Non-symmetric Linnik distributions

Probabilites/Probability Theory (Analyse complexe/Complex Analysis>

Non-symmetric

Linnik

distributions

Mehmet Burak ERDOGAN and Iosslf Vladimirovich OSTROVSKII

M. B. K: Department of Mathematics, Bilkent University, 06533 Bilkent, Ankara, Turkey; E-mail: erdogan@fen.hilkent.edu.tr

I. V. 0.: Department of Mathematics, Bilkent University, 06533 Bilkent , Ankara, Turkey and Verkin Institute for Low Temperature Physics and Engineering, 310164Kharkov, Ukraine. E-mail: iossif@fen.hilkent.edu.tr.ostrovskii@ilt.kharkov.ua

Abstract.

The aim of this Note is to study the probability density with characteristic function

where 0

<

a

<

2,

181

$ min(lI'a/2,1I' - lI'a/2). and 1/

>

O. This density. first introduced by Linnik for 8

=

0, 1/

=

1, received several applications later.Itdoes not have any explicit representation. We consider here its integral and series representations and its analytical properties.

US

distributions de Linnik non-symmetriques

Resume.

Nous etudions dans cette Note la densite de probabilite de fonction caracteristique

ou0

<

a

<

2,

181

$ min(ll'a/2,1I' - ll'a/2) et1/

>

O. Cette densite, introduite pour la premiere fois par Linnik avec 8

=

0, 1/

=

I, a trouve quelques applications par la suite. Il n'existe aucune representation explicite de cette densite. Nous considerons ses representations integrales et en series, et ses proprietes analytiques.

Version irenceise abregee

En 1953, Linnik (voir [8]) a demontre que la fonction (1) est la fonction caracteristique (f.c.) correspondant

a

la densite de probabilite (d.p.) Po:. Les proprietes de d.p. sont etudiees en detail

dans [2] et [5], OU on peut trouver les references concernant leurs applications. En 1984, Klebanov, Maniya et Melamed (voir [3]) ont introduit Ie concept de stabilite geometrique stricte, et ils ont dernontre que la famille des d.p.

a

stabilite geometrique stricte coincide avec la famille des d.p. dont les f.c. sont de la forme (2). Cette famille est plus vaste que la famille {Po : Q E (0,2)}; son etude a ete detaillee dans [1] (quelques resultats ont ete obtenus auparavant dans [6]). Notons que pour

181

=

min(1rQ/2,1r -

1rQ/2).

les d.p. de cette famille sont apparues pour la premiere fois

dans [7] et [11]. En 1992, Pakes (voir [to]) a trouve que les distributions de probabilite avec des f.c. de formes plus generales (3), jouent un role important dans des problemes de caracterisation de statistique mathematique. L'etude des proprietes des distributions avec des f.c. (3) est done un

probleme interessant,

D'autre part, on ne peut pas appliquer les methodes de [1], [5] et [6] it ce probleme, En effet, dans Ie cas ou v

=

1, la representation (9) rend impossible la reduction du problerne itl'etude d'une integrale de type de Cauchy. II semble qu'une telle reduction soit impossible dans Ie cas general ou u

>

O. Nous utilisons done une methode differente, basee sur les idees de [II].

TIIEOREME 1. - Pour les triplets (4), la distribution de probabilite avee La fe. (3) est absoLument continue, et on peut representer sa densitePa,fJ,v sous La forme (5)-(6).

Nous utilisons la representation (5)-(6) pour etudier Ie developpement de Pa,fJ,v en serie et son comportement asymptotique quandx -+ 0 et x -+ 00.Le theoreme suivant est la base de l'etude des proprietes analytiques de Pa,fJ,v' Notons que, par (5)et (6), it est clair que Pa,fJ,v(x)= Pa,-fJ,v( -x). On peut done se restreindre it l'etude des triplets (7).

TIIEOREME2. - Pour les triplets (a, (), v) EE PD* definis par(8), La representation(9) est valable. Remarque. - Les theoremes I et 2 ont ete demontres dans [5] pour ()

=

0, v

=

I, et dans [1] pour 0

:f.

0, u = 1. Le theorerne 2 pour 1

<

a

<

2, ()

:f.

0,v = 1, est aussi demontre dans [4]. La representation (10), qui n'est pas couverte par Ie theoreme 2, a ete obtenue dans [I] pour ()

=

7r-7ra/2.

