =: d. л-f+oo \ h sez s < u < s + h J (ii) A sequence d satisfies conditions srf(d) and 38(D) for any D > limsup tkoo dt. Proof, (i) Let 8 > 0 and D' = d + 8, D" = d According to the definition, for any 8 > 0 there exists К < oo such that for all s < t, t~s> K, <*s,t := ^ ~ < - ( 2-4 ) s<u<

9 Time-varying fractionally integrated processes 775 Then, for all s < t, t s > K, we have s<u<t s<u<t s<u<t where J' 3t > oo as t s -» oo. On the other hand, using (2.4) and summation by parts, t-s~2 / -i -i \ t 3-2 _ where K" t-s-2 ^-^г(г + 1) +^ + 1 ф + 1)- :^ + ^ and where i^t < С is uniformly bounded in for any К < oo fixed while J" it < 0 by (2.4). This proves condition sf{d) with any D >d. The proof of condition 38(D) follows similarly. (ii) Follows from (i) and the fact that d < limsupj^^ d t. 3. Invertibility. In this section, we prove that, for a sequence d such that the two operators are well defined, A(d) and B(-d) are inverse to each other. Theorem 3.1. (i) Let a sequence d = (d t, t E Z) satisfy condition /(D 0 ) with D 0 < f, and let the sequence d = {-d t) t E Z) satisfy 2 condition 38(D X ) with D x < 0. ТЛеп A(d) ma^ ^T 0 to ^Г 2 С &\ B(-d) maps 3 x to itself and B(-d)oA(d) = J, (3.1) where I is the identity operator on Ж (ii) Let a sequence d = (d u t E Z) satisfy condition 38(D 2 ) with D 2 <, and Zet sequence d = ( d, i Z) satisfy condition g/(d 3 ) with 2 D 3 < 0. T/ien B(d) maps & 0 to 5C 2 С A(-d) maps X 1 to itself and A(-d) о 5(d) = I. (3.2) As an illustration of this theorem, consider sequences d having limits d± (0, 5) at ±00. Then Theorem 3.1 together with Proposition 2.2 imply that A(-d) о B(d) and B(~d) о A(d) both are the identity operator on the space &Q of all L 2 zero-mean orthogonal sequences. Proof of Theorem 3.1. The first parts of (i) and (ii) are directly deduced from Corollary 2.1. Let us prove (3.1) and (3.2). To prove (3.1), it is enough to show that for every n ^ 1 n An(*) = b 7(*) a -j(t - J) = 0, (3-3) 3=0

10 776 Philippe A., Surgailis D., Viano M.-C. where ^ bj(t) = ( - 1 ) ^ П -*-1 - j + fc), j > 1, are the coefficients defined in (1-5), with d replaced by d. We obtain n-l Д»(<) = a n (t) + b~(t) a _i(t - 1) + 3=2 &;(<) ^(t - j) + b~(t) + * - E TTSV П и -» + * -i) n W < +' -1) Next, n-l + d t " 1!,7 1 ) n n^-i- f e +fc-»)- j-1 n-j n-l n^-fc-i + к - j) п +1-1) = П +* - j) fc=l *=1 fc=l leads to *«(*) = (-1)' f (1Л + n - j) (Уп-1 + n - j) with y k := dt_fc_i n + fc, & = 1,..., n 1. The result is now deduced from Lemma 3.1 below. To prove (3.2), it is enough to show that for every n ^ 1 A n (t) :=itaj(t)b n4 (t-j) = 0, (3.4) where the coefficients aj(t) are defined by (1.4) with (d tj t 6 Z) replaced by (-d i 6 Z). Using (1.4) and (1.5), for j = 1,..., n - 2 we have ^ j n-j-l a~(t) b n4 (t - j) = ^ П (^ + dt-n+^-i). Using the notation ж л := d t - n +k-i + can be rewritten as j & ^ 1,..., n 1, the last product n-j-l J[(k-1 -dt^d^j^ Yl (^ + dt-n+*-i) ( n-j-l \ n-l Д x«j(n- j П (n-x/). 1=1 / Ып-j + l

