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Semistable domain’s a. s. limit theorem

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An almost sure functional limit theorem for the domain of geometric partial attraction of semistable laws ´n Fazekas1 and Alexey Chuprunov2 Istva Abstract An almost sure functional limit theorem is obtained for variables being in the domain of geometric partial attraction of a semistable law.

Key words and phrases: almost sure limit theorem, functional limit theorem, semistable law, domain of partial attraction, slowly varying function. 2000 Mathematics Subject Classification: 60F17, 60F15, 60E07.

1

Introduction

Let ζn , n ∈ N, be a sequence of random elements defined on the probability w space (Ω, A, P). Almost sure limit theorems state that Hn [ζn ](ω) −→ µ, as n → ∞, for almost every ω ∈ Ω, where n

1 X1 Hn [ζn ](ω) = δζ (ω) , ln n k=1 k k

(1.1)

w

δx is the unit mass at x and −→ µ denotes weak convergence to the probability measure µ. In the simplest form of the almost sure central limit theorem (a.s. CLT) √ ζn = (X1 + · · · + Xn )/ n, where X1 , X2 , . . . , are i.i.d. real random variables with mean 0 and variance 1, and µ is the standard normal law N (0, 1); see [7], [27], [21]. Almost sure versions of several known usual limit theorems were 1

Faculty of Informatics, University of Debrecen, P.O. Box 12, 4010 Debrecen, Hungary, e-mail: [email protected], tel: 36-52-316666/22825 2 Department of Math. Stat. and Probability, Chebotarev Inst. of Mathematics and Mechanics, Kazan State University, Universitetskaya 17, 420008 Kazan, Russia, e-mail: [email protected]

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Istv´an Fazekas and Alexey Chuprunov

proved, see e.g. [20], [4], [1]. Almost sure functional limit theorems are also widely studied (see e.g. [14] and the references therein). The concept of a semistable distribution appeared first in 1937 in Paul L´evy’s fundamental work [22]. The semistable distributions were described by Kruglov in [19]. A description of the domain of geometric partial attraction of a semistable law was obtained in Grinevich and Khokhlov [18]. In Cs¨org˝o and Megyesi [13] the theory of semistable laws was inserted into the framework of the ‘probabilistic’ approach of Cs¨org˝o [11] and Cs¨org˝o, Haeusler, and Mason [12]. In Berkes, Cs´aki, Cs¨org˝o, and Megyesi [3] an almost sure limit theorem was obtained for laws being in the domain of geometric partial attraction of a semistable law. In that case ordinary convergence in distribution takes place only along some subsequences. However, the almost sure version of the limit theorem is valid. In [10] an integral analogue of the a.s. limit theorem of [3] was obtained. We mention that in [26] an almost sure limit theorem was proved for distributions being in the domain of geometric partial attraction of a max-semistable law. In this paper we prove an almost sure functional limit theorem for laws being in the domain of geometric partial attraction of a semistable law (Theorem 3.1). The limit in our theorem is a mixture of distributions of semistable homogeneous random process with independent increments. We mention some earlier results in this direction. The main result in Major [24] is an a.s. functional limit theorem for variables being in the domain of attraction of a stable law. Its proof is based on the continuous time ergodic theorem (see Major [23]). Major’s approach was followed in Fazekas and Rychlik [15], where the discrete time ergodic theorem was used to obtain an a.s. functional limit theorem for semistable distributions. In the present paper we turn back to the usual approach, so here the proof is based on the method described e.g. in [1] and [14]. In Section 2 we describe the intuitive background of our results. Precise definitions and statements are given in Section 3. Section 4 contains the complete proof. We shall use the following notation. IA is the indicator function of the set A. Let ρ denote the usual metric on D[0, 1] (see [5]). The distribution of a d random element ξ will be denoted by Lξ . Sign −→ denotes the convergence in distribution.

Semistable domain’s a. s. limit theorem

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3

The description of the phenomenon

In this section we show that the mixture type limiting distribution in the a.s. limit theorem arises in a natural way. Let X, Xj , j ∈ N, be independent identically distributed random variables. Let {kn } be a sequence of positive integers with the property kn+1 = c > 1. n→∞ kn

(2.1)

lim

Pn Let Skn = b1n kj=1 Xj − an , n = 1, 2, . . . , where bn and an are real numbers. Assume that the sequence Skn converges in distribution to W as n → ∞. As c > 1 in (2.1), therefore the sequence kn is eventually strictly increasing. For the sake of simplicity in this section we assume that kn is strictly increasing. For kl−1 < j ≤ kl let Bj = bl and Aj = jal /kl . Let Sn =

n 1 X Xj − An , Bn j=1

n = 1, 2, . . . .

