Geometric Stable Laws Through Series Representations

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provided by Bulgarian Digital Mathematics Library at IMI-BAS Serdica Math. J. 25 (1999), 241-256

GEOMETRIC STABLE LAWS THROUGH SERIES REPRESENTATIONS

Tomasz J. Kozubowski, Krzysztof Podg´orski

Communicated by S. T. Rachev

Abstract. Let (Xi) be a sequence of i.i.d. random variables, and let N be a geometric random variable independent of (Xi). Geometric stable distributions are weak limits of (normalized) geometric compounds, SN = X + + XN , when the mean of N converges to inﬁnity. By an appro- 1 · · · priate representation of the individual summands in SN we obtain series representation of the limiting geometric stable distribution. In addition, we [Nt] study the asymptotic behavior of the partial sum process SN (t) = Xi, i=1 and derive series representations of the limiting geometric stable processP and the corresponding stochastic integral. We also obtain strong invariance principles for stable and geometric stable laws.

1. Introduction. An increasing interest has been seen recently in geometric stable (GS) distributions: the class of limiting laws of appropriately normalized random sums of i.i.d. random variables,

(1) S = X + + X , N 1 · · · N 1991 Mathematics Subject Classification: 60E07, 60F05, 60F15, 60F17, 60G50, 60H05 Key words: geometric compound, invariance principle, Linnik distribution, Mittag-Leffler distribution, random sum, stable distribution, stochastic integral 242 Tomasz J. Kozubowski, Krzysztof Podgórski where the number of terms is geometrically distributed with mean 1/p, and p 0 → (see, e.g., [7], [8], [10], [11], [12], [13], [14] and [21]). These heavy-tailed distributions provide useful models in mathematical finance (see, e.g., [1], [20], [13]), as well as in a variety of other fields (see, e.g., [6] for examples, applications, and extensive references for geometric compounds (1)). Gnedenko and Fakhim [5] in their classical transfer theorem gave the conditions for the weak convergence and the form of the limiting distribution of (1). Our approach, which is in the spirit of LePage et al. [17], is fundamentally different from that of Gnedenko and Fakhim [5] and other authors. It utilizes the following representation of a vector of symmetric i.i.d. random variables:

d (X ,...,X ) = (X ,...,X ) (2) 1 n 1,n n,n def −1 −1 = π((δ1G (Γ1/Γn+1),...,δnG (Γn/Γn+1)).

In the above equation, (Γi) is a sequence of arrival times of a standard Poisson process, (δi) is a sequence of independent symmetric signs, and π is a random uniform permutation of coordinates in Rn, all three mutually independent, while

(3) G−1(y) = inf x 0 : G(x) y , { ≥ ≤ } where G(x)= P ( X x). | 1|≥ After preliminary deﬁnitions of Section 2, in Section 3 of the paper we generalize (2) to the case of random number of variables and then use it to prove almost sure convergence of the obtained version of the random sum SN . Further, in Theorem 3.1, we establish LePage type series representation of the limiting distribution of (1). We also derive an almost sure version of the well-known stability property of symmetric GS random variables (Proposition 3.1). Then, we extend the representations to processes and consider

[Npt] (4) S (t)= X , 0 t 1, p i ≤ ≤ Xi=1 where Np is geometric with mean 1/p (independent of an i.i.d. sequence (Xi)). Rachev and Samorodnitsky [21] define a GS Lévy motion as the week limit of (4) in D[0, 1] (as p 0). We study almost sure asymptotic behavior of a weak → representation of (4) using the method of LePage et al. [17]. Our approach leads to a series representation of the limiting process and allows for a natural definition of the stochastic integral with respect to a GS Lévy motion. We conclude with Section 4, where we use the representation (2) to obtain strong invariance principles for stable and GS laws. In a series of papers concluded Geometric stable laws 243 by Berkes and Dehling [2], a sum of symmetric, i.i.d. random variables from n the normal α-stable domain, Xi, was approximated almost surely by a sum i=1 P n of i.i.d. symmetric α-stable r.v.’s, Yi. We follow a different approach, and i=1 n n P n d n approximate Xi,n strongly by Yi,n, where (Yi,n)i=1 = (Yi)i=1. While we i=1 i=1 obtain a weakerP form of approximation,P the rate of convergence is always at least o(n1/α), the rate that may not always hold for other strong invariance principles (see examples [2]). Our proofs, which are essentially different and simpler, allow for an extension to the general stable domain (not necessary normal or symmetric). We also derive analogous strong invariance principle for random Np Np Np sums Xi,Np , by approximating them by Yi,Np , where (Yi,Np )i=1 is a vector i=1 i=1 of i.i.d.P symmetric GS r.v.’s. P