11 Time-varying fractionally integrated processes 777 Similarly, n-l a 0 (*) b n(*) = ~^~r П х ь n! ti n-l a n _ x {t)b x (t-n+l) = ^ J](n-^), n-l This yields 1=1 Д (*) - dt-i ra! n-l n-2 / \ /n-j-l П x n-l * + J П ^ ) ( N - i - хъ-з) П ( N - ^) i n + F n - I) ( 1 " X L ) 1=2 П ( N " X < } " П Ы1 ( П " X L ) We claim that for any m = 1,..., n 2 Indeed, for m = 1, 4 = Г п-г ' 71 ТП 1 \ П 1 П X < П ( n ~ X k)- ^=1 / k=n m 7? ^ j=0 (3.5) ' n-2 *=1. =1 'n-2 \ 'n-l N, 1, Y[xA(n-X n -i). к 1=1 I The induction step m 1 > m easily follows: m / 1 \ / n ~ m \ n 1 g^=(m-l)(n^), П n 1=1 / k=n-m+l 'n m 1 n-l + 1 II П ff/j (n - m - x n - m ) YL ( n ~ 'ra- 1 n m \ n 1 > П x 4 П ( n "- x fc) 771 ч ^=1 / k=n m fc=n m+1 х ь) This proves (3.5). Using it, we obtain n-i / _ j \ n-i f n \ n~ 1 3 = 1 V" "/ k=2 This proves (3.4) and Theorem 3.1 too

12 778 Philippe A.j Surgailis D., Viano M.-C. Let n, fc ^ 1 be integers. Introduce the polynomial P n,k(xu in x u...,xk by PnA*u > *ь) -= E(-l) n_i Г) (J + xi) (j + x k ). (3.6) Lemma 3.1, For every к {1,..., n}, ue polynomial function defined in (3.6) satisfies P n,k( x i> - = 0. Proof. For fc < ra, it can easily be checked that Pn+ij^i,, я*) = Pn,*(^i + 1,..., ж* + 1) - Pn,*0&i,..., ж*), (3.7) P n+ i Jfc+ i(xi,..., х к +х) = (n + 1) Pn f *(ffi + 1,..., x k + 1) + x f c + ip + i, f c (&i,..., x k ). (3.8) First, Р2д(ж) = (2 + x) 2(1 + x) + x = 0. Then the lemma is proved by induction on n. Assume that P n,/c(#i> = 0 for some n ^ 1 and all 1 < fc < n. Then from (3.7), we deduce that P n+1^(x 1,... = 0 for all 1 ^ к < п. Finally, (3.8) ensures that P n +i,n(#i> > x n ) = 0, or the validity of the induction step n > n + 1. Remark 3.1. Lemma 3.1 is not true for fc ^ ra. In fact, for > s*~< )-л&- 1Г '6)^-{-} is the Stirling number, or the number of ways fc different objects can be partitioned into ra groups. 4. A class of self-similar processes with asymptotically stationary or asymptotically vanishing increments. Definition 4.1. Let W = (W(r), r > 0) be a stochastic process. We say that (i) W is self-similar with index H > 0 (Я-self-similar for short) if, for any a > 0, {W{ar), т > 0) = fdd (а я^(т), r > 0), where the subscript «fdd» means that the property holds for finite dimensional distributions; (ii) W has stationary increments if, for any T > 0, ( W (T + r) - W(T), r > 0) = fdd (W(r) - W(0), т > 0); (iii) ТУ has asymptotically stationary increments if as T goes to +oo (W(T + r) - W(T), r > 0)» ш (W(T), r z 0). (4.1) where W is a nontrivial stochastic process;