To obtain an almost sure limit theorem for Sn we can apply the general method of the a.s. limit theory (see Theorem B in Section 4). First we have to find P the limit of ln1n nk=1 k1 LSk . Let ϕn (x), ϕX (x), ϕW (x) denote the characteristic d

function of Sn , X, and W , respectively. The convergence Skn −→ W implies   x kn ϕX e−ixan = ϕkn (x) → ϕW (x) (2.2) bn as n → ∞. Now let kn∗ = min{km : km ≥ n}. Let γn = n/kn∗ . Then for the characteristic function of Sn we have        γ n x x x −ix k n∗ an∗ kn∗ γn kn∗ n −ixAn −ixa ∗ n n ϕn (x) = ϕX e = ϕX e = ϕX e . Bn bn∗ bn∗ Now (2.2) implies that limn→∞ |ϕn (x) − ϕγWn (x)| = 0. Therefore n kn kn 1 X1 1 X 1 1 X 1 γl lim ϕl (x) = lim ϕl (x) = lim ϕ (x) = n→∞ ln n n→∞ ln kn n→∞ ln kn l l l W l=1 l=1 l=1 n km n 1 X X 1 γl 1 X 1 = lim ϕ (x) = lim n→∞ ln kn n→∞ ln kn l W k m=1 l=k m=1 m +1 m−1

km X l=km−1 +1

l km

ϕW

. l (x) = km

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Istv´an Fazekas and Alexey Chuprunov

Z 1 1 1 s = ϕ (x)ds. (2.3) ln c 1/c s W P So we obtained that limn→∞ ln1n nk=1 k1 LSk = µ where µ has characteristic R1 function ln1c 1/c 1s ϕsW (x)ds. Now to prove that n

1 X1 δS (ω) = µ lim n→∞ ln n k k k=1 for almost all ω we have to check the remaining conditions of Theorem B. For k < l let l 1 X Bk ζkl = Xj − Al + Ak . Bl j=k+1 Bl Using notation ζk = Sk , we see that ζk and ζkl are independent for k < l. To check condition (4.1) one can use the equality E|ζl − ζkl |p = (Bk /Bl )p E|Sk |p . In [2] the special case when X is the gain in the St. Petersburg game is studied. In [3] a detailed description is given for the general case when X belongs to the domain of geometric partial attraction of a semistable law. For the functional version the above considerations lead to the following. Let Zn (t) be the next step function corresponding to the partial sums Sk (k = 1, 2, . . . ) [nt] 1 X [nt] am , t ∈ [0, 1], Zn (t) = Xj − bm j=1 km where km−1 < n ≤ km and [.] denotes the integer part. Here we shall consider only the limit of the increments of this process. Let 0 ≤ t1 < t2 ≤ 1. Then 1 Zn (t2 ) − Zn (t1 ) = bm

[nt2 ] X

Xj −

j=[nt1 ]+1

[nt2 ] − [nt1 ] am . km

The characteristic function of the increment is   [nt2 ]−[nt1 ] x [nt2 ]−[nt1 ] e−ix km am = ϕZn (t2 )−Zn (t1 ) (x) = ϕX bm  =

ϕkXm



x bm



−ixam

e

 [nt2k]−[nt1 ] m

[nt2 ]−[nt1 ] km

≈ ϕW

(x).

Semistable domain’s a. s. limit theorem

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Now for the logarithmic average of the distributions of the increments we obtain n n km 1 X1 1 X X 1 ϕZl (t2 )−Zl (t1 ) (x) = lim ϕZl (t2 )−Zl (t1 ) (x) = n→∞ ln n n→∞ ln kn l l m=1 l=k l=1 +1

lim

m−1

n 1 X 1 n→∞ ln kn k m=1 m

= lim

n 1 X 1 = lim n→∞ ln kn k m=1 m

km X l=km−1 +1

km X

[lt2 ]−[lt1 ] l l km

ϕW

l=km−1 +1

. l (x) = km

Z 1 . l 1 1 (t2 −t1 )s = ϕ (x)ds. km ln c 1/c s W

(t2 −t1 ) k l m (x) ϕW

The above simple calculation shows that the limit distribution in the a.s. functional limit theorem will be given by an integral. In the next sections we shall present the precise statement and the complete proof. There we shall give explicit formulae for the centering and norming constants. As we want to prove the functional version of Theorem 1 of [3], in the following sections we shall use the same centering and norming constants as the ones in [3] which are slightly different from the constants applied in this section.

