Analytic and asymptotic properties of Linnik's probability densities

Probabilites/ Probability Theory

Analytic and asymptotic properties

of Linnik's probability densities

Azize HAYFAVI, Samuel KoTZ and Iosif Vlarimirovitch OsrnovsKII

Abstract - The analytic and asymptotic properties of the probability density Jin (:r:) introduced in 1953 by Ju. V. Linnik and defined by the characteristic function 1/(1

+

It I"), 0 < n < 2, are studied. Expansions of Jin (:r:) into convergent and asymptotic series in terms of log

I

:r:

I , I

x

I

k n,

I

,r

I

k

(k = 0, 1, 2, ... ) are obtained. It turns out that the analytic structure of Pc, (:r:) depends substantially on the arithmetical nature of the parameter n.

Les proprietes analytiques et asymptotiques des densites de probabilites de Linnik

Resume - Nous etudions /es proprietes analytiques et asymptotiques des densites de probabilites f!a (:r:) introduites par Ju. V. Linnik en 1953 et definies a ['aide de leur fonction caracteristique, 1/ ( 1

+

It I"'), 0 < a < 2. Nous ob tenons des developpements en series convergentes et asymptotiques de termes log

I

:r:

I, I

:r;

I

k O ,

I

x

I

k, k = 0, 1, 2, ... La structure analytique des Jlc, ( :r;) depend de far,;on substantielle de la nature arithmetique du parametre n.

Version franraise abregee - En 1953, Ju. V. Linnik [1] a demontre que la fonction

1Pa (t)

=

1/(1

+

It I"), 0

<

a<

2, est la fonction caracteristique d'une densite de probabilite. Cette densite, notee p" (x), peut etre vue comme une generalisation de la densite de Laplace

p2 (x)

=

e-1 x I /2 qui correspond au cas a

=

2. Recemment, Jes densites de Linnik Pa (x) ont connu un regain d'interet et des travaux ([2]-[7]) ont explore leurs proprietes probabilistes. Comme dans le cas des densites du type stable, la simplicite analytique des fonctions caracteristiques ne se transmet pas aux densites et nous ne connaissons pas de formules explicites pour !es densites de Linnik. Le comportement asymptotique

a

l'infini des Pa (x) et des densites stables presente des similitudes. Cependant la structure analytique des Pa ( x) differe de celle des densites stables. Elles dependent de fas;on substantielle de la nature arithmetique du parametre a.

Le theoreme suivant decrit le comportement asymptotique

a

l'infini des Pa ( X).

THEOREME 1. - Pour tout a E (0, 2), le comportement asymptotique d l'infini de Pa (x) est dunne par la serie (2). De fafon plus precise, pour tout N

=

l, 2, ... , la formule (3) est vraie

Ult RN,a (x) verifie la condition (4).

II s'ensuit done que Pa (x) a un taux de decroissance polynomiale

a

l'infini et de fas;on plus precise, la formule ( 5) est vraie.

Nous formulons maintenant des theoremes lies

a

la structure analytique des Po: (x).

THEOREME 2. - Pour tout a E (0, 2), la densite pO! (x) est une fonction completement monotone de :z:

>

0.

Puisque Pa ( x) est une fonction paire, elle est done absolument monotone pour x

<

0. Notons par Q0_dd !'ensemble des a rationnels representables sous la forme a

=

m/n ou m et n sont des entiers avec de plus m impair. Notons L !'ensemble des nombres transcendants de Liouville. Le theoreme suivant decrit la structure analytique des Pa (x) pour presque tous

Note presentee par Jean-Pierre KAHANE.

Jes a (pour la mesure de Lebesgue), bien que !'ensemble exceptionnel soit un ensemble partout dense dans (0, 2) qui de plus a la puissance du continu.

THEOREME 3. - Pour tout a E (0, 2)\{Qodd UL}, la representation (6) est vraie, 01i An (z) et N°' (z) sont desfonctions entieres d'ordrefini respectivement l/a et 112 donnees par /cs/im1111/es

(7) et (8).

