A mixture representation of the Linnik distribution

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¢ w ' . - " < " r

E L S E V I E R Statistics & Probability Letters 26 (1996) 61 64

STATISTICS&

PROBABILITY

LETTERS

A mixture representation of the Linnik distribution

S a m u e l K o t z a, I . V . O s t r o v s k i i b ' *

a Universi~ of Maryland at College Park, Department of Management Science and Statistics, College Park, MD 20742, USA bBilkent Universi~ at Ankara, Department of Mathematics, 06533 Bilkent, Ankara, Turkey, and Institute for Low Temperature Physics and

Engineering at Kharkov, 310164 Kharkov, Ukraine Received April 1994; revised October 1994

Abstract

Linnik distribution with the characteristic function

~o,(tl = 1/(1 + Itl=), 0 < ~ < 2,

is shown to possess the following property.

Let X , , X p be random variables possessing the Linnik distribution with parameters ~ and (0 < ~ < fl ~< 2). Denote by Y~ an independent of X~ non-negative random variable with the density

,q(s;~,fl) = sin 1 + s 2~ + 2s c o s ~ Then

X, -'- X e Y~p,

fl respectively

where - denotes the equality in the sense of distributions.

Infinite divisibility of mixtures of Linnik distributions with respect to the parameter ~ and scale is obtained as a corollary.

A M S 1980 Subject Classification." Primary 62H05, 60El0; Secondary 33A40

Keywords: Characteristic function; Mixture; Infinite divisibility; Contour integration

1. Introduction and statement o f the theorem

Recently, the L i n n i k d i s t r i b u t i o n - o r i g i n a l l y i n t r o d u c e d b y Ju.V. L i n n i k in 1953 (Linnik, 1963) - h a s a t t r a c t e d a t t e n t i o n o f a n u m b e r o f r e s e a r c h e r s (see e.g. A r n o l d , 1973; D e v r o y e , 1986, 1990; A n d e r s o n , 1992;

* Corresponding author.

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62 S. Kotz, I.V. Ostrovskii/Statistics & Probability Letters 26 (1996) 61-64

Anderson et al., 1993; Devroye, 1993). Although the characteristic function of this distribution is of a simple form, a general expression of the distribution is not easily attainable. 1 In this connection, any properties of the distribution such as a mixture representation which facilitates generation of Linnik random variables ought to be of interest. One such property is proved in the present note.

Recall that the generic definition of a Linnik random variable is given in terms of the characteristic function ~ p ~ ( t ) = l / ( l + l t l ~ ) , 0 < ~ < 2 .

We shall denote the corresponding density by p,(x). This density can be viewed as a generalization of the well-known Laplace (double exponential) density p2(x) = c-Ixl/2 for the case c~ = 2 (see, e.g. Johnson and Kotz, 1970). The main result of the paper is the following theorem.

Theorem. For any 0 < ~ < fl <~ 2, the following equality is valid ~p~(t) = r ~ ~ ~p~(t/s)g(s;~,fl)ds, - 0o < t < oc,, ~u where 9 ( s ; ~ , f l ) = ( ~ s i n ~ t ~ ) s~-I . l + s z~ + 2s ~cos (1)

Noting that the equality (1) is equivalent to the following

p,(x)=ffp~(sx)g(s;~,B)ds, - ~

< x < ~ ,

and taking into account that g(s; ~, fl) is a genuine density function, we arrive at the representation of the form X~ - X~ Yap,

as stipulated in the abstract. This representation allows us to generate Linnik variables of different parameters starting from convenient base, e.g. from the Laplace distribution corresponding to fl = 2. That is, our theorem yields immediately the following corollary.

Corollary 1. For any ~ ~ (0, 2), Linnik distribution with characteristic function q~,(t) is a scale mixture of

Laplace distributions with characteristic functions ~02(t/s) = s2/(s 2 + t2), 0 < s < ~ .

By Steutel's theorem (Steutel, 1970), any scale mixture of Laplace distributions is infinitely divisible. Therefore, Corollary 1 yields infinite divisibility of Linnik distributions. This fact is not new and was proved in Devroye (1990). However, noting that any mixture of scale mixtures of Laplace distributions is again a scale mixture of Laplace distributions, we obtain a stronger result: any mixture of Linnik distributions is infinitely divisible. More precisely, the following result is valid.

Corollary

2. Let P be a probability measure on the half-strip S = {(~,s): 0 < ~ ~< 2, 0 ~< s < ~ } . Then the distribution with the characteristic function

~o(t) = f s (o~(st)P(d~ ds) is infinitely divisible.