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The a. s. functional limit theorem

We shall use the description of the p-semistable distribution, 0 < p < 2, and the domain of the geometric partial attraction of a p-semistable distribution given in Cs¨org˝o, Haeusler and Mason [12], Cs¨org˝o [11], Megyesi [25], and Cs¨org˝o and Megyesi [13]. Let Nj , j = 1, 2, be standard left-continuous independent Poisson processes. Suppose that gj (s) = −Mj (s)s−1/p , s > 0, j = 1, 2,

(3.1)

are nondecreasing functions, where M1 , M2 are nonnegative, right-continuous functions on (0, ∞), either identically zero or bounded away from both zero and infinity, such that M1 + M2 is not identically zero, moreover Mj (cs) = Mj (s) for all s > 0, j = 1, 2, for some constant c ≥ 1. Let Wj (Mj ) = Wj (Mj , p) =

(3.2)

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Istv´an Fazekas and Alexey Chuprunov Z

∞

 [Nj (s) − s]d

= 1

−Mj (s) s1/p



Z

1

 Nj (s)d

+ 0

−Mj (s) s1/p

 + Mj (1), j = 1, 2,

and W (M1 , M2 ) = W2 (M2 ) − W1 (M1 ).

(3.3)

A random variable W is a p-semistable random variable with 0 < p < 2 if and d only if W = W (M1 , M2 ) + b for some M1 , M2 and b ∈ R. We will denote by ψ(x) = ψ(x, M1 (y), M2 (y)) = EeixW (M1 ,M2 ) the characteristic function of W (M1 , M2 ). Let X, Xi , i ∈ N, be independent identically distributed random variables. Recall that X is said to belong to the domain of geometric partial attraction of a semiatable law if for some sequence {kn } of positive integer numbers with property (2.1) for some norming numbers Bkn and some centering numbers Pkn d Akn , the sequence Skn = B1k i=1 Xi − Akn −→ W , as n → ∞, where W is a n p-semistable random variable. In this case we will write X ∈ Dgp (M1 , M2 , p). Denote by F the distribution function of X and by Q the quantile function Q(s) = inf{x ∈ R : F (x) ≥ s},

0 < s < 1.

Denote by Q+ the right-continuous version of the quantile function Q. Consider a subsequence {kn }∞ n=1 ⊂ N satisfying (2.1). As c > 1, the sequence {kn } is eventually strictly increasing. So for all s ∈ (0, s0 ), with s0 ∈ −1 (0, 1] small enough, there exists a unique kn∗ (s) such that kn−1 ∗ (s) ≤ s < kn∗ (s)−1 . We define γ(s) = skn∗ (s) for s ∈ (0, so ) and γ(s) = 1 for s ∈ [s0 , 1). So 1 ≤ γ(s) < c + ε for any fixed ε > 0 and all s ∈ (0, 1) small enough for the limiting c > 1 from (2.1). Then X ∈ Dgp (M1 , M2 , p), p ∈ (0, 2), if and only if for all s ∈ (0, 1) small enough Q+ (s) = −s−1/p l(s)[M1 (γ(s)) + h1 (s)],

(3.4)

Q(1 − s) = s−1/p l(s)[M2 (γ(s) + h2 (s)],

(3.5)

where l(·) is a right-continuous function, slowly varying at zero, and the error functions h1 and h2 are right-continuous such that limn→∞ hj (t/kn ) = 0, for every continuity point t > 0 of Mj , j = 1, 2.