Quand a E Qodd la structure analytique des p°' (x) est plus compliquee. Notons Q1 _le sous-ensemble de Q0dd forme des a = l/n, n = l, 2, ...

THEOREME 4. - Si a E Qodd\Q1, la representation (9) est vraie, ou M°' (z) et Nn (z) sont des fonctions entieres d'ordrefini respectifl/a, 1 et 1/2. De plus si a= m/n ou met n son/ premiers entre eux, ces fonctions peuvent etre decrites d !'aide des formules (10)-(13). Si a

=

1/n E Q1 ,

a/ors la representation ( 14) est vraie.

Les cas particuliers sont donnes par (15) et (16).

Pour a E L nous n'avons pu trouver une representation de p°' ( x) similaire

a

celle des theoremes 3 et 4. Cependant, nous sommes en mesure de demontrer qu'il existe des Yaleurs de a pour lesquelles Jes theoremes 3 et 4 ne sont pas vrais. De plus, !'ensemble de tels er est partout dense et a la puissance du continu. Neanmoins, le comportement asymptotique de Pn (x)

a

l'origine peut etre decrit

a

!'aide de la formule (18) pour tout a

rt.

Q0_{clcl et} N

=

l, 2, ... (la

premiere somme est nulle si a

>

1). Des exemples de formules asymptotiques pour p()/ (x),

a E Qodd, x --. 0, sont donnes par (19)-(21) ou "/

=

0, 5772157 ... est la constante d'Euler.

1. INTRODUCTION. - In 1953, Ju. V. Linnik proved [1] that the function (1) 'POI

(t)

=

1/(1

+

It I°'),

0

<a<

2,

is the characteristic function of a probability density. We shall denote this density by p°' ( x ). It can be viewed as a generalization of the well-known Laplace density p2 (x)

=

e-1 '" I /2,

which corresponds to the case a

=

2. Recently, Linnik's probability densities p°' (x)

attracted attention of a number of researchers who discovered some interesting probabilistic properties and applications (see, e.g. [2]-[7]). In this connection, the study of analytic and asymptotic properties of p°' ( x) seems to be of interest. Moreover, the study of the structure of these densities is of importance in connection with classification of continuous long-tailed distributions. This Note is devoted to such a study.

As for stable densities, analytic simplicity of characteristic functions (1) is not inherited by the corresponding densities and these densities elude closed form representations. The asymptotic behaviour of p°' ( x) at oo is similar to some extent to those of stable densities. However, the analytic structure of p°' ( x) is quite different from those of stable ones. It depends substantially on the arithmetic nature of the parameter a.

2. ASYMPTOTIC BEHAVIOUR OF p°' (x) AT 00.

THEOREM 2.1. - For any a E (0, 2), the asymptotic behaviour of p°' (x) at x can be described by the following asymptotic (divergent) series

(2) POI (x) "-'

~

f {

(-1l-l

r

(1

+

O! k) sin 7r~

k} IX

1-

1

-°'\

k=l

More precisely, for any a E (0, 2) and N

=

1, 2, 3, ... , the following formula is valid

(3)

pc,(x)=;

1

I:{(-1)

N k-1 f(l+ak)sin-7f(¥

2-

k }

lxl-l-c,k+RN,c,(x),

k=l

where

(4)

COROLLARY. - For any a E (0,

2),

the density

pc, (x)

decreases at oo at the rate of a power function, more precisely

(5)

_{pc, (x) ,.._,}

~

_{

r

(1 +

a)

sin

7l"t}

IX

1-l-0!'

X ---t 00. 3. ANALYTIC PROPERTIES OF

pc, (x).

THEOREM 3.1. - For any a E (0, 2), the density

pc, (x)

is a completely monotonic function of x on the positive ray.

COROLLARY. - For any a E (0, 2), the density

pc, (x)

is a restriction to the po_sitive ray of a function analytic in the right half-plane.

It is evident that

pc, (x)

is an even function of

x,

therefore the analogous properties are valid on the negative ray.