2. Geometric stable laws. Let (Xi) be a sequence of i.i.d. random vec- d tors in R , and let Np be a r.v. independent of (Xi) and geometrically distributed with parameter p (0, 1), ∈ (5) P (N = k)= p(1 p)k−1, k = 1, 2,.... p − We say that Y (and its distribution) is geometric stable (GS) if and only if

Np d (6) a(p) (X b(p)) Y, p 0, i − → → Xi=1 for some a(p) > 0 and b(p) Rd. As shown in Mittnik and Rachev [19], Y is ∈ GS if and only if its characteristic function (ch.f.) has the form

(7) Ψ(t) = (1 log Φ(t))−1, t Rd, − ∈ where Φ is a ch.f. of some multivariate α-stable distribution. A random vector Y is strictly GS if it corresponds to a strictly α-stable r.v. via (7). In the univariate case, GS laws form a four parameter family given by ch.f. (8) ψ(t)=[1+ σα t α ω (t) iµt]−1, | | α,β − where 1 iβsign(x) tan(πα/2), if α = 1, (9) ωα,β(x)= − 6 ( 1+ iβ 2 sign(x) log x , if α = 1. π | | 244 Tomasz J. Kozubowski, Krzysztof Podg´orski

The parameter α (0, 2] is the index of stability, determining the tail of the ∈ distribution, β [ 1, 1] is the skewness parameter, and µ R and σ 0 ∈ − ∈ ≥ control the location and scale, respectively. A GS distribution given by (8) will be denoted as GSα(σ,β,µ). Similarly, the stable law corresponding to (8) via (7) will be denoted S (σ,β,µ). Strictly GS laws have ch.f. (8) with α = 1 and α 6 µ = 0, or α = 1 and β = 0. In the symmetric case (µ = 0 and β = 0), the ch.f. simplifies to ψ(t) = 1/(1 + σα t α), and is known in the literature as the | | Linnik characteristic function (see, e.g., [9], [3]). Positive and completely skewed GS distributions (0 <α< 1 and β = 1) are Mittag-Leffler distributions, studied in [23] in relation to relaxation phenomena. Relation (7) leads to the fundamental representation of a GS r.v.’s in terms of stable and exponential variates (see, e.g., [10]). Proposition 2.1. Let Y GS (σ,β,µ). Then, ∼ α µW + W 1/ασX, if α = 1, (10) Y =d 6 ( µW + W σX + 2π−1σβW log(Wσ), if α = 1, where X Sα(1, β, 0) and W , independent of X, has the standard exponential distribution.∼ A GS stochastic process can be defined through its finite-dimensional distributions: X(t), t T is geometric stable if all of its finite-dimensional { ∈ } distributions, (11) (X(t ),...,X(t )), t ,...,t T, d 1, 1 d 1 d ∈ ≥ are GS. It is strictly GS (symmetric GS) , if all of its finite-dimensional distributions are strictly GS (symmetric GS), respectively. Note that if the finite- dimensional distributions are GS, then by consistency, they must all have the same index α.

3. Series representations. The series representations of stable and, in general, inﬁnitely divisible random variables, were studied on various occa- sions (see [18], [4], [15], [16], [22]). It is well-known that a symmetric α-stable distribution has the representation

∞ −1/α δiΓi , Xi=1 where (Γi) is a sequence of arrival times of a Poisson process with intensity one and (δi) is a Rademacher sequence independent of (Γi). By Proposition 2.1, the Geometric stable laws 245 corresponding symmetric GS distribution can be written as ∞ 1/α −1/α W δiΓi , Xi=1 where W has the standard exponential distribution and is independent of (δi) and (Γi). We derive similar representation for the general non-symmetric case through almost sure convergence of an appropriately scaled and centered representation of Np Xi. Our approach is based on the results of LePage et al. [17] and illustrates i=1 probabilisticP structure of the weak convergence of normalized random sums. Let (Xi) be a sequence of i.i.d. random variables. Deﬁne G (x)= P (X x X 0), + 1 ≥ | 1 ≥ G (x)= P ( X >x X > 0), − − 1 |− 1 G(x)= P ( X x) | 1|≥ p = P (X 0), q = P ( X > 0). 0 1 ≥ 0 − 1 Then,