Semistable domain’s a. s. limit theorem

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We will use the norming and centering constants Bn = n

1/p

l

1 n

,

Ank

k = Bn

1−1/n Z

Q(s)ds, k = 1, . . . , n, n = 1, 2, . . . . (3.6) 1/n

Let Snk

k 1 X Xi − Ank , k = 1, . . . , n, n = 1, 2, . . . . = Bn i=1

(3.7)

We will denote Snn and Ann by Sn and An , respectively. The aim of this paper is to obtain the functional version of the following result. Theorem A. (Theorem 1 in Berkes, Cs´aki, Cs¨org˝o and Megyesi [3].) Let X ∈ Dgp (M1 , M2 , p) along a subsequence {kn } satisfying (2.1), 0 < p < 2. Then we have w Hn [Sn ](ω) −→ µ, as n → ∞, for almost all ω ∈ Ω, where µ is a distribution with characteristic function Z c 1 ψ (x, M1 (zy), M2 (zy)) φµ (x) = dz, x ∈ R. ln(c) 1 z To obtain the functional version of Theorem A consider the random processes Zn (t) = Sn[nt] , t ∈ [0, 1], (3.8) with sample paths in the Skorokhod space D[0, 1]. Also we will consider the random process Z(t) with finite dimensional distributions defined as follows. Let 0 ≤ t0 < t1 < · · · < tm ≤ 1 and x1 , x2 , . . . , xm ∈ R. Let ∆ti = ti − ti−1 . Then let the characteristic functions of the increments of the process Z(t) be  Xm  φZ (x1 , . . . , xm ) = E exp i xj (Z(tj ) − Z(tj−1 )) = j=1



1 = ln(c)

Zc Y m ψ (∆t )1/p x , M j j 1 1



zy ∆tj



, M2



zy ∆tj



eixj f (∆tj ,z)

z

j=1

where Z f (∆tj , z) = −∆tj 1

1/∆tj

s−1/p (M2 (sz) − M1 (sz)) ds.

dz,

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Istv´an Fazekas and Alexey Chuprunov

Theorem 3.1. Let X ∈ Dgp (M1 , M2 , p) along a subsequence {kn } satisfying (2.1), 0 < p < 2. Let Zn be defined in (3.8). Then for almost all ω ∈ Ω, as n → ∞, w

Hn [Zn ](ω) −→ LZ , in D[0, 1].

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Proof

The proof of Theorem 3.1 is based on a method widely applied in a.s. limit theory which is explicitly described e.g. in Berkes and Cs´aki [1], Chuprunov and Fazekas [9], or Fazekas and Rychlik [14]. Theorem B. (Theorem 1.1 in [14].) Let (B, %) be a complete separable metric space and ζn , n ∈ N, be a sequence of random elements in B. Assume that there exist C > 0, ε > 0, an increasing sequence of positive numbers cn with limn→∞ cn = ∞, cn+1 /cn = O(1), and B-valued random elements ζkl , k, l ∈ N, k < l, such that the random elements ζk and ζkl are independent for k < l, and  β ck E{%(ζkl , ζl ) ∧ 1} ≤ C , (4.1) cl P for k < l, where β > 0. Let 0 ≤ dk ≤ log(ck+1 /ck ), assume that ∞ k=1 dk = ∞ Pn and set Dn = k=1 dk . Then for any probability distribution µ on the Borel σ-algebra of B the following two statements are equivalent n 1 X w dk δζk (ω) −→ µ , as n → ∞, for almost every Dn k=1 n 1 X w dk Lζk −→ µ , as n → ∞ . Dn k=1

ω ∈ Ω;

(4.2)

(4.3)

To apply this theorem we shall construct an auxiliary sequence {Zjk } with appropriate properties. Moreover we shall prove an ordinary (i.e. a non almost sure) functional limit theorem for the logarithmic averages. Therefore the following result will be a by-product of the proof of Theorem 3.1.

Semistable domain’s a. s. limit theorem

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P Proposition 4.1. Let L(n) = nk=1 1/k, k = 1, 2, . . . . Under the conditions of Theorem 3.1 n 1 X1 w LZ −→ LZ L(n) k=1 k k as n → ∞, in D[0, 1]. The convergence of the finite dimensional distributions is a consequence of the merge theorem of Cs¨org˝o and Megyesi [13]. To prove the tightness we shall apply the method described in Sect 6. Ch. 9 of [16]. The following lemma is a simple corollary of the merge theorem (Theorem 2 in [13]). Lemma 4.1. Let F ∈ Dgp (M1 , M2 , p) along a subsequence {kn } satisfying (2.1), 0 < p < 2. Then we have sup |P{Sn ≤ x} − P{W (M1 (γ(1/n)y), M2 (γ(1/n)y)) ≤ x}| → 0, x∈R