Soit

[z]"

Ie plus grand entier strictement inferieur it x.

THEOREME 3. - (i) Si oa» ~ 2, 0 ~ min(7ra/2, 7r/v - 7ra/2), aLors Les deux fonetions Pa,8,v(±X) sont completement monotones sur (0,00). Cette assertion n'est pas valable pour toutes les vaLeurs de

((t,fJ,

/1)

EE PD*. Dans taus les cas, pour chaque triplet(a, (), v) EE P D*,Les fonctionsPa,8,v (±x ) sont la difference de deux fonctions completementmonotones sur (0,00).

(ii) Si Hill(I/(()

±

7rn/2))

:f.

0, alors pour k

=

[av]*, [av]*

+

1, ... , les fonnules (11) et (12) sont valables. Si ()

=

7rn/2, alors Po,fJ,v(x)

=

0 pourx

<

O.

Si(w

>

1,aLors la fonctionPo,lJ,v est[av]· - 1fois continument differentiable surR "de plus, pour

k

=

I), 1, ... ,[nv]· - 1, la formule (13) est valable.

TIIEOREME 4. - Pour ehaque (a, (), v) E EPD*, Les d.p. Pa,8,v sont unimodaLes. De plus, pour

IfJl

~ maxlf), 7r

1/1 -

7ra/2), Ie mode s'annule, et pour

I()I

>

max(O, 7r /v - 7ra/2), Ie mode est positif. TIIEOREME 5. - (i) Si (n, (),v) E EPD*, av ~ 1, l'egalite est valabLe. (ii) Comme fonetion de (),

n

~ fJ ~ rnin(7rn/2,7r/(2v) - 7ro:/2), Pa,IJ,v(x) est croissante, et Pa,IJ,v(-x) est decroissante pour chaque :1:

>

0 et 0:, v fixes tels que oa/ ~ 1.

Remarque, - Le theoreme 3 a ete demontre pour ()= 0, v = 1 dans [5], et pour ()

:f.

0, v

=

1 dans [I]; dans Ie deuxierne cas. la partie (ii) a ete demontree un peu auparavant dans [6] avec quelques restrictions additionnelles. Le theoreme 4 a ete demontre pour 0

<

a

<

2,v

=

1 dans [1], et un peu auparavant dans [6] sous une forme plus faible. Pour Ie cas v

=

1, un resultat plus fort que Ie thcorerne 5 a ete demontre dans [I]. .

TIIEOREME6. - Pour chaquetripLet

(a, (},

v) E

E PD,

Le developpemeruen serieasymptotique

(15)

est valable.

Remarque. -

Le theoreme 6 a ete demontre pour ()

=

0,

v

=

1 dans [5], pour ()

:f.

0,

v

=

1 dans [I] et (6) et, independamment, sous la condition additionnelle 1

<

a

<

2 dans [4].

Les theorernes 7, 8 et 10 ci-dessous traitent du developpernent de Po,lJ,v en series convergentes. Pour

v

=

1, i1 est demontre dans [I] et [5], que la structure de ces developpements depend de la nature arithmetique de

a.

Un phenomene semblable se manifeste dans le cas general, it condition qu'on considere la nature arithmetique de deux parametres a et u.

THEOREME 7. - Supposons qu'une des conditions suivantes soit satisfaite : (i) a

¢

Q et u E Q; (ii) a E Q et v

tJ.

Q; (iii) a

tJ.

Q, v

tJ.

Q et v

tJ.

{a

=

(2q

+

1)/a - P; q E l.+,p EN}; (iv) 0 E Q

et v E Q, ou a et v sont representables sous laformea

=

min, v

=

kllavecm, netk,1des entiers relativement premiers, 1 ne divisant pas

m.

Alors, La representation (16) est valable, ou La limite est uniforme par rapport

a

x sur chaque

compact de IR+.

Notons L l'ensemble de tous les reels transcendants au sens de Liouville.

THEOREME8. - Supposons qu'une des conditions suivantes soit satisfaite : (i) 0:

¢

Q

u

Letv E Q; (ii) a E

Q

et v

tJ.

Q

u

L; (iii) la condition (iv) du Theoreme 7. Alors la representation (17) est valable, ou les termes de la partie de droite de (17)sont les memes que dans (16),et ou les deux series convergent absolument et uniformement par rapport

a

x sur chaque sous-ensemble borne de R+.