d (12) S = X + + X = δ G−1(Γ /Γ )+ + δ G−1(Γ /Γ ), n 1 · · · n 1 δ1 1 n+1 · · · n δn n n+1 where (δ ) is an i.i.d. sequence of random signs: P (δ =+)= p and P (δ = )= i i 0 i − q0. If the distribution of X1 is in an α-stable domain of attraction with α< 2, the representation on the right hand side of (12), appropriately normalized, converges almost surely to a stable law: n (δ G−1(Γ /Γ ) b ) ∞ i=1 i δi i n+1 n a.s. −1/α (13) lim − = (ziΓ Ci), n→∞ a i − P n i=1 X where a = G−1(1/n), b = p A 1 G−1(x)dx q B 1 G−1(x)dx, n n 0 1/n + − 0 1/n − δi+1 δi−1 i+1 −1/α zi = A 2 + B 2 R, Ci = Ezi i x R dx, −1 −1 −1 −1 with A = lim G (1/n)/G (1/n), B = limn→∞RG (1/n)/G (1/n) (see [17]). n→∞ + − We extend (13) to the case where n is a random variable. In the following result, Np is a geometric r.v. (5) independent of (Xi) and (14) G (x)= P (N x), G−1(y) = inf x : G (x) y . p p ≤ p { p ≥ } 246 Tomasz J. Kozubowski, Krzysztof Podg´orski

Theorem 3.1. The following representation holds

∗ Np d (15) S = δ G−1(Γ /Γ ∗ ), Np i δi i Np +1 Xi=1 ∗ −1 where Np = Gp (U) and U is uniformly distributed on (0, 1), independent of (δi) and (Γi). Moreover,

∗ Np (δ G−1(Γ /Γ ∗ ) b ) ∞ i=1 i δi i Np +1 p a.s. 1/α −1/α (16) lim − = W (ziΓi Ci) p→0 P ap − Xi=1 α 2 2 1/α (p0A q0B )(W W ), if α = 1, + α−1 − − 6 2 2 ( (p0A q0B )W (ln W 1), if α = 1, − − where b = p A 1 G−1(x)dx q B 1 G−1(x)dx, a = G−1(p), and W = ln U. p 0 p + − 0 p − p − R R ∗ P r o o f. First, by (12), the right hand side of (15), conditionally on Np = n, has the same distribution as Sn. Thus, the equality of distributions follows, ∗ since Np is independent of Sn and Np is independent of (δi) and (Γi). Next, write

∗ Np −1 (δiG (Γi/ΓN ∗+1) bp) δi p − i=1 = P ap ∗ Np −1 (δiG (Γi/ΓN ∗+1) bN ∗ ) a ∗ δi p p b ∗ b Np  i=1 − ∗ Np p  = + Np − , ap P aN ∗ aN ∗  p p        ∗ −1 ∗ −1 1/α and note that aNp /ap = G (1/Np )/G (p) converges almost surely to W , since pN ∗ converges almost surely to ln U. Further, since N ∗ with p − p → ∞ probability one, (13) implies

∗ Np −1 (δ G (Γ /Γ ∗ ) b ∗ ) ∞ i=1 i δi i Np +1 Np a.s. −1/α lim − = (ziΓ Ci). p→0 a ∗ i − P Np Xi=1 Geometric stable laws 247

Finally, note that −1 ∗ −1 ∗ G+ (u/Np ) −1/α G− (u/Np ) −1/α lim −1 ∗ = Au , lim −1 ∗ = Bu , p→0 G (1/Np ) p→0 G (1/Np ) and apply the dominated convergence theorem to obtain p −1 p −1 b ∗ b ∗ G+ (x)dx ∗ G− (x)dx ∗ Np p ∗ 1/Np ∗ 1/Np lim Np − = p0ANp lim q0BNp lim p→0 ∗ p→0 −1 ∗ p→0 −1 ∗ aNp R G (1/Np ) − R G (1/Np ) pN ∗ −1 ∗ pN ∗ −1 ∗ p G+ (u/Np ) p G− (u/Np ) = p0A lim du q0B lim du p→0 G−1(1/N ∗) − p→0 G−1(1/N ∗) Z1 p Z1 p W = (p A2 q B2) u−1/αdu, 0 − 0 Z1 which concludes the proof. Symmetric GS laws possess the well-known stability property: if Xi are i.i.d. symmetric GS random variables with index α (0, 2), then for any p > 0 ∈ we have (17) p1/α(X + + X ) =d X , 1 · · · Np 1 where Np is geometrically distributed and independent of (Xi). Using our series representation we can obtain an almost sure version of (17). Suppose that the arrival times of Np Poisson processes are combined to define a new process. More precisely, let , ,... be a sequence of independent standard Poisson N1 N2 processes with corresponding arrival times (Γ1,i), (Γ2,i),.... The observed processes are (W ),..., (W ), where (W ) is a sequence of intensities, and N1 1· NNp Np · i N is a geometric r.v. (5) independent of ( ). The observed arrival times, p Ni Γ /W ,..., Γ /W , i N, when combined and ordered, define a new count- 1,i 1 Np,i Np ∈ ing process, ((W + + W ) ). N 1 · · · Np · Thus, is the combined process with time scaled by the divisor (W + +W ). N 1 · · · Np Consequently, if Γ ’s are arrival times of , then the observed arrivals have the i N form Γ /(W + +W ). Assuming that the intensities W ,W ,... are standard i 1 · · · Np 1 2 exponential random variables, mutually independent, and independent of ( ), Ni we have the following almost sure version of (17). Proposition 3.1. The process defined above is a standard Poisson process, and with probability one N