as n → ∞. Denote by L0 the space of all random variables endowed with the topology of convergence in probability. Proof of Theorem 3.1. We start with some general properties of the sequence {Sn } and of an auxiliary sequence {Sn0 }. Let ρ be the ordinary metric ρ(x,y) in D[0, 1]. Then ρ1 (x, y) = 1+ρ(x,y) , x, y ∈ D[0, 1] is a metric on D[0, 1], which is equivalent to ρ. 0 By Theorem 1 of [13], the sequence {Sn }∞ n=1 is bounded in L .

To define the sequence Sn0 , we need the notation 1 EXI{|X|≤Bn } , A0nk = kA0n1 , k = 1, 2, . . . , n, A0n = A0nn , Bn P for n = 1, 2, . . . . Let Sn0 = B1n ni=1 Xi − A0n . A0n1 =

By Megyesi [25] (see the proof of Theorem 3) max{|Q+ (s)|, Q(1 − s)} ≤ Cs−1/p l(s),

(4.4)

for s > 0 small enough, where C < ∞. Now, using (4.4) and the slowly varying property of l, we can find κ0 > 0 so that max{|Q+ (1/n)|, Q(1 − 1/n)} < Bm ,

(4.5)

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Istv´an Fazekas and Alexey Chuprunov

for m ≥ κ0 n and n being large enough. We shall use the following facts. Let a > 0. If a > Q(1 − s), then 1 − F (a) ≤ s.

(4.6)

If − a < Q+ (s), then F− (−a) ≤ s,

(4.7)

where F− is the left-continuous version of F . Introduce the notation F (x) = P(|X| > x) = 1 − F (x) + F− (x), for x > 0. Then (4.5), (4.6)–(4.7) give   X sup nP > 1 = sup nF (Bn ) = C1 < ∞. (4.8) Bn n∈N n∈N P Now let Un = B1n ni=1 |Xi |I{|Xi |>Bn } , n = 1, 2, . . . . Integrating by parts, we obtain Z 1 nF (Bn ) P(|Un | > K) ≤ nF (KBn x)dx. + K 1/K R1 Then, using (4.5), (4.6)–(4.7), we can see that 1/K nF (KBn x)dx < ε if K > Kε . This fact and (4.8) imply that Un is bounded in probability. Then Un0 = Pn 1 i=1 Xi I{|Xi |>Bn } , n = 1, 2, . . . , is also bounded in probability. Bn P Let Vn = B1n ni=1 Xi I{|Xi |≤Bn } , n = 1, 2, . . . . Then Vn − An = Sn − Un0 , n = 1, 2, . . . , is also bounded in probability. Using the same method (symmetrization and stopping times) as in the proof of Theorem 10.1.1 in [8], we can prove that   X 2 sup nD I{|X|≤Bn } = C2 < ∞. (4.9) Bn n∈N Using this fact and Tchebychev’s inequality, we see that the sequence {Sn0 } is bounded in probability. Therefore the sequence {An − A0n } is bounded. So we obtain that 2  X I{|X|≤Bn } − An1 = C20 < ∞. (4.10) sup nE B n∈N n Introduce the auxiliary sequence of processes {Zjk (t)}. For j < k let  [tk](A0k1 − Ak1 ), 0 ≤ t < kj , P Zjk (t) = j Zk (t) − B1k ji=1 Xi + jA0k1 , ≤ t ≤ 1. k Then Zjk (t), 0 ≤ t ≤ 1, and Zj (t), 0 ≤ t ≤ 1, are independent random processes for j < k. Using the inequality ρ(x, y) ≤ kx−yk∞ , where kx−yk∞ =