13 Time-varying fractionally integrated processes 779 (iv) W has asymptotically vanishing increments if (4.1) holds with W[T) = 0. It is easy to show that if a process W has asymptotically stationary increments (respectively, is ij-self-similar), then the limit process W in (4.1) has stationary increments (respectively, is Я-self-similar). The characterization of all Gaussian self-similar processes with asymptotically stationary (asymptotically vanishing) increments seems to be an interesting problem. One may expect also that if a process W has asymptotically stationary increments and the process (K~ 1 W(ar) J r ^ 0) tends to some limit process Q as a > oo, in the sense of weak convergence of finite dimensional distributions, under some (nonrandom) normalization K a > oo, then Q is Я-self-similar with some H > 0 and also has stationary increments. The last fact is true and is well known (due to [11]) if W is assumed to have stationary increments. Let W+(r) := f T Z(ds) Ai-^-Mt, (4.2) J0 Js U d+4 Ar) := f Z(ds) fl**-*-(t-s)*-4t, (4.3) J o Jo V d+<d _(r) := f (-s)'-** Z(ds) f\t - s)**- 1 dt, (4.4) J-oo JO where 6 (0, ) 3 and Z(ds) is a standard Gaussian white noise, with zero mean and variance ds. The process W% is called a type II fbm (see [13]). Note that W d is independent of U d+}d _ and V d+fd^} that U d)d = V d4, and that W2{r) + U d4 {r) = W M / 2(T) (r > 0) (4.5) is a fbm with the self-similarity (or Hurst) index H = d+\. Proposition 4.1. Let d, d+, GL (0,\). Then (i) The processes W d, U d+id _, and V d+4 _ are well defined; they are selfsimilar with respective indices d + \, d+ +, and GL + ; (ii) the processes U d+id _ and V d+fd _ have asymptotically vanishing increments, while W d has asymptotically stationary increments tending to those of a fbm W d+1 / 2 in (4.5); (iii) trajectories of U d + >d _ and V d+4 _ are a.s. infinitely differentiable on (0, oo). The proof of the above proposition can be found in Appendix (Section 6). 5. Convergence of partial sums processes. Let d be a sequence such that condition (1.11) holds, i.e., we assume the existence of the limits

14 780 Philippe A., Surgailis D., Viano M.-C. d± (0, ) at ±00. Moreover, we shall assume that d t Z for all t and that there exists a 6 > 0 such that dt = d ± + 0(\t\~ 5 ) (t-+±oo). (5.1) We also assume that the short-memory filter G satisfies condition (2.1) and, moreover, 00 Note that it is without loss of generality that the exponents in (5.1) and in (2.1) are the same. By Proposition 2.2 (ii), condition (1.11) implies both conditions srf(d) and 38(D) with any D > max(d +,d_) so that the processes X t and Y t in (1.8) and (1.9) are well defined. The first question to study concerns the asymptotic behavior of the coefficients a v - u (v) and b v u (v) as v and v~~и tend to infinity. In the simple case when the sequence d assumes two values, d t Г d+, 0, \cl, t<0, the filter coefficients a t - s (t) and b t _ s (t) can be obtained explicitly from definitions (1.4) and (1.5): and ' Vt-«(d+), a t-s{t) = {,, f ч il> t (d+) ^t-s{d-) f V>t- e (d+), 6t-a(t) = < d +^i_ 5 (d_) d-i> x - s (d+y 0 < s ^ i, 5 < 0 ^ t, 0 ^ s *S t, 5 < 0 t. It turns out that these relations remain valid asymptotically also in the general case of d, as stated in Lemma 6.1 in Appendix (Section 6). The presence of the short-memory filter G preserves this asymptotic behavior up to a multiplicative constant, as stated below. Lemma 5.1. Let (d u t Z) be a sequence such that (1.11) and (5.1) hold. Assume that the short memory filter G satisfies (2.1) and (5.2). Then there exist C,8 X > 0 independent of s,t and such that g^t-s{d+) + e { tl 0 ^ s < t, ( a *^' ( t ) H #,-.(d.)^# + e<«, s < 0 < t, (5 ' 3) (b*g) t -,(t) = < 0 ^ s < t,, s < 0 ^ t, (5.4)