Le theoreme 8 n'est pas valable si on omet L dans les conditions (i) ou (ii).

TfffioREME 9. - Dans l'ensemble de tous les triplets (0, (),v) satisfaisant les conditions (i) et(ii) du

Theoreme 7, ilya un sous-ensemble dense ayant La puissance du continu pour lequel chaque serle de la partie de droite de (17) est divergente pour tous les x E IR+.

Si Ie triplet(a, (),v) ne satisfait pas les conditions du theoreme 7, alors la structure du developpement en series de Pa,6,v est beaucoup plus cornpliquee que celie de (16) et (17). On a alors des termes en log

x

dans les developpements,

Exemple. - Pour a

=

2/3, ()

=

1,v

=

3/2, la formule (18) est valable.

1.

Introduction

In 1953, Linnik (see [8]) proved that the function

(1)

is the characteristic function (ch.f) of a symmetric probability density (p.d.) Po. Properties of the p.d. 's were studied in detail in [2] and [5], where one can find several references on some applications. In 1984, Klebanov, Maniya, and Melamed (see [3)) introduced the concept of geometric strict stability and proved that the family of geometrically strictly stable p.d.'s coincides with the family of p.d.'s with ch.f.'s

(2)

This family is larger than {PO! : a E (0,2)}, and its detailed study was done in[I] (some of the results had been obtained before in [6]). Note that for

181

=

min(1f0:/2,1f - 1f0:/2),the p.d.'s of this family first appeared in [7] and [II]. In 1992, Pakes (see [10)) showed that the probability distributions with ch.f.'s more general than (2), as given in (3),

1 .

(3)

CPO!,6,v(t)

=

(1

+

e-i6sgntltla)v ' 0: E (0,2),

lei

~ mm(1fo/2,7r - 7ro:/2) ,

u

>

0,

play an important role in characterizing some problems in mathematical statistics. Therefore, the study of properties of distributions with ch.f.'s (3) seems to be of interest.

However, the methods used in [I], [5], and [6] are not applicable to such a problem. The reason is that, in the case 1/

=

1, the representation (9) below makes it possible to reduce the problem to the study of a Cauchy type integral. Such a reduction seems to be impossible in the general case

v

>

O.

(5)

2. Absolute continuity and integral representations

THEOREM 1. - For any

(4) (a,O ,v) EEPD:= {(a ,B,v): a E(0 ,2) ,

IBI

~ min(7ra/2,7r -7ra/2) ,v

>

OJ,

the probability distribution with chf. 's (3) is absolutely continuous and

1

l

c

+

i oo POI9,,(±x)

= - .

f±9 (z) xZ-1dz ,

-a

<

c

<

min(av,1),x

>

0, , 1 21r1. c - i CXJ 0' , , V where

(6) f± (z)

=

_1_sin(7rz/2

±

Bz/a) r(z/a)f(v - z/a).

01 ,9,,,

of'(v) sin 7rZ I'(z)

We use the representation (5) to study the series expansions ofPOI,9,,, and, hence. its asymptotic behavior as

x

~

°

and

x

~ 00. The next theorem is a base for studying analytical properties of pn,9 ,,,' Note that from (5) and (6). it is obvious that POI ,8,v{X)

=

POI,-8,,,( -x). Therefore. we can restrict our study to triples of the form

(7)

THEOREM 2, - For

(a ,B ,v) EEPD+:= {(a ,0,v) EEPD: B~OJ.

(8) (a ,B,v)E EPD*:= {(a,B,v) E EPD+: B

<

7r-7ra/2} , the following representation holds:

1

1

00 e-yrdy

(D) Po,8 ,,,(±X)

=

:;Im

0

(1

+

e'f i8- l1ro/2yo ),,' x>0.

Remark. - Theorems 1 and 2 had been proved in [5] forB

=

0, u

=

1.and in [1] for 0

i:-

0, v

=

1. Theorem 2 for 1

<

a

<

2, ()

i:-

0,v = 1 is also proved in [4] . The following representation. which is not covered by Theorem 2, has been obtained in [I] :

sin(7ra)

1

00

eYXyOdy

(10) Pn,8,l(X)

= -

11 i11'O

°1

_{2 '} X

<

0, Po,B,l(X)

=

e:" [a,

X> 0,

7r 0 -e y

for ()

=

7r - 7ra/2.