Semistable domain’s a. s. limit theorem

11

x sup0≤t≤1 |x(t)−y(t)|, x, y ∈ D[0, 1], and that the function h(x) = 1+x , x > 0, is 0 0 increasing; then intoducing notation ak1 = Bk Ak1 and applying the definition x of Zjk ; then using the subaddivity of the function h(x) = 1+x , x > 0, we obtain

sup0≤x≤1 |Zk (x) − Zjk (x)| ≤ 1 + sup0≤x≤1 |Zk (x) − Zjk (x)| P 0 ) (X − a sup1≤m≤j B1k m i k1 i=1 ≤ P ≤E 1 m 0 1 + sup1≤m≤j Bk i=1 (Xi − ak1 ) P
1 m 0 sup1≤m≤j Bk i=1 Xi I{|Xi |≤Bk } − ak1 P ≤E
+ 1 m 0 1 + sup1≤m≤j Bk i=1 Xi I{|Xi |≤Bk } − ak1 P 1 m sup1≤m≤j Bk i=1 Xi I{|Xi |>Bk } . P +E X I 1 + sup1≤m≤j B1k m i {|Xi |>Bk } i=1 Eρ1 (Zk , Zjk ) ≤ E

|x| Now using the inequality 1+|x| ≤ |x|, x ∈ R, Holder’s inequality; then Doob’s maximal inequality and (4.8); finally applying (4.9), we obtain that v !2 u m u 1 X t Eρ1 (Zk , Zjk ) ≤ E sup (Xi I{|Xi |≤Bk } − a0k1 ) + jP{|X| > Bk } ≤ Bk i=1 1≤m≤j

v u u ≤ t4E

!2 r j p j 1 X j 0 (Xi I{|Xi |≤Bk } − ak1 ) + C1 ≤ ( 4C2 + C1 ) . (4.11) Bk i=1 k k

So conditon (4.1) of Theorem B is satisfied. Now we turn to condition (4.3). Let RN be a process with distribution PN 1 w 1 LRN = L(N n=1 n LZn . We will prove that LRN −→ LZ , as N → ∞, in ) D[0, 1]. We start with the convergence of the finite dimensional distributions. Let 0 ≤ t0 < t1 < · · · < tm ≤ 1 and x1 , x2 , . . . , xm ∈ R. Let ∆tj = tj − tj−1 , j = 1, . . . , m. Consider the multidimensional characteristic functions  Xm  ϕ eN (x1 , . . . , xm ) = E exp i xj (RN (tj ) − RN (tj−1 )) = j=1

=

1 XN 1 ϕn (x1 , . . . , xm ), n=1 n L(N )

12

Istv´an Fazekas and Alexey Chuprunov

where  Xm  ϕn (x1 , . . . , xm ) = E exp i xj (Zn (tj ) − Zn (tj−1 )) = j=1

Ym

=

j=1

E exp (ixj (Zn (tj ) − Zn (tj−1 ))) . Bl

l

(l )1/p l(1/l )

Let lj = [ntj ] − [ntj−1 ]. We have nj ≈ ∆tj , Bnj = nj 1/p l(1/n)j ≈ (∆tj )1/p . Using Lemma 4.1, we obtain ϕn (x1 , . . . , xm ) ≈        m Y kn∗ (1/lj ) kn∗ (1/lj ) 1/p y , M2 y eixj dnj , ≈ ψ (∆tj ) xj , M1 l l j j j=1 where dnj lj =− Bn

Z

Z

lj =− Bn

1/lj

1−1/n

! Q(s)ds

1/n

1−1/lj n/lj

Z 1

Q(s)ds

=

1/lj

1−1/n

Q(s)ds +

!

1−1/lj

Q(s)ds − 1/n

Z

lj =− n

Z

lj =− Bn

Z

1/lj

(Q(s) + Q(1 − s))ds = 1/n

(Q(s/n) + Q(1 − s/n)) ds. Bn

Observe that       kn∗ (1/lj ) kn∗ (1/n) n kn∗ (1/lj ) kn∗ (1/lj )+1 Mh y = Mh · · ··· y ≈ lj lj kn∗ (1/lj )+1 kn∗ (1/lj )+2 n  ≈ Mh

kn∗ (1/n) n 1 1 · · ··· lj c c n

     1 kn∗ (1/n) y ≈ Mh · y , ∆tj n

h = 1, 2.

Using (3.4), (3.5) and the properties of h1 , h2 , we obtain Z dnj ≈ −∆tj 1

Z ≈ −∆tj

1/∆tj

1/∆tj

(Q(s/n) + Q(1 − s/n)) ds ≈ n1/p l(1/n)

s−1/p (M2 (γ(s/n)) − M1 (γ(s/n))) ds =

1

Z = −∆tj 1

1/∆tj


s−1/p M2 ((s/n)kn∗ (s/n) ) − M1 ((s/n)kn∗ (s/n) ) ds ≈

Semistable domain’s a. s. limit theorem  ≈f

13

kn∗ (1/n) ∆tj , n

 .