15 Time-varying fractionally integrated processes 781 where eg < c(\t- s \- 5 > + \s\- s > + \t\- s i) j(t-s) d +-\ 0^s<t, \t d+ - d -(t-s) d \ s<0^t, eg I < c(\t-s\- 5 * + \s\- s > + \t\- 5 >) ((t-s) d +-\ 0^s<t, (d--d+)vo^ _ sy+-i S < 0 < t. (5.5) (5.6) The proof of this lemma is given in Appendix (Section 6). In the sequel, >D[o t i] denotes the weak convergence of random elements in the Skorokhod space D[0,1] endowed with the sup-topology. Theorem 5.1. Let Y t := A(d)Ge t, where (e t, t Z) is a stationary ergodic martingale difference sequence, with unit variance. Moreover, assume that the assumptions of Lemma 5.1 hold. Then [Nr] N-*+-li/2) Y f _ + D m _ _ ( w + ( r ) + c/ d+jd _( r) ), (5.7) where W d (r) and U d+yd _{r) are defined in (4.2) and (4.3). Proof. To prove the convergence of finite dimensional distributions, we use the so-called scheme of discrete stochastic integrals [17]. We shall restrict the proof to the case r 1. Let Ipj stand for the left-hand side of (5.7), where r = 1. We shall write I N as a discrete stochastic integral with respect to a stochastic measure with orthogonal increments. Namely, IN- Jn f N {x)z N (dx), where the piecewise constant function f N = f N (x), x R, is defined by Ms) := N d + I ( a *0)t-«(*)> t=i o, otherwise. a-1 N ' N The stochastic measure Z N is defined on finite intervals (x f, x"], x f Z N ((x',x"])~n- 1 / 2 е.. x'<s/n^x" x E (-oo, 1], (5.8) < x", by The discrete stochastic integral f n q(x) Z N (dx) = fqdz N is defined by S s 9s^(^s) f r e a c h 8 р1 е function q(x) taking nonzero constant values q s on a finite number of intervals Д 5 {s/n, (s + 1)/N], s e Z, and

16 782 Philippe A., Surgailis D., Viano M.-C. vanishing elsewhere. Because of the orthogonality of the measure Z N (which is a consequence of the martingale difference property of the s 's), E (JqdZ N y = \\q\\l = j^q\x)dx. (5.9) According to the classical martingale central limit theorem [2], for any m < oo and any disjoint intervals (a^, x"], i = 1,..., ra, ( ((<, *'/]),...,Z N ((x' m, X^})) -> l a w (^((xi.xil),..., Z(0C *«])). (5.10) where Z(dx) is a standard Gaussian noise with mean zero and variance dx. The right-hand side of (5.7) (corresponding to r = 1) is written as / = f R f(x) Z(dx) = ffdz, where / {t-xf^dt, x e [0,1], f { x ) : = W J ) [\t-x) d - 1 t d +- d -dt, xe(-oo,0), JO ( 5, 1 1 ) K 0, otherwise. Then convergence I N -» law I follows from (5.8) (5.11) and from /лг-/ 2->0 (N -> 00); (5.12) see [17] for details. With Lemma 5.1 in mind, set ( N ]TV> t-s(d+)> x e s-1 s N ' N xe[o, 1], s-1 s N ' N x (-oo,0), K 0, otherwise. Clearly the convergence (5.12) follows from /лг-/лг 2-»0 and from fn(x)^f(x), (ЛГ^оо) \f N (x)\<7(x), (5.13) (5.14) where / L 2 (R). By Lemma 5.1, (5.3), (5.5), and the dominated convergence theorem, /лг(ж) /N(X) tends to zero almost everywhere. Indeed, since 5(t,s) ;= \t-s\-" 1 + s ~ dl + \t\~ dl

17 Time-varying fractionally integrated processes 783 tends to 0 as > oo, s» oo and is uniformly bounded in 5 <, for any x < 0, x e ((s - 1)/N, s/n], we obtain \IN(X) - f N (x)\ < C ^ E l t - ^ - 1 ^ ^ ^ ) < CN~ d + [ (t-nx) d 4 d +- d -6([t], Jo [Nx])dt = C I {T-XY ^-^S^NT}, Jo [Nx})dr = o(l) and the case 0 < x < 1 follows similarly. Prom Lemma 5.1, (5.3), (5.5), it also follows that /iv(ar) /jv(#)i < C/^(x), with С < 00 independent of AT, я, and therefore (5.13) is a consequence of (5.14) and the dominated convergence theorem. In turn, relations (5.14) can easily be shown using well-known properties of FARIMA(0, d, 0) weights This proves finite dimensional convergence in (5.7). Let us check the tightness in D[0,1]. By a well-known criterion, it suffices to show that there exists 7 > 0 such that for0^r<r + /i<l /[N(r+h)} \ 2 E Y s \ ^C/i 1+7 JV 1+2d +. (5.15) V *HNT] J Using Lemma 5.1, the proof of (5.15) reduces to N [N(r+h)] E E ^СЛ 1 +^ 1 + м +, (5.16) e = l t l ft 2 = [JVr] 0 [N(r+/i)] E s=-oo i 1,t 2 =(Arr] ^-Да-) Vta-Да-) >tl (d_) ^t2(d_) <C7i 1+7 iv 1+2d+, (5.17) Let us check (5.17); the proof of (5.16) follows similarly and is omitted. By using integral approximation and the elementary inequality: l*i - sir - s\-/ < C\t x - t 2 i- e -" (5.18) which is valid for any 0 < a < 1, 0 < /3 < 1, a + /3 > 1 and any i b t 2 Z, inequality (5.17) follows from / r+/i гт+h j \и г - u 2 \ 2d -- l ui + - d -u 2 +~ d - du ± du 2 < Ch 1+ \ (5.19) If d + ^ d_, (5.19) holds with 7 2cL > 0. This is immediate, since 0 ^ ^!,гх 2 ^ 1. For d + < dl, the left-hand side of (5.19) is monotone in r

