A Note on the Linnik Distributions

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.

217, 701]706 Ž1998.

AY975736

NOTE A Note on the Linnik Distributions Gwo Dong Lin Institute of Statistical Science, Academia Sinica, Taipei, 11529, Taiwan, Republic of China Submitted by N. H. Bingham Received March 28, 1997

We prove that Linnik distributions are geometrically infinitely divisible, and clarify a characterization theorem for Linnik distributions concerning the stability of geometric summation. An explicit expression for absolute moments of Linnik distributions is also given. Q 1998 Academic Press

1. INTRODUCTION In 1953, Linnik established a distribution Fa with characteristic function Žch.f.. fa Ž t . s Ž1 q < t < a .y1 , t g R ' Žy`, `., where a g Ž0, 2x. Recently, this distribution Fa was named after Linnik and attracted a great deal of attention. For example, using the Linnik distribution Anderson and Arnold w3x introduced some stable processes as models for temporal stock prices. Kotz, Ostrovskii, and Hayfavi w10x investigated the density function of Fa . Lin w11x and Alamatsaz w1x proved independently the self-decomposibility of Fa by different approaches. The existence of finite moments of Fa was discussed by Lin w11x and Anderson w2x as well as Anderson and Arnold w3x. In the next section we prove that each Fa is geometrically infinitely divisible Žg.i.d.. ŽTheorem 1., and clarify a characterization theorem for Linnik distributions concerning the stability of geometric summation ŽTheorems 2 and 3.. The absolute moments of Fa and of its extension are given explicitly ŽTheorems 4 and 5..

701 0022-247Xr98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.

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NOTE

2. MAIN RESULTS We first consider the g.i.d. property of Fa . A random variable X Žor its ch.f.rdistribution. is said to be g.i.d. if for any p g Ž0, 1. there is a sequence of independent identically distributed Ži.i.d.. random variables Z p, n , n s 1, 2, . . . , such that Np

Xs

d

Ý Zp , n ,

Ž 1.

ns1

where Np , independent of  Z p, n4`ns1 , is a geometric random variable having mass function P  Np s n4 s p Ž 1 y p .

ny 1

,

n s 1, 2, . . . ,

Ž 2.

and the notation sd expresses the equality in the sense of distributions Žsee, e.g., Klebanov, Maniya, and Melamed w9x, henceforth referred to as KMM.. The right-hand side of Ž1. is called a geometric summation. Also recall that the family of g.i.d. distributions is a proper subclass of the family of infinitely divisible Ži.d.. distributions Žsee, e.g., Pillai and Sandhya w15x.. Hereafter, denote by Xa a random variable obeying the Linnik distribution Fa . Any scale transform of Fa is also named after Linnik. THEOREM 1.

For each a g Ž0, 2x, the Linnik distribution Fa is g.i.d.

Proof. Recall that a distribution F is g.i.d. if and only if its ch.f. f satisfies the condition that the function f Ž t . s exp 1 y 1rf Ž t .4 , t g R, is an i.d. ch.f. ŽKMM w9x.. For each a g Ž0, 2x, we have fa Ž t . s exp 1 y 1rfa Ž t .4 s expŽy< t < a ., t g R. The latter is the ch.f. of a symmetric stable distribution with exponent a , and hence i.d. Therefore Fa is g.i.d. It is known that the family of Linnik distributions is closed under geometric summation Žsee Anderson and Arnold w3x, Lin w11x as well as Kakosyan, Klebanov, and Melamed w8x.. Specifically, let Xa , n , n s 1, 2, . . . , be a sequence of i.i.d. random variables having distribution Fa , and let Np , independent of  Xa , n4 , be a geometric random variable defined by Ž2.. Then the following stability identity is valid: Np

Xa s p d

1r a

Ý

Xa , n

for each p g Ž 0, 1 . .

Ž 3.

ns1

It is natural to ask the question whether the property Ž3. characterizes the Linnik distribution Fa . The answer is in general negative due to the following ramification of KMM w9, Theorem 3x.

703

NOTE

THEOREM 2. Let a g Ž0, 2x and let X, X n , n s 1, 2, . . . , be i.i.d. nondegenerate random ¨ ariables. Assume further that Np , independent of  X n4 , is a geometric random ¨ ariable defined by Ž2.. Then the stability condition Np

Xs p d

1r a

for each p g Ž 0, 1 .