Therefore we obtain ϕn (x1 , . . . , xm ) ≈ m    1 k ∗    1 k ∗   Y n (1/n) n (1/n) ≈ ψ (∆tj )1/p xj , M1 · y , M2 · y × ∆t n ∆t n j j j=1

 kn∗ (1/n)  × exp ixj f ∆tj , . n 



We have ϕ eN (x1 , . . . , xm ) ≈

kn∗ (1/N )

X 1 ϕn (x1 , . . . , xm ) = ln(cn∗ (1/N ) ) n=k +1 n 1

1

=

n∗ (1/N )−1

1

X

ln(cn∗ (1/N ) )

r=1

∗

n (1/N )−1 X 1 X 1 ϕn (x1 , . . . , xm ) = ∗ Dr . n n (1/N ) ln(c) r=1 n=k +1 kr+1

r

Here kr+1

Dr =

X 1 ϕn (x1 , . . . , xm ) ≈ n n=k +1 r

kr+1

      m   X 1Y 1 kr+1 1 kr+1 1/p ≈ y , M2 y × ψ (∆tj ) xj , M1 n ∆t ∆t j n j n j=1 n=k +1 r





 kr+1 × exp ixj f ∆tj , = n “ ”        k 1 ixj f ∆tj , r+1 1 kr+1 1 kr+1 kr+1 n m p y , M2 y e X Y ψ (∆tj ) xj , M1 ∆tj n ∆tj n 1 ≈ = n kr+1 kr+1 n=k +1 j=1 r

Z ≈

m 1 Y

 −1 ψ (∆tj ) xj , M1 ((∆tj z) y), M2 ((∆tj z) y) eixj f (∆tj ,z )

1/c j=1

=



1 p

−1

−1

z

  1 Z cY m ψ (∆t ) p x , M ((∆t )−1 zy), M ((∆t )−1 zy) eixj f (∆tj ,z) j j 1 j 2 j 1 j=1

z

dz =

dz.

14

Istv´an Fazekas and Alexey Chuprunov

Therefore we obtain ϕ eN (x1 , . . . , xm ) ≈ ≈

1 ln(c)

  1 Z cY m ψ (∆t ) p x , M ((∆t )−1 zy), M ((∆t )−1 zy) eixj f (∆tj ,z) j j 1 j 2 j 1 j=1

z

dz.

Consequently, the finite dimensional distributions of RN converge to the finite dimensional distributions of Z, as N → ∞. Now we turn to the proof of tightness. Consider the random processes Zn0 (t)

[nt] 1 X = Xi − [nt]A0n1 = Zn (t) + [nt](An1 − A0n1 ), t ∈ [0, 1]. Bn i=1

We know that the numerical sequence cn = A0n − An is bounded. We will show that the sequence of functions fn (t) = [nt](A0n1 − An1 ), t ∈ [0, 1], belongs to a set being compact in the uniform norm k · k∞ . Let 0 < ε < 1. We will construct a finite ε-net fi0 , 1 ≤ i ≤ r, such that {fn } ⊂ ∪ri=1 Gε (fi0 ). Here Gε (f ) = {g ∈ D[0, 1] : kf − gk∞ < ε}. Since cn is a bounded sequence, there exsist d0 , d00 ∈ R such that d0 < cn < d00 for all n ∈ N. 00 0 |cn | Choose m ∈ N such that d m−d < ε/2 and n0 ∈ N such that maxi∈N < ε/2. n0 d00 −d0 0 0 0 Put di = d + i m , 1 ≤ i ≤ m. Define fn as fn = fn for n ≤ n0 and fn0 (t) = tdn−n0 , t ∈ [0, 1], n0 < n ≤ n0 + m. Let n > n0 and let i ∈ {1, . . . , m} be such that |cn − di | < ε/2. Then for all t ∈ [0, 1] we have |fn (t) − fn0 0 +i (t)| ≤ |cn − di | + (nt − [nt])

|cn | < ε. n

0 +m Therefore we have {fn } ⊂ ∪ni=1 Gε (fi0 ). Consequently, there exists a set K1 ⊂ D[0, 1] which is a compact in the norm k · k∞ and {fn } ⊂ K1 .