18 784 Philippe A., Surgailis D. f Viano M.-C. and therefore does not exceed 2h \2d -1 d+-d- j 2d +- x fv^-'-dv! f V \v 1 -v 2 ) 2d - l v d 2 Jo Jo = 2h 2d +~ l f v 2d+ Av 1 (\l-w) 2d - l w d +- d -dw = 0{h 2d+ - 1 ), Jo Jo which yields again (5.19) with 7 = 2d+ > 0. This concludes the proof of the theorem. Theorem 5.2. Let X t := B(d)Ge t. Theorem 5.1 are satisfied. Then INr) Assume that the assumptions of N- d+ -i/2 >p X t ^ d [ o i ] _A_ w+ ( r ) i f d + > d i ( ) [Nr] N-*-i/2J2x t -* D M -^±-V^4_(T) if d + < d_, (5.21) where Vd +i d_(r) is defined in (4.4). Proof. It is similar to the proof of Theorem 5.1, so we just sketch it. Let J N, J stand for the left- and the right-hand sides of (5.20), respectively, when т = 1. Then = / h N dz N) J = / hdz, where Z N) Z are the same as in the proof of Theorem 5.1, and where /ijv = h N (x), h = h(x), x E R, are defined by and N 0, otherwise, л м, 9 \ f\t-x) d +- l dt, x [0,l], 4 х ) Then, the convergence :, Ж (-00,1], (5.22) = щ^у < Jo (5.23) {0, otherwise. ~haw J follows from \\h N - h\\ (5.24) First, using Lemma 5.1, (5.4) and (5.6), it is easy to establish that II(hps fo)l[o,i] 2 ~* 0- Then it remains to prove Ллг1(-оо,о) 2-^0. (5.25) Now, by definition of h N and using Lemma 5.1, (5.4), (5.6), (5.18), we obtain Лдг1<-оо.о)Н1 < CN-^Yl E l(b*ffk-(*i)(b*5)t 3 -.(*2) 5<0 ti,t 2=l

19 Time-varying fractionally integrated processes 785 x (\s\ 2 ( d - d^ + ( s -* + M^XM - ' 1 + \t 2 \- 51 )) = О (N 2^ **) + N- 2Sl ) =o(l) in view of cl < d +. This proves (5.25) and the one-dimensional convergence in (5.20) for r = 1. We turn now to the proof of (5.21) for r = 1. We need again to verify (5.24), with and h N (x) := N~ d -{ {rc У s 1 s j I>*fl)t-.(t), *e(, x (-oo,l], / W ' V N ' N. ( 0, othen otherwise, U Л _Jd ± _l (-x) d - d+ ['(t-xft-'dt, я е(-оо,0), ^ 0, otherwise. Using cl > d+ and Lemma 5.1, (5.4), and (5.6), it follows easily that H^N1[O,I] 2 ~~* 0 and \\(h N - Л)1(_оо,о) 2 ~* 0- The remaining details including the proof of tightness in both convergences (5.20) and (5.21) are similar to the proof of Theorem 5.1. Theorem 5.2 is proved. 6. Appendix: Proofs of Proposition 4.1 and Lemma 5.1. Proof of Proposition 4.1. In the sequel, С denotes a generic constant which may be different at each occurrence. (i) Clearly, f Q r(fs(t - s) d ~ l dt) 2 < oo for d G (0,±), implying that the stochastic integral W d (r) is well defined. Next, where ^ - ( t + sy-'dtj =/!+/ 2, ^ ^ " ^ - ^ M ^ ^ ^ ^ - d t ^ o o, I 2 ~ ds(^j\ d+ - d -(t + s) d -- l dt^\ (/ T * d "" ldt ) <0 - Therefore, U d+id _(r) is well defined. It is also easy to check that / 0 s 2^d-- d^ ds{q\t- s) d +~ l dt) 2 < oo, or the fact that V d+td _(r) is well defined. The statement about self-similarity follows from the scaling property