Xn

Ý

Ž 4.

ns1

holds if and only if X has a ch. f. of the form f Ž t . s 1 q l < t < aexp yi

½

ž

p 2

ua sgn t

y1

/5

,

t g R,

where the constants l ) 0 and < u < F ua s min 1, 2ra y 14 . In the above theorem if we assume further that X obeys a symmetric distribution, then the stability condition Ž4. does become a characterization of the Linnik distribution. In fact we have the following result on slightly greater generality, in which it suffices to consider two values p1 , p 2 of p with the quotient Žlog p1 .rŽlog p 2 . irrational. THEOREM 3. Let a , X,  X n4 , and Np be defined as in Theorem 2. Assume further that the distribution of X is symmetric at 0 and that p1 , p 2 are two ¨ alues of p with quotient Žlog p1 .rŽlog p 2 . irrational. Then the stability condition Np

X sd p1r a

Ý

Xn

for p s p1 , p 2

Ž 5.

ns1

holds if and only if X obeys a Linnik distribution F Ž x . s Fa Ž xrl., x g R, for some constant l ) 0. Proof. Apply KMM w9, Theorem 4x or Lin w11, Theorem 3x. For the case a s 2, the Linnik distribution Fa reduces to the Laplace distribution, and hence the absolute moment EŽ< X 2 < d . equals G Ž1 q d . or ` according to whether d ) y1 or d F y1. In the next theorem we carry out the absolute moments of Fa for a g Ž0, 2.. To simplify the representation, we have applied the identity G Ž1 y s . G Ž1 q s . s sprsinŽ sp . for s g Ž0, 1. Žsee, e.g., Widder w17, p. 411x.. THEOREM 4.

Let a g Ž0, 2. and Xa obey the Linnik distribution Fa . Then

Ži. EŽ< Xa < d . s  2 d G ŽŽ1 q d .r2. G Ž1 y dra . G Ž1 q dra .4 r G Ž1r2. G Ž1 y dr2.4 s  2 dd'p G ŽŽ1 q d .r2.4 r a G Ž1 y dr2.sinŽ dpra .4 if d g Žy1, a . l Žya , a . ' Ia ; Žii. EŽ< Xa < d . s ` if d g R y Ia Ž the complement of Ia .. To prove Theorem 4 we need the following lemma.

704

NOTE

LEMMA. Let a g Ž0, 2. and Ya be a random ¨ ariable obeying the symmetric stable distribution with exponent a , namely Ya has a ch. f. fa Ž t . s expŽy< t < a ., t g R. Then Ži. EŽ< Ya < d . s  2 d G ŽŽ1 q d .r2. G Ž1 y dra .4 r G Ž1r2. G Ž1 y dr2.4 if d g Žy1, a .; Žii. EŽ< Ya < d . s ` if d g R y Žy1, a .. Proof. The first part is due to Shanbhag and Sreehari w16, Theorem 3x. The second part can be proved by the Monotone Convergence Theorem and by the fact that EŽ< Ya < d . ª ` as d ­ a or x y1. The proof is complete. Proof of Theorem 4. Let Ya be a random variable obeying the symmetric stable distribution with exponent a Ždefined as in the Lemma above., and let Z be a standard exponential random variable independent of Ya . Then Xa and Ya Z 1r a have the same ch.f., and hence are equally distributed Žsee, e.g., Feller w7, p. 596x or Devroye w6x.. This implies that EŽ< Xa < d . s EŽ< Ya < d . EŽ Z d r a . for d g R. In view of the fact that EŽ Z d r a . s G Ž1 q dra . or ` according to whether d ) ya or d F ya , we obtain the desired result by the Lemma above. The proof is complete. Remark 1. There is an alternative approach for carrying out the moments EŽ< Xa < d ., d g Ž0, a .. Recall that for a random variable X with ch.f. f, we have EŽ < X < d . s

GŽ1 q d .

ž

p

sin

dp 2

/

`

Hy`

1 y Re Ž f Ž t . . < t < dq1

for d g Ž 0, 2 . ,

dt

in which ReŽ f Ž t .. denotes the real part of f Ž t . Žsee, e.g., Chung w5, p. 159x.. From this formula it follows that E Ž < Xa < d . s

2

p

G Ž 1 q d . sin

ž

dp 2

`

/H

0

1 y fa Ž t . t dq1

dt

2 sin Ž dpr2 .

G Ž 1 q d . G Ž dra . G Ž 1 y dra . . ap Therefore, for d g Ž0, a ., Theorem 4Ži. is valid by the identity s

GŽ x. s

2 xy1

'p

G Ž xr2. G Ž Ž 1 q x . r2 . ,

x)0

Žsee, e.g., Apostol w4, p. 341x.. Finally, we consider an extension of the Linnik distribution. For each a g Ž0, 2x, the ch.f. fa is i.d.; this implies that for each s ) 0 the function fa , s Ž t . s Ž fa Ž t .. s, t g R, is a bona fide ch.f. Denote by Xa Ž s . a random

705

NOTE

variable with ch.f. fa , s . We now investigate the absolute moments of Xa Ž s ., s ) 0. For the special case a s 2, we have EŽ< X 2 Ž s .< d . s  G Ž1 q d . G Ž s q dr2.4 r G Ž1 q dr2. G Ž s .4 or ` according to whether d ) max y1, y2 s4 or d F max y1, y2 s4 . The remaining results for a g Ž0, 2. are given below. THEOREM 5.