Now we will prove the tightness of the distribution family {LZn0 : n ∈ N}. To this end we shall apply Theorem 15.3 of Billingsley [5]. Let 0 < ε < 1 and

Semistable domain’s a. s. limit theorem

15

0 < u < 1. Using similar calculations as in (4.11), applying (4.8) and (4.9), we obtain   0 0 0 lim lim sup P{ sup |Zn (t)| > ε} + P{ sup |Zn (1) − Zn (t)| > ε} ≤ u→0

n→∞

≤ lim lim sup u→0

n→∞

1−u≤t≤1

0≤t≤u

8([un] + n − [n − un]) ε2

 E

X I{|X|≤Bn } − A0n1 Bn

(4.12) !

2

+ P{|X| > Bn }

= 0.

Using the inequality in Sect. 6 Ch. 9 of the book [16] by Gikhman and Skorokhod, we obtain ) ( m X min{|Zn0 (t2 ) − Zn0 (t1 )|, |Zn0 (t3 ) − Zn0 (t2 )|} > ε ≤ lim lim sup P sup m→∞

n→∞

k−1 k ≤t1 ≤t2 ≤t3 < m m

k=1

≤ lim lim sup m→∞

n→∞

m X

( sup

P

k=1

k−1 k ≤t< m m

)!2 0 k ε Zn ( ) − Zn0 (t) > ≤ 4 m

k o2 o n] − [ k−1 n]) n n X C([ m 0 m E I −A +P{|X| > B } × n {|X|≤B } n n1 m→∞ n→∞ 1≤k≤m ε2 Bn m k o2 o X C([ m n] − [ k−1 n]) n n X 0 m E × I − A + P{|X| > B } = 0. n {|X|≤Bn } n1 2 ε B n k=1 (4.13)

≤ lim lim sup max

0 0 (t), t ∈ [0, 1], where (t) + Z2n Let Zn0 (t) = Z1n

0 Z1n (t)

[nt] [nt] 1 X 1 X 0 0 = Xi I{|Xi |≤Bn } − [nt]An1 and Z2n (t) = Xi I{|Xi |>Bn } , Bn i=1 Bn i=1

for t ∈ [0, 1]. By Kolmogorov’s inequality, we have

0 sup P {kZ1n k∞ > K} ≤ sup n∈N

nE



1 XI{|X|≤Bn } Bn

K2

n∈N

− A0n1

2 ≤

C2 → 0, (4.14) K2

as K → ∞. We know that the family of random variables {Un } is bounded in L0 . Therefore 0 k∞ > K} ≤ sup P {|Un | > K} → 0, as K → ∞. sup P {kZ2n n∈N

n∈N

(4.15)

16

Istv´an Fazekas and Alexey Chuprunov

From (4.14) and (4.15) we obtain sup P {kZn0 k∞ > K} → 0, as K → ∞.

(4.16)

n∈N

By Theorem 15.3 in Billingsley [5], (4.12), (4.13), and (4.16) imply the tightness of the distribution family {LZn0 : n ∈ N}. Therefore for any ε > 0 there exists a compact set Kε0 ⊂ D[0, 1] such that LZn0 (Kε0 ) > 1 − ε for all n ∈ N. Since K1 is a compact set in the norm k · k∞ , therefore Kε = Kε0 + K1 is a compact set in D[0, 1]. For all n ∈ N we have LZn (D[0, 1] \ Kε ) ≤ LZn0 (D[0, 1] \ Kε0 ) < ε. So, for all N , we have LRN (D[0, 1] \ Kε ) < ε. Therefore the family {RN , N ≥ 1} is tight. By Prohorov’s theorem, the tightness of RN , N ∈ N, and the weak convergence of the finite dimensional distriw butions of RN to the finite dimensional distributions of Z imply LRN −→ LZ , as N → ∞. By Theorem B, this and (4.11) imply our result. The proof is complete. ACKNOWLEDGEMENTS Partially supported by the Hungarian Foundation of Scientific Researches under Grant No. OTKA T047067/2004 and Grant No. OTKA T048544/2005. The research was partially realized while A. Chuprunov was visiting Faculty of Informatics, University of Debrecen, Hungary. The authors are indebted to the referee whose suggestions helped to improve the presentation of the results.

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