20 786 Philippe A., Surgailis D., Viano M.-C. (Z(au),u) = fdd (a l / 2 Z(u), u) of the white noise and the scaling properties of the kernels of integrals in (4.2)-(4.4). This proves (i). (ii) The statement about U d+, d _ and V d+id _ follows from respective relations poo / pt+r \ 2 J s 2^d- d^ds^j^ (t + s^dt) >0, (6.1) rt+r \2 td + -d^t + s y-~l d t \ Q j (g 2 ) for each r > 0, as Г > oo. It suffices to consider т = 1 only. Then the inner integral in (6.1) does not exceed (T + s)^' 1 and therefore the left-hand side of (6.1) does not exceed P oo T2d-l / 52(d_-d + ) ( 1 + s ) 2d + -2 d s ^ C T2d -l = o ( l ) ) proving (6.1). The proof of (6.2) follows similarly. To prove the statement about W d (r), write W+(r) = W d+1/2 (T)-U d4 (r), see (4.5), where the process U d)d has asymptotically vanishing increments as shown above, while W d +i/ 2 has stationary increments. Therefore, has asymptotically stationary increments (W}(T + T) W}(T), т > 0) >ш + 1 / 2 (т), r ^ 0). This proves (ii). (iii) For a Gaussian process of the form W(r) = f Q g{r,s) Z(ds), the almost sure differentiability on (0, oo) follows if we show that for any к ^ 1, the partial derivatives #^(т, s) = d k g(r,s)/dr k exist for each r, 5 > 0 and are square integrable in s G (0,oo) and the integrals f 0 (g^k\r, s)) 2 ds are bounded on compact intervals of (0,oo) (see, e.g., [10]). The process U d+ - d _{r) admits this form of representation with We have g{ T) s)~ Г t d+ - d -(t + s) d ~- l dt, r,5>0. Jo where Then \g[ e) (T)\^Cr d +- d - e and 9< F C ) (T + *) cc(t + S) D - F C + ',

21 Time-varying fractionally integrated processes 787 implying / (gw(t,s))4s <CY, ^=0 fc-i & 1 =0 r2(d+ 2(rf + -d_-^) poo / (r + S) 2 ( d - f c +^d S Jo />oo / (r + s) 2^- 1 ) da Уо 1 /"(l + s^-^ds, Уо where the last expression is bounded on r (e, oo) for any > 0 fixed. This proves the a.s. differentiability of Ud +t d_. For the process Vd +i d_ the proof is similar by taking the function д(т, s) = s d ~~ d + f Q T(t + 5) d+ *" 1 dt. This proves part (iii) and the proposition. Proof of Lemma 5.1. Introduce the following notation. For any integers s < t, put a t - s {t) a_ a t - s {t)^t{d-) b t -s(t) V> _,(d_)v>t(d + )' bt_,(t)vi-.(rf+)<i- To prove Lemma 5.1, we need the following result. Lemma 6.1. Let (d u t G Z) be a sequence such that (1.11) and (5.1) /io/d. ГЛеп йеге exists 5 2 > 0 such that sup K -1 = 0{r & % t -> +oo, (6.3) s: 0<3<t sup -1 = 0{V S *), t -> +oo, (6.4) s: s^o sup t: s<t *t -1 = 0(s- s *), 5 +СЮ, sup t:t>0-1 = 0(\s\-% s > -oo. Proof. Let us prove (6.3). By definition, (6.5) (6.6) Note that for t large enough, /3^ := (d k -~d + )/(d + +t k 1) is arbitrary small uniformly infc < t. Indeed, by boundedness of (d t, t G Z), for any e > 0 one can find К < oo such that < б for any t G Z,fc < t K. On the other hand, max t _ K^fc<t -> 0 (i -> +oo), because тах^.^^ц^ d fc - d + -> 0 (i -> +oo). Since е~ 2,ж ^ 1 + x ^ е 2 ' х for x small enough, we obtain for t sufficiently large exp{-2 A + f c W s^exp{2 # fc