1 See Hayfavi, A., S. Kotz and I.V. Ostrovskii (1994), Analytic and asymptotic properties of Linnik's probability densities, C,R. Acad. Sci. Paris, S6rie I, 319, 985 990.

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S. Kotz, L V. Ostrovskii / Statistics & ProbabiBty Letters 26 (1996) 61-64 63

2. Proof of the theorem

Note, that the equality (1) is equivalent to the following one:

1 - ~ t ° - 1 f ~ s 7~-~+tpO(s;~,[l)ds, ~ 0 < ~ < f l ~ < 2 , t>~O.

T o prove the latter, we denote

__

/I;

f:

S'

.

= f:

S/~+°- lds

O /tO¢

I flsin-~ s-g--~+t ~9(s'ct'fl)ds (s p + t~)(1 + s 2° + 2s cos~-) It is sufficient to establish that

rc 1

I = - -

f l s i n ~ 1 + t °

Transforming the integral I by means of the change of variables z = t p s -~, we have

z(°/tJ)-'dz

I = ~ (1 + z)(z 2°/p + t E° + 2z°/#t°cos~)

Utilizing the identity

we have

1 { f ~ z('/')- t dz I

2 i f l s i n ~ (1 + z)(z °/~ + t°e -i~/~) Consider the function

1 q(z) = (1 + z)(z °/~ + t°ei~/P) ' (2)

I]

}

- (1 + z)(T °/# + t°e i'°/e) ' (3)

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in the complex z-plane cut along the positive ray. Define the branch of z °/~ in (4) by the conditions

z ~/~ = r "/p e i~/l~, z = re i~°, 0 < (p < 2re.

Since

when 0 < ~o < 2re, we have

zo/; + t • em/B = ei~/~(rO/~ + t, eiO/p(~- o)) 4= O.

Therefore the function q(z) is analytic in the cut out plane. D e n o t e by GR,~ the simply connected region

GR,,:

= {z: ~ < Izl < R}\{z: s < z < R}, 0 < e < R ( > 1)

and denote by t?Gg.~ its b o u n d a r y traversed in the direction which leaves GR., from the left (the line interval {z: e < z < R } is being traversed twice in the opposite directions). By the C a u c h y Residue Theorem, we have

~,~ 2~i

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64 S. Kotz, I.V. Ostrovskii / Statistics & Probability Letters 26 (1996) 61-64

Taking the limit of the last integral as R ~ oo, e, ~ 0, we arrive at

fo

v(,/p)- 1 dr ~]

f ;

~'~/~'-ldz

_ 2rri

(1 + z)(z "/¢ +

t'e ~'/~) -

e2ni[(~///) (1 + "c)(z'~/Pe 2ni~//~ -+-

t'e i~'/~)

1 + t ~"

It is evident that this equality can be rewritten in the form

f ; r~'/~)-ldr

_ f ;

rl'/')- 1 d~ _ 2~zi

(1 + z)(v "/a + t'e i~'/e) (1 + z)(z "/p + t'e -i~/~) 1 + t "

Thus, the difference of the integrals appearing in the braces of (3) has been calculated to be equal to 2hi/(1 + t'). Substituting this value into (3), we arrive at (2). The t h e o r e m is thus proved.

R e f e r e n c e s

Anderson, D.N. (1992), A multivariate Linnik distribution, Statist. Probab. Lett. 14, 333-336.

Anderson, D.N. and B.C. Arnold (1993), Linnik distributions and processes, J. Appl. Probab. 30, 330-340.

Arnold, B.C. (1973), Some characterizations of the exponential distribution by geometric compounding, SIAM J. Appl. Math. 24, 24~244.

Devroye, L. (1986), Non-Uniform Random Variable Generation (Springer, New York). Devroye, L. (1990), A note on Linnik's distribution, Statist. Probab. Lett. 9, 305-306.

Devroye, L. (1993), A triptych of discrete distributions related to stable law, Statist. Probab. Lett. 18, 349 351.

Johnson, N.L. and S. Kotz (1970), Distributions in Statistics, Continuous Univariate Distributions, II (J. Wiley, New York).

Linnik, Ju.V. (1963), Linear forms and statistical criteria, I, II, Selected Translations in Math. Statist. Probab. 3, 140. (Original paper appeared in: Ukrainskii Mat. Zhournal 5 (1953) 207-290.)

Anderson, D.N. and B.C. Arnold (1993), Linnik distributions and processes, J. Appl. Probab. 30, 330-340.

Steutel, F.W. (1970), Preservation of Infinite Divisibility under Mixino and Related Topics, Vol. 33 (Mathematical Center Tracts, Mathematisch Centrum, Amsterdam).