Now, we shall consider representations of

pc, (x)

by convergent series in terms of

Ix

lkc,,

Ix

lk

(k

=

0, 1, 2, ... ) and log 1/

Ix I-

Denote by Qodd the set of all rational numbers a E (0, 2) representable in the form a

=

m/n, where m and n are integers

and, moreover, m is odd. Denote by

L

the set of all Liouville transcendental numbers o: E (0, 2). It is well-known that the set L is of zero Lebesgue measure.

The following theorem describes the analytic structure of

pc, (x)

for almost all a E (0, 2) in the sense of Lebesgue measure, though exceptional values of a form an everywhere dense in (0, 2) set of the power of the continuum.

THEOREM 3.2. - For any a E (0, 2)\{Q0_dd UL} the following representation is valid

(6)

where Ac, ( z) and N c, ( z) are entire functions of the finite orders 1 / a and 1 /2

correspondingly. These functions admit the following explicit power series representations

(7)

(8) 1

00

(-it

_zk

N Cl! (

z)

= ;-

L

r (

2 k + 1) sin (

x (

2 k + 1)

/a)"

k=O

In the case

a

E Q0_dd, the analytic structure of pc, ( x) is more complicated. Denote by

THEOREM 3.3. - If a E Qodd\Q 1, then the following representation is valid

where Ao, (z), Mo, (z), N°' (z) are entire functions of the orders 1/a, 1, 1/2 correspondinglr. Letting a

=

m/n,

where m and

n

are relatively prime integers, we can describe the pmrer series representation of these functions in the following way

(10) (11) (12) (13) \ (o,) -l\k -00 Ao, (z)

=

L

.xt)

zk, k=l 2r(ak) cos ('rmk/2)' k for non-integer n (-1/m+n)s [r'(ms) . 1rms 1 1rms] sm cos -r (ms) r (ms) 2 2 2 ' for k

=

ns, s

=

1, 2, ... oo ( )(m+n) k . (

k/ )

M ( )

=

.!_

""°"'

-1 sm 1rm 2 rnk-l Di z 7r ~

r (

mk) z ' No,

(z)

=

.!_

f

(-l)k

zk

a ( )/ r(2k+l)sin(1r(2k+l)/a)· k=O, 2 k+l m~N

If a

=

1/n E Q 1, then the following representation is valid

(14) 1 0, (-1r=

1 ₁

po, ( x)

=

R

A°' (

I

x

I )

+ 1r · cos x · log

R

where the entire function Ao, (z) is defined above.

We mention the following particular cases of representations of p°' (x) by theorem 3.3:

(15) - 1 1 1 . 1

00

k I'' (2 k + l) 2 /c

p1

(x) - ;

cos

x ·

log

R

+

2

sm

Ix I

+ ; ; (-1)

r2 (

₂ k + l) :i: ,

1 00 (-l)[(k+1)/2

llxlk

1 1 1 .

(16) Pl/2

(x)

=

J2lxT ;

r

(k

+ (1/2)) - ; cos X • log

R

+

2

Sill

I

:r

I

_ .!_

f

(-1) r'(2k+ 1) x2k_

7r k=O I'2(2k+l)

In the case

a

E L we could not find any representation of po,

(x)

similar to (6). (9), (14), therefore we have to restrict ourselves to the following less convenient one.

THEOREM 3.4. - If a E (0, 2) \ Q0_<l<l, then the following representation is valid ( J 7) p (

X) - -

1 hm . { - " 1 s ----'----'--'---'---( -1

t+l

I

x

I

k " a - IX I S--->00 2 6 r (a k) cos (1ra k/2) 1 (-lllxl2k+l } +

~

L

r ( 2 k + 1) sin ( 1r ( 2 k + 1) /a) · 1~2 k+l<a (s+(l/2))

The limit is uniform with respect to x on any compact set of R.

Note that the values of a such that the separate limits of the each of the two sums in the right hand side of (17) do not exist form an everywhere dense in (0, 2) set of the power of the continuum. It is possible to give a more detailed description of this set.