∞ 1/α ∞ 1/α ∞ 1/α 1/α W1 WNp W p δ1,i + + δNp,i = δi , Γ1,i · · · ΓNp,i ! Γi Xi=1 Xi=1 Xi=1 248 Tomasz J. Kozubowski, Krzysztof Podg´orski where W = p(W + + W ) has the standard exponential distribution and (δ ) 1 · · · Np i is an independent of ((Γi),W ) Rademacher sequence deﬁned by

Γ Γ δ = δ if and only if i = j,k . i j,k W + + W W 1 · · · Np j

P r o o f. Conditionally on W1 = λ1,W2 = λ2,..., and Np = n, the observed process ((W + + W ) ) is a Poisson process with intensity λ + N 1 · · · Np · 1 + λ . Thus, under the same conditioning, is a standard Poisson process · · · n N whose distribution does not depend on the conditioning. Consequently, the un- conditional distribution of must be the same as that of a standard Poisson N process. Next let us consider symmetric GS processes. Let (Xi) be a sequence of i.i.d. random variables independent of a geometrically distributed (5) random variable Np. The partial sum process on [0, 1],

[Npt]

(18) Sp(t)= Xi, Xi=1 has càdlàg trajectories with jumps at points k/N for k 1,...,N and is con- p ∈ { p} stant between the jumps. The values of jumps are equal to Xk, k 1,...,Np . ∈1 { } It does not have independent increments. Note that the integral 0 vdSp is well defined for any measurable function v on [0, 1]. R Rachev and Samorodnitsky [21] introduced a GS Lévy motion S(t) as the weak limit (as p 0) of an appropriately normalized Sp(t). We study the limits 1 → of Sp(t) and 0 vdSp through their almost sure representations. We use a sequence (Ui) of i.i.d. random variables uniformly distributed R n n n n on [0, 1] to define a random permutation (Jk )k=1 of (k/n)k=1. Let J1 be the right end of the interval ((k 1)/n, k/n] which includes U . For l> 1, define J n as the − 1 l right end of the shortest interval of the form (j/n,k/n] and containing Ul, with n n n k/n = Jj for j < l. It can be shown that (Jk )k=1 is uniformly distributed over 6 n all possible permutations of (k/n)k=1. Thus, we have the distributional equality

[nt] n d Xi = δi X 1 n , | |(i,n) {Ji ≤t} Xi=1 Xi=1 where ( X ) are order statistics of X and δ ’s are 1’s (see [17] for details). | |(i,n) | i| i ± Geometric stable laws 249

The above equality also produces

[Npt] Np d Xi = δi X (i,Np)1 Np . | | {Ji ≤t} Xi=1 Xi=1 The random vector ( X ) has the well-known representation in distri- | |(i,n) bution: d ( X ,..., X ) = (G−1(Γ /Γ ),...,G−1(Γ /Γ )). | |(1,n) | |(n,n) 1 n+1 n n+1 Consequently, assuming mutual independence of (Γn), (δn), (Un), and U, we can deﬁne a process S˜p as follows