Let a g Ž0, 2. and s ) 0. Then

Ži. EŽ< Xa Ž s .< d . s  2 d G ŽŽ1 q d .r2. G Ž1 y dra . G Ž s q dra .4 r  G Ž1r2. G Ž1 y dr2. G Ž s .4 if d g Žy1, a . l Žya s, a . ' Ia , s ; Žii. EŽ< Xa Ž s .< d . s ` if d g R y Ia , s . Proof. Let Ya be a random variable obeying the symmetric stable distribution with exponent a , and let Z s , independent of Ya , be a gamma random variable having density function gsŽ z . s

1 GŽ s.

z sy1 eyz ,

z ) 0.

Then Xa Ž s . sd Ya Z s1r a and hence EŽ< Xa Ž s .< d . s EŽ< Ya < d . EŽ Z sd r a . for d g R. In view of the fact that EŽ Z sd r a . equals G Ž s q dra .rG Ž s . or ` according to whether s q dra ) 0 or s q dra F 0, we obtain the desired result by the Lemma above. This completes the proof. Remark 2. Assume in Theorem 3 that a g Ž0, 1x and that X is positive instead of obeying a symmetric distribution. Then the stability condition Ž5. holds if and only if X has a Mittag]Leffler distribution with Laplace transform fa Ž s . s Ž1 q l s a .y1 , s ) 0, for some constant l ) 0. The Mittag]Leffler distribution has been investigated by Pillai w14x and by Lin w12x.

REFERENCES 1. M. H. Alamatsaz, A note on an article by Artikis, Acta Math. Hungar. 45 Ž1985., 159]162. 2. D. N. Anderson, A multivariate Linnik distributions, Statist. Probab. Lett. 14 Ž1992., 333]336. 3. D. N. Anderson and B. C. Arnold, Linnik distributions and processes, J. Appl. Probab. 30 Ž1993., 330]340. 4. T. M. Apostol, ‘‘Mathematical Analysis,’’ 2nd ed., Addison]Wesley, Reading, MA, 1974. 5. K. L. Chung, ‘‘A Course in Probability Theory,’’ Academic Press, New York, 1968. 6. L. Devroye, A note on Linnik’s distribution, Statist. Probab. Lett. 9 Ž1990., 305]306. 7. W. Feller, ‘‘An Introduction to Probability Theory and Its Applications,’’ Vol. II, 2nd ed., Wiley, New York, 1970. 8. A. V. Kakosyan, L. B. Klebanov, and J. A. Melamed, ‘‘Characterization of Distributions by the Method of Intensively Monotone Operators,’’ Lecture Notes in Mathematics, Vol. 1088, Springer-Verlag, New York, 1984.

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9. L. B. Klebanov, G. M. Maniya, and I. A. Melamed, A problem of Zolotarev and analogs of infinitely divisible and stable distributions in a scheme for summing a random number of random variables, Theory Probab. Appl. 4 Ž1984., 791]794. 10. S. Kotz, I. V. Ostrovskii, and A. Hayfavi, Analytic and asymptotic properties of Linnik’s probability densities, I, J. Math. Anal. Appl. 193 Ž1995., 353]371; II, 497]521. 11. G. D. Lin, Characterizations of the Laplace and related distributions via geometric compound, Sankhya Ser. A 56 Ž1994., 1]9. 12. G. D. Lin, On the Mittag]Leffler distributions, submitted for publication. 13. Yu. V. Linnik, Linear forms and statistical criteria, I, II, in, Selected Translations Math. Statist. and Probab., Vol. 3, pp. 1]90, Amer. Math. Soc., Providence, RI, 1963; original paper, Ukrain. Mat. Zh. 5 Ž1953., 207]290. 14. R. N. Pillai, On Mittag]Leffler functions and related distributions, Ann. Inst. Statist. Math. 42 Ž1990., 157]161. 15. R. N. Pillai and E. Sandhya, Distributions with complete monotone derivative and geometric infinite divisibility, Ad¨ . Appl. Probab. 22 Ž1990., 751]754. 16. D. N. Shanbhag and M. Sreehari, On certain self-decomposable distributions, Z. Wahrsch. Verw. Gebiete 38 Ž1977., 217]222. 17. D. V. Widder, ‘‘Advanced Calculus,’’ 2nd ed., Prentice Hall, Englewood Cliffs, NJ, 1961.