Asymptotics of weighted random sums

Transcription

1 Asyptotics of weighted rando sus José Manuel Corcuera, David Nualart, Mark Podolskij arxiv:402.44v [ath.pr] 6 Feb 204 February 7, 204 Abstract In this paper we study the asyptotic behaviour of weighted rando sus when the su process converges stably in law to a Brownian otion and the weight process has continuous trajectories, ore regular than that of a Brownian otion. We show that these sus converge in law to the integral of the weight process with respect to the Brownian otion when the distance between observations goes to zero. The result is obtained with the help of fractional calculus showing the power of this technique. This study, though interesting by itself, is otivated by an error found in the proof of Theore 4 in [2]. Introduction Let Ω,F, P be a coplete probability space. Fix a tie interval [0, T] and consider a double sequence of rando variables ξ = {ξ i,, Z +, i [T]}. For any we denote by g t the stochastic process defined as the distribution function of the signed easure on [0, T] which gives ass ξ i, to the points t i = i, for i [T], that is, g t := [t] ξ i,. Notice that ξ i, = g t i g t i. Throughout this paper we assue the following hypotheses on the double sequence ξ: H The sequence of processes g t satisfies f.d.d. g t wt F-stably, University of Barcelona, Spain. E-ail: jcorcuera@ub.edu University of Kansas, USA.E-ail: nualart@ku.edu University of Heilderberg, Gerany. E-ail:.podolskij@uni-heidelberg.de

2 where wt is a standard Brownian otion independent of F, and the latter denotes convergence of finite diensional distributions F-stably in law see the definition below. H2 The faily of rando variables ξ satisfies the tightness condition k 4 E ξ i, k j 2 C, i=j+ for any j < k [T]. Notice that these hypotheses iply that g t L wt, F-stably in the Skorohod space D[0, T] equipped with the unifor topology. Under these assuptions, the purpose of this note is to establish the following result. Theore Let ft be a α-hölder continuous process with index α > /2. Suppose that ξ = {ξ i,, Z +, i [T]} is a faily of rando variables satisfying hypotheses H and H2. Set Then, X t := [t] t ft i ξ i, = fsdg s. 0 X t L t fsdws, 0 F-stably in D[0, T], wherewt is a standard Brownian otion independent off. Recall that a sequence of rando vectors or processes Y n converges F-stably in law to a rando vector or process Y, where Y is defined on an extension Ω,F, P of the original probability Ω,F, P, if Y n, Z L Y, Z for any F-easurable rando variable Z. If Y is F-easurable, then we have convergence in probability. We refer to [5] and [] for ore details on stable convergence. Reark 2 The conclusion of Theore also holds for the forward-type Rieann sus X t := [t] ft i ξ i,. Indeed, it suffices to put the rando weights ξ i, at the points t i. 2

3 This theore has been otivated by a istake in the proof of Theore 4 of the reference [2]. More precisely the fact that li n li sup P B n, t > ǫ = 0 in page 724 of [2] is a particular case of the convergence 4. The rest of this note is devoted to the proof of Theore. First we establish a basic decoposition, which reduces the proof of Theore to the proof of the convergence 4. In section 3 we discuss different attepts to prove this convergence using p-variation nors and artingale ethods. Finally in Section 4 we provide a proof of 4 using techniques of fractional calculus. 2 The ain decoposition The basic idea of the proof of Theore is the classical Bernstein s big blocks/sall blocks technique. For this purpose we set u j = j n, n, and decopose the process X t as follows X t = = [t] ft i ξ i, [nt]+ i I n j fti fu j ξ i, + [nt]+ fu j ξ i, 2 i I n j with I n j := {i : i [T], i [ j n, j n }. According to our hypothesis it holds that g L w F-stably on D[0, T] with wt being a Brownian otion independent of F. This iplies, in particular, the F-stable convergence [nt]+ We also have that [nt]+ fu j ξ i, i I n j [nt]+ L fu j wu j wu j. fu j wu j wu j u.c.p. t fsdws. n 0 where u.c.p. stands for unifor convergence in probability. Now we treat the first ter of 2, but before we consider, separately, the last suand. We clai that P- li li sup sup n i I n [nt]+ ft i fu [nt] 3 ξ i, = 0. 3