Np ˜ −1 Sp(t)= δiG (Γi/ΓNp+1)1 Np . {Ji ≤t} Xi=1 The process S˜p has identical distribution in D[0, 1] as the randomized partial sum process Sp. We show below that the normalized S˜p is almost surely convergent, and derive almost sure series representation for the limiting process. We assume (14), where Np is geometric with mean 1/p. −1 Theorem 3.2. With the above notation, denoting cp = G (p) and W = ln(U), we have − ∞ d ˜ a.s 1/α 1/α lim Sp/cp = lim Sp/cp = W δi/Γ 1[U ,1] p→0 p→0 i i Xi=1 in the Skorohod metric in D[0, 1]. Moreover, if a real function v on [0, 1] has bounded variation, then

1 1 ∞ d ˜ a.s 1/α 1/α lim vdSp/cp = lim vdSp/cp = W δi/Γi v(Ui). p→0 0 p→0 0 Z Z Xi=1 Proof. Since N = G−1(U) and pG−1(u) converges to ln u as p 0, p p p − → Np diverges almost surely to inﬁnity, and with probability one

(19) lim pNp = W. p→0 Moreover, Np −1 δ G (Γ /Γ )1 Np S˜ CN i=1 i i Np+1 {J ≤t} p = p i , cp cp P CNp 250 Tomasz J. Kozubowski, Krzysztof Podg´orski

−1 where CNp = G (1/Np). The function G−1 is regularly varying at zero with index 1/α. Thus, by − (19), CN lim p = W 1/α. p→0 cp Since N , it follows from the corresponding result for the stable case given p →∞ in [17] that, with probability one,

N p −1 ∞ i=1 δiG (Γi/ΓNp+1)1 Np {Ji ≤t} 1/α lim = δi/Γi 1[Ui,1] p→0 P CNp Xi=1 in the Skorohod metric in D[0, 1]. This establishes the ﬁrst part of the theorem. The second part follows by almost identical arguments.

4. Strong invariance principles. Let (Xi) be a sequence of symmetric, i.i.d. random variables from the normal domain of attraction of a stable law. It is well-known that X1 satisﬁes the following tail condition (20) G(x)= cx−α + β(x)x−α, or, equivalently, (21) G−1(y) =cy ˜ −1/α + γ(y)y−1/α, where β(x) 0 as x and γ(y) 0 as y 0. Without loss of generality, → → ∞ → → we may assume thatc ˜ = 1. n n We shall exploit the representation (Xi,n)i=1 of (Xi)i=1 given in (2) to n n Np derive an almost sure approximation of Xi,n by Yi,n, as well as Xi,Np i=1 i=1 i=1 Np P P P by Yi,Np , where (Yi,n) is a sequence of symmetric, i.i.d. α-stable (respectively, i=1 GS)P random variables. Using (2) and the deﬁnitions of (Γ1,j), (Γ2,j), . . ., and (δ1,j), (δ2,j), . . . given in Section 3 we have n n n d −1 Xi = Xi,n = δiG (Γi/Γn+1) Xi=1 Xi=1 Xi=1 and n n ∞ def X =d Y = δ Γ−1/α , i,n  i,j i,j  Xi=1 Xi=1 Xj=1   Geometric stable laws 251

∞ 1/α −1/α where the stable r.v. X = n δjΓj . The following theorem holds in this j=1 setting. P Theorem 4.1. Let M be a sequence of integers such that M and n n →∞ M /n 0 as n . Then, with probability one, n → →∞ n n X Y = i,n − i,n i=1 i=1 X XM n √log log M log log n = O(n1/α) δ Γ−1/αγ(Γ /Γ )+ n + . i i i n+1 1/α−1/2 n " Mn r # Xi=1

Remarks.

1. The exact rate of convergence in the above approximation depends on the rate of convergence of γ at the origin: the faster γ tends to zero, the faster Mn can tend to inﬁnity, increasing the rate of convergence of 1/2−1/α Mn √log log Mn to zero.

2. Regardless of the rate of convergence of γ, we obtain almost sure convergence with the rate at least o(n1/α). This rate could not always be obtained without additional assumptions, when another form of strong invariance n n principle for Xi Yi was considered in [2] (see examples therein). i=1 − i=1 P P For the random summation, we have the following representation (see Theorem 3.1), ∗ ∗ Np Np Np d ∗ −1 ∗ Xi = Xi,Np = δiG (Γi/ΓNp +1). Xi=1 Xi=1 Xi=1 ∞ 1/α By Proposition 3.1 of Section 3, a GS r.v. Y = δi(W/Γi) has the i=1 representation P

Np Np ∞ d def Y = p1/α Y = p1/α δ (W /Γ )1/α i,Np  i,j i i,j  Xi=1 Xi=1 Xj=1   (see the deﬁnitions in Section 3). 252 Tomasz J. Kozubowski, Krzysztof Podg´orski