22 788 Philippe A., Surgailis D., Viano M.-C. where, by (5.1), E #J<C E k- s {t-k)-\ s^k<t l^k<t As t tends to infinity, the right-hand side is dominated by rt-i r l ~ 1 / 1 С f x~\t - x)~ l dx = СГ 6 [ x' 8 (l - x)~ l dx ^ C(t~ 5 lnt +1" 1 ). Ji ii/t Ji/t This proves (6.3). Next, let us prove (6.4). We have for 5 < 0 < t * - ( l + r & b ) n, ^ > As above, for any 5 < 0 and some t > t 0 > 0 large enough, exp {-2 /? tlp -2 1/^lU^expb \f3 t, p \+2 /3+ fc Here, by (5.1), E IA.PI < с E bi-'(*-p)- l <cx;fc-'(*+*r 1 and s^p<0 s^p<0 k>0 poo / со x~ s {t + x)' 1 dx ^ С(Г* lni + Г 1 ) E Ю < Ko\ + c *r*(t - fc)- 1 < C(t~ l + t~ s lni), 0^k<t k=l where the last sum was estimated in the proof of (6.3). This proves (6.4). Let us prove (6.5). By definition, Note that for 5 large enough, \+ k := (d k d+)/(d+ + fc s + 1) is arbitrary small uniformly in fc ^ s. Consequently, and since d t -i/d+ > 1 as t У +oo, for any б > 0 and all 5 sufficiently large we obtain (l_ e ) exp{-2 E АЬ }<<й<(1 + е)«р{2 E Ш}> ^ s^k^t-2 ' ^ s^k^t-2 * where, by (5.1), s^k^t-2 s^k proving (6.5).

23 Time-varying fractionally integrated processes 789 Finally, let us prove (6.6). We have for s < 0 < t,9- - d -$-»( d +) TT (л л. d k~ d + d + + k-s + l As above, for any \s\ large enough, s < 0, and alh > 0 we have { 2 t~2 ^ ( 2 t 2 "\ -2E A s, P -2 A+ fc [ <Л:. < exp 2E A s, P + 2 А+Л P=S = J I P=5 fc = -l J Here, by (5.1), and 2 2 El A *.pl < c E M ~ ' ( P - t-2 oo s + < ^(I'l" 1 + N" 5 inh) E Щ < + IA+OI+cE*-'(*+иг 1 < с(и -1 + м - ' ln I*Dfc=-l k=l This proves (6.6) and Lemma 6.1 too. Return to the proof of Lemma 5.1. Denote tyf := i>j(d±), <j>f ' F(d±)~ 1 j+ ± ~ 1 (j = 0,1, 2,...). We shall use the well-known relation \Ф?-Ф?\<С#- 2 (j>0). (6.7) Let us prove (5.3). Let first 0 < s < t. Then в$ = (a*g) t - 8 (t) -g^t- 8 = J 2 (M) := E 9i{<l>t.-i- 0^i<t-s h(t,s):= E 9iW-.-i-<l>t i), Q^i<t-s h(t )S ) := (фи-ф а ) E 7 5 (t,a) := 0^i<t-s E Qitf ivts+i- 1 )- Then it suffices to show that there exists ^ > 0 such that \I k (t, 5) ^ C(t - s)^" 1^ -, fc = 1,..., 5. (6.8) For fc = 1 and fc = 4, (6.8) is immediate by (2.1) and (6.7), with 5 г = 5 and S x = 1, respectively. Next, split I 2 (t,s) = E ( t- e )/ e < < <t- e + Eo«(t- e )/ e ='