4 In fact, using the Hölder continuity of f we can write ft i fu [nt] ξ i, f αn α ξ i,, i I n [nt]+ where i I n [nt]+ fu fv f α := sup u v T u v α <. Then, 3 follows fro Hypothesis H2 taking into account that the cardinality of the set I n [nt]+ is bounded by n + and α > 2. Then, in order to finish the proof, we need to show that [nt] P- li li sup sup n fti fu j ξ i, = 0. 4 i I n j In fact, this is a key step of the proof. In particular situations, such as e.g. in the artingale fraework, there are various specific techniques of the proof. We will present soe of the in the next section. However, proving convergence 4 in a general setting turns out to be not quite easy. A first straightforward attept is as follows. We set R n, t := Then we deduce sup [nt] i I n j fti fu j ξ i,. [T] R n, t n α f α ξ i,, but in general we have that [T] li ξ i, = as the following siple exaple shows. Exaple 3 Consider the case where ξ i, = X i, where X i i are i.i.d with PX i = = PX i = = /2. Then g t = [t] X i, 4

5 and g L w on D[0, T] and [T] ξ i,. 3 Young s calculus A ore sophisticated approach for proving 4 is to use Young s integral. Consider the interval Ij n := [ u j, u j. Then [nt] i I n j fti fu j ξ i, = [nt] fs fu j dg s. Ij n By the Love-Young inequality and for > α, [nt] [nt] fs fu j dg s Ij n C α, υ j fυ j g α with υ j ph := sup π N hs i hs i p /p, where the supreu runs over all partitions π = {s 0,..., s N } of the interval [u j, u j ], and C α, is a positive constant; see [8]. Notice that υ j α f n α f α. The proble is then to bound υ j L g under the hypothesis g w on D[0, T]. Unfortunately, the strong p-variation υ p is not a continuous functional on D[0, T] equipped with the unifor topology. Nevertheless, we suppose for a oent that the convergence υ j g L υj w, holds. Then, recalling that α > 2, we can choose = 2 ε with ε < α 2 and we obtain that υ j w w n <. Consequently, υ j α fυ j g L υj α fυ j w f α w n α. 5

6 Thus, we deduce the desired convergence li n li sup P { sup [nt] I n j fs fuj } dg s > ε = 0, since α+ >. This would coplete the proof of our central liit theore. Unfortunately, there are only few results about the asyptotic behaviour of υ g the latter denotes strong /-variation on the interval [0, ]. Below, we shall ention soe of the. Denote by W p [0, ] the space of functions on [0, ] such that υ p < p with the nor [p] := υ p /p +. Proposition 4 Norvaiša-Račkauskas, 2008, [4]. Let X i be an iid sequence, set S nt := [t] X i, t [0, ]. Then, for p > 2, S L w, inw p [0, ] iff EX = 0 and EX 2 =. As a consequence, if the rando variables ξ i, are iid with Eξ i, = 0, E ξi, 2 =, L we iediately deduce that υ p g υ pw. Another result that can help is the following one. Theore 5 Lépingle 976, [3]. If p > 2 there exists a positive constant C depending on p, such that Eυ p M /p CE M for all artingales M. Assue that g is a artingale for a fixed, which is equivalent to say that{ξ i,, i [T]} is a artingale difference. Then we have and if Eυ p g /p CE g, g L w 6

7 in D[0, T], we also have that g L w. Using the tightness condition on g and Doob s inequality, we obtain that E g < C. Now, by the Skorohod representation theore and doinated convergence theore, we have that li sup Eυ j pg /p Cn w. and we can obtain the central liit theore. So, in the both cases entioned above we need additional conditions on the process g, in order to get the desired result. However, other interesting exaples are not covered by above ethods. For instance, consider a fractional Brownian otionb H t with Hurst paraeter H 0, 3/4 and define g t = [t] 2H Bt H i Bt H i 2. L It is well known that g w on D[0, T], but, to the best of our knowledge, there is less known about the asyptotic behaviour of the strong p-variation of g. For this reason we will develop a new technique, which does not rely on p-variation concepts. 4 Proof of the convergence 4 In this section we are going to prove 4 and, therefore, coplete the proof of Theore, using techniques of fractional calculus. We refer the reader to [6] for a detailed exposition of fractional calculus. Proof. Fix γ 0, such that /2 < γ < α. Throughout the proof all positive constants are denoted by C, although they ay change fro line to line. Denote by the sallest integer greater or equal than u j. Then, an integer i belongs to I n j if and only if u j i <. Let Jn, j be the interval write R n, t = [nt] i I n j fti fu j ξ i, = [ u j,. With this notation we can [nt] fs fu j dg s. J n, j 7