Theorem 4.2. Let the assumptions of Theorems 4.1 and 3.1 hold. If M and pM 0 as p 0, then, with probability one, p →∞ p → → ∗ Np Np X ∗ Y = i,Np − i,Np Xi=1 Xi=1 Mp −1/α −1/α log log Mp log log(1/p) = O(p ) Γ γ(Γ /ΓN ∗+1)+ + .  k k p 1/α−1/2 (1/p)1/2  k=1 pMp p X   We prove Theorem 4.1. Similar proof of Theorem 4.2, based on Theo- rem 3.1, is omitted. P r o o f o f T h e o r e m 4.1. It follows directly from the deﬁnition of (Xi,n) and (Yi,n) that n n Xi,n Yi,n − ≤ i=1 i=1 X X n n ∞ −1 1/α −1/α 1/α −1/α δiG (Γi/Γn+1) n δiΓi + n δiΓi ≤ − i=1 i=1 i=n+1 X X X I + I + I , ≤ 1,n 2,n 3,n where I = n δ G−1(Γ /Γ ) (Γ /Γ )−1/α , 1,n i=1 i i n+1 − i n+1 P 1/α n −1/α ∞ −1/α I = Γ n1/α δ Γ , I = n1/α δ Γ . 2,n n+1 − i=1 i i 3,n i=n+1 i i

First, by the Law P of Iterated Logarithm (LIL) and P integration by parts formula,

∞ I O(1) n log log n + x−1/α−1/2 log log xdx 3,n ≤ ZΓn+1 ! p√log log n p = O(n1/α) . n1/α−1/2

Then, again by LIL,

1/α log log n I2,n O(n ) . ≤ r n Geometric stable laws 253

Equation (21) and the discrete version of integration by parts formula produce

Mn 1/α −1/α I1,n = O(n ) δiΓi γ(Γi/Γn+1) " Xi=1 n Mn i=1 δi i=1 δi +γ(Γn/Γn+1) + γ(ΓM /Γn+1) n1/α n 1/α P PMn N i−1 N i−1 −1/α + n δk∆i + δk∆˜ i , !# i=XMn+1 Xk=1 i=XMn+1 Xk=1 −1/α −1 where ∆i and ∆˜ i are increments of x and G taken at Γi/Γn+1, i = 1,...,n+ 1. The approximation (from above) of the last two sums, coupled with the LIL, yields

M n √log log M I = O(n1/α) δ Γ−1/αγ(Γ /Γ )+ n 1,n i i i n+1 1/α−1/2 " Mn Xi=1 1 +n1/2−1/α x log log(2nx)d(x−1/α) ΓMn /Γn+1 Z p 1 + x log log(2nx)dG−1(x) . ΓMn /Γn+1 !# Z p 1 1/α −1 Denote H(u) = u t dG (t) and apply integration by parts formula, to obtain R 1 x log log(2nx)dG−1(x) ΓM /Γn+1 Z n p H(1) log log(2n)+ H(ΓMn /Γn +1) log log(ΓMn n/Γn+1) ≤ 1 1 1 1 + O(1)p + p + H(x)dx log log(2nx) log(2nx) x ZΓMn /Γn+1 p √log log Mn O(1) log log n + n−(1/2−1/α) + x−1H(x)dx . ≤ 1/α−1/2 " Mn ΓMn /Γn+1 # p Z 1/2−1/α The last integral is of order O((Mn/n) ), since the function 1 u x−1H(x)dx is regularly varying at zero with index 1/α 1/2. The same 7→ u − R 254 Tomasz J. Kozubowski, Krzysztof Podg´orski arguments produce

x log log(2nx)d(x−1/α) ΓMn /Γn+1 ≤ Z p √log log M O(1) log log n + n−(1/2−1/α) n . 1/α−1/2 ≤ " Mn # p Thus, M n √log log M I = O(n1/α) δ Γ−1/αγ(Γ /Γ )+ n , 1,n i i i n+1 1/α−1/2 " Mn # Xi=1 which concludes the proof.

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Tomasz J. Kozubowski Department of Mathematics The University of Tennessee at Chattanooga Chattanooga, TN 37403, USA e-mail: [email protected]

Krzysztof Podg´orski Department of Mathematical Sciences Indiana Univ.-Purdue Univ. Indianapolis Indianapolis, IN 46202, USA e-mail: [email protected] Received May 23, 1999