4. ASYMPTOTIC BEHAVIOUR OF Pa (x) AT 0.

THEOREM 4.1. - If a E (0, 2)\Q0_<l<l, then for any N

=

1, 2, ... the following asymptotic

formula is valid

(18)

1 [1/a] (-l)k+llxlka-1 1

Pa (x)

= -

_{2 k=l r a}

L ( ) ( / )

_{k cos 7f(Y_ k 2}

+

_{a sm}

. ( / )

_{1r a}

[3/a] (-l)k+llxlka-1 x2

+

L

r(ak) cos (1rak/2) - 2a sin (31r/a) + · · · k=[l/a]+l l [(2N-1)/a] (-l)k+l IX jka-1 +2

L

r(ak)cos(1rak/2) k=[(2 N-3)/a]+l (-1t-1 x2N-2 ( I 12N 2 + r(2N -1) sin ((2N - l)1r/a) + 0 _{X - ),} X---> 0.

(The first sum vanishes for a

>

1.)

Thus, the case a E

L

is not considered here separately.

For a E Q0_{<l<l, the asymptotic formula of Pa (x) as x---> 0 is a corollary of theorem 3.3.}

However, its general form is rather complicated. Therefore we restrict our attention to a few examples:

{PI (x)

=

_!._ log - 1- - i +~Ix I -

__!__

x 2 log - 1- + 0 (x2),

(19) 7f lxl ,.. 2 21r !xi X---> 0, 1 n-1 (-l)k+l IX I (k/n)-1 (-1)"+1 1 n '"Y PI/ ( X) - - "

+

log -

+ (

1) -n -26r(k/n)cos(k1r/2n) 7f lxl 7f (20) (-l)"+l n IX j l/n 2/n + 2r(1/n) sin (1r/2n) +O(lxl ), (21) X ---> 0, n

=

2, 3, ... , 4

l

1/2 p3;2(x)

= -

-

- ·

!xi

3v'3

7f 1 1 3 - 2'"'( + - x 2 log - + + 0 (Ix 1712), 21r lxl 41r X---> 0,

where , -

r'

(1)

=

0.5772157 ... is the Euler constant.

The proofs of the main results of this Note are based on the following representation of Po: ( x) in terms of Cauchy type integral

where

x1_fo:Po:(x1_fo:)

=

_{(l/a)Imfa(-xe-i1ro:fz), x}

_>

_0,

1

1

00

!a

(z)

=

-7f 0 e-v'la vlfo: - - - d v v-z

and on the explicit solution of a Riemann-Hilbert boundary problem. Note remise le 12 fevrier 1994, acceptee le 29 aoGt 1994.

REFERENCES

[I] Ju. V. LINNIK, Linear forms and statistical criteria, I, II. Selected Translations in Mathematical Statistics and Probability, 3, 1963, pp. 1-90. American Mathematical Society, Providence, RI (original paper appeared in: Ukrainskii Mat. Zhurnal, 5, 1953, pp. 207-290).

[2] B. C. ARNOLD, Some characterizations of the exponential distribution by geometric compounding. SIAM Journal of Applied Mathematics, 24, 1973, pp. 242-244.

[3] L. DEVROYE, Non-Uniform Random Variable Generation, Springer, New York, 1986.

[4] L. DEVROYE, A note on Linnik's distribution, Statistics and Probability Letters, 9, 1990, pp. 305-306. [5] D. N. ANDERSON, A multivariate Linnik Distribution, Statistics and Probability Letters, 14, 1992, pp. 333-336.

[6] D. N. ANDERSON and B. C. ARNOLD, Linnik distributions and processes, Journal of Applied Probability, 30, 1993, pp. 330-340.

[7] L. DEVROYE, A triptych of discrete distributions related to stable law, Statistics and Probability Le1ters, 18, 1993, pp. 349-351.

S. K. : University of Maryland, College Park, MD 20742-1815, USA; I. V. 0. : Bilkent University, 06533 Bilkent, Ankara, Turkey and Institute of Low Temperature Physics and Engineering, 47 Lenin ave., Kharkov 310/64, Ukraine, e-mail: ostrovskii@ilt.klwrbov.ua; A. H. : Bilkent University, 06533 Bilkent, Ankara, Turkey and Middle East Technical University, Ankara, Turkey.