8 Set R n,,j := fs fu j dg s. J n, j We have that for any 0 a < b T, the identity [a,b fs fadg s = b a D γ a+ f asd γ b g b sds 5 holds, where f a s = fs fa,g b s = g s g b for s [a, b and D γ a+ and D γ b are the fractional derivative operators defined by D γ a+ f fs fa s fs fy as = Γ γ s a γ + γ dy, a s y γ+ and = D γ b g b s g s g b Γγ b s γ b + γ s g s g y y s 2 γ dy. Notice that these operators are well defined by the α-hölder continuity of f, with α > γ and since g is piecewise constant. The identity 5 can be found e.g. in [7, Theore 3. v]. Now, if we take a = u j, we can estiate D γ a+ f as by the Hölder nor of f and in this way we obtain that D γ a+ f as C f α n α+γ = Gn α+γ, for soe rando variable G. As a consequence, if we put a = u j and b =, we deduce the inequality b R n,,j : = D γ a+ f asd γ b g b sds a Gn α+γ Γ γ j / u j D γ b g b s ds Gn α+γ tk Γ γ D γ b g b s ds, k= t k j [ The last inequality follows fro the inclusion u j, ] j g b = g t j. [t j, t j ]. Notice also that 8

9 Suppose that t k < s < t k. Then we obtain the identity, = = = D γ b g b s Γγ Γγ Γγ l=k+ g t k g t j tj γ + γ s t j s g t k g t j j γ + γ t j s g t k g t j γ t j s l=k+ g t k g y y s 2 γ tl dy t l g t k g t l y s 2 γ g t k g t l [t l s γ t l s γ ]. dy Therefore, D γ b g b s Γγ l=k+ g t k g t l [t l s γ t l s γ ] + g t k g t j γ t j s. Integrating in the variable s yields, tk t k D γ b g b s ds C γ l=k+ g t k g t l l k γ 2l k γ +l k+ γ +C γ g t k g t j [ k+ γ k γ ]. Then, for any constant K > 0, and ε > 0 P sup R n, t > ε PG > K+ ε E sup R n, t {G K}. 9

10 Now, we have E sup R n, t {G K} KCn α+γ γ [nt] k= [nt] E R n,,j {G K} E g t k g t l l=k+ l k γ 2l k γ +l k+ γ +KCn α+γ γ [nt] E k= Due to tightness condition we obtain that E g t k g t l C t k t l /2, g t k g t j [ k+ γ k γ ]. hence, E sup R n, t {G K} KCn α+γ γ [nt] l k γ 2l k γ +l k+ γ +KCn α+γ γ [nt] k= KCn α+γ γ 2 KCn α+γ γ 2 [nt] k= l k l=k+ k[ k+ γ k γ ] k= [nt] γ+ j 2 j. k γ k= k γ 2 Then, since n +, we conclude that E sup R n, t {G K} KCn /2 α 0, since α > /2. Therefore, by letting K goes to infinity, we obtain li li P sup R n, t > ε = 0. n 0

11 References [] Aldous D. and Eagleson, G. K On ixing and stability of liit theores. Annals of Probability, 6, [2] Corcuera, J.M. Nualart, D., Woerner, J. H. C Power variation of soe integral fractional processes, Bernoulli [3] Lépingle, D La variation d ordre p des sei-artingales. Z. Wahrsch. verw. Geb, [4] Norvaiša, R. and Račkauskas, A Convergence in law of partial su processes in p-variation nor. Lithuanian Matheatical Journal, [5] Renyi, A On stable sequences of events. Sankhyā Ser. A 25, [6] Sako, S. G., Kilbas, A. A., Marichev, O.I Fractional integrals and derivatives. Theory and applications. Gordon and Breach Science Publishers, Great Britain. [7] Zähle, M On the link between fractional and stochastic calculus. In: Stochastic dynaics, Eds. H. Crauel and M. Gundlach Springer, New York. [8] Young, L. C An inequality of the Hölder type, connected with Stieljes integration. Acta Matheatica