24 790 Philippe A., Surgailis D., Viano M.-C. I 21 (t,s) + I 22 (t,s), Then \I 2 i(t, s)\ < Cits)- 1-5 ((ts-iy+^+it-s)^- 1 ) < Cits) 4 *- 1-6 (t~s)/s<i<t-s satisfies (6.8) and, assuming without loss of generality that 8 < 1, Ы*, s) < С E г; 1 " 5 ((* - в - г)^- 1 - (t - s)^" 1 ) (Ki<(t-*)/2 0<t^(t-a)/2 So, (6.8) is proved for fc = 2. For fc = 3, (6.8) follows by (6.7) in a similar way. Finally, by Lemma 6.1, (6.3), 0^i<t-s Now consider the case 5 < 0 < t. Similarly as above, we shall split = (a*g) t - s {t) -дфг-зф^/фг = ELi^(M) and show, in order to prove (5.5), that there exist <5 3 > 0 and 5 4 > 0 such that \r k {t, s)\ < C(t - s) d -4 d +- d - (Г* + (s)- 6 *), к = 1,..., 6. (6.9) Here Ii(t,s) I 2 {t,s) I' 3 (t,s) := (ФТ-а-ФГ-з)^ Y 9i> ^l 0^.i<-s У* 0^г<-з ~ Y 9i<^t-3-i{t). Let us first estimate the last term. By Lemma 6.1, (6.3), we have \I' 6 (t, s)\ < CE E <<<T+ e + N ~ i) d+ ~ l. There are two cases: \s\ ^ t and S < t. For \s\ ^ t, im^u^t + H)- 1 -* (t+и-г)^- 1 < c^ + L S I ) - 1 M«*<T+ *l - * ^ <ct d+ -,1 -(t + H) d ^t + H ) - '.

25 Time-varying fractionally integrated processes 791 For s < t, by splitting the sum Ew^<t+M - w^<(t+ W )/2 + Е( 4 +м)/2^+ ф we obtain \r 6 (t, s)\ < c(t+m)^- 1!*!"' + c(t + is])**- 1-8 < ct d +~ d - (t + isi)"- 1! thus proving (6.9) for к 6. Next, by Lemma 6.1, (6.4), we have \V b {t,s)\ ^Ct d +~ d - x r 5 a <Ct d +- d -(t+\s\) d -4- s^ 0^i<t+ s proving (6.9) for к 5. In the case A; = 4, use (6.7) to obtain \I' 4 {t, s)\ < C(t+ \s\) d 4 d +~ d - ^ Ct d +- d -(t + \s\) d 4-\ and, similarly, 5) < ct d+ ~ d - J2 ч 1 ~ 6 (*+M - *) d ~~ 2 < ct d+ - d ~(t + \s\y- г ^i<t+\s\ For к = 1, 2, (6.9) follows analogously. This proves (5.3) and (5.5). Let us prove (5.4) and (5.6). For 0 ^ 5 < t the proof is similar by Lemma 6.1, (6.5), and is omitted. Consider the case 5 < 0 < t. Split S = (b*g)t-s(t) - ^t-sid+ipt-s/d-ipt-s) = ELi where 0^г<-5 ^ Yl-s-iS Using (1.11), (2.1), (6.7), and Lemma 6.1, (6.6), the proof of the bound (5.6), with factor 5 (*--rf+)vo replaced by \s\ d -~ d + } for terms J fc (t,s), к = 1,2,3, is similar to the estimation of the corresponding terms in the decomposition м = Efc=i Д(^> 5 ) a bove, and is omitted. Consider the last term J 4 (, s). Here 0 ^ t s г and we use the bound 6t-*-i(t) < Сг/ _ в _* which follows from Lemma 6.1, (6.5). Together with (2.1), it implies J fc (t, s)uc^ i- 1-6^ + \s\ - i)^' 1 = Ei + E 2, where Sj := E N <^<(t+ s )/2> ^ 2 := E<t+w)/2*i<t+w T h e n w e o b t a i n 51 < C(t + \s\) d +~ l г~ 1_<5 ^ C(t - s)**- 1^- 6, (t+ * )/2 5 2 ^ C(t 4- к))" 1-5 i d +~ l ^C{t-s) d , г=1 thus proving the bound (5.6) for J 4 (, s) and Lemma 5.1 too.

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