3, Analytic properties of

POI,B,,,

Let

[xl*

denotes the greatest integer strictly less than

z .

THEOREM 3. - (i) If Q:V

~

2, B

~

mine 7ra/2, 7r/v - 7ra/2), then both functions POI ,8,,,(±x)

are completely monotonic on

(0,00).

This statement fails to be true for all other values of (a , B, v) E EP D*. Nevertheless,Jor any (a, B,v) EEPD*, both functions pOI ,8 ,,,(±X) are differences of two completely monotonic functions on

(0,00).

.

(ii) Ifsin(v(fJ

±

7ra/2))

f;

0, then for k

=

[av] .. [av]*

+

1, . .. , we have:

(11) (_I)k+[,,(0/2+8/1I')]' lim p(k ) (x)

=

+00,

x-o+ a ,8 ,v

(12) (_1)["(0/2-8/11')]' lim p(k) (x)

=

+00.

x--+o - a,8 ,£1

If

av

>

I, then the function

p""e,,,.

is

[av]* -

1times continuously differentiable on R and. moreover.

(13)

p(k)

_"',9,'"

(0)

=

sin[(1r/2 -

_1ra

O/a)(k

+

1)]B(k

+

l)/a, u _

(k

+

1)/a)

for k

=

0 , 1, ... ,

[av]* -

1.

THEOREM 4. - For any

(a ,O,v)

E

EPD*,

p.d. :s

Po,o,,,.

are unimodal. Moreover. for

101

~

max(O ,1r/v

-1ra/2),

the mode is zero. andfor

1°1>

max(O,1l"/v

-1l"a/2),

the mode is positive.

THEOREM 5. - (i)

If

(a,

(J,

v)

E

EP

D* and

av

~ 1, then

(14)

Po,II,,,.(X)

~

Pcr,lI ,v(-x), x>

O.

(ii) As a function of 0,

°

~

()

~ min(1l"a/2,1l"/(2v) -

1ra/2). p""o,v(x)

increases and

Pa,e,,,.( -x)

decreases for any fixed x

>

0 and a. v such that oa/ ~ 1.

Remark. - Theorem 3 has been proved for ()

=

0,v

=

1 in [5], and for ()

f;

0,v

=

1 in [1]; in the latter case, part (ii) had been proved previously in [6] under some additional restrictions. Theorem 4 has been proved for 0

<

a

<

1,

v

=

1 in [6], and for 0

<

a

<

2,

v

=

1 in [1]. For the case

v

=

1, a stronger result than Theorem 5 has been proved in [1].

4. Series expansions of

Pa,O,,,.

THEOREM 6. - For any

(a,

0,

v)

E

EPD,

(15)

_{Pa,B ,v(±X) '" 2f(v)}

1 ~(-1)qsin(1l"aq/2±Oq)r(v+q)

~

_sin(1rqa)f(

_-qa)f(l

₊

_q)

_[z]

-l-qa

_,

Ixl

--+00.

Remark. - Theorem 6 has been proved for

°

=

0,v

=

1 in [5], for

°

f;

0, u

=

1 in [1] and [6], and, independently, under the additional condition 1

<

a

<

2 in [4].

Theorems 7, 8, and 10 below deal with convergent series expansions of

Pa.B,,,.'

For

v

=

1 it has been shown in [1] and [5] that the structure of such expansions depends on the arithmetic nature of a. It turns out that a similar phenomenon remains true in the general case, but one has to take into account the arithmetic nature of two parameters a and u,

THEOREM 7. - Suppose that one of the following conditions is satisfied:

(i) a

rt.

Q andt/ E Q; (ii) a E Q andv

rt.

Q; (iii) a

rt.

Q, v

rt.

Q, andv

rt.

{a

=

(2q

+

1)/0: - P; q E

r»

EN}; (iv) a E

Q

and v E Q, where a and v are representable in the form 0:

=

min.

lJ

=

kll. with m,nandk,l relatively prime integers and lnot dividing m. Then

1 I' { 1

i:

(-I)qsin«v+q)(1l"a/2±O))f(v+q) o(v+q)

IG 0 0 " .

±x

= -

1m - - . x

( ) p " ( )

xs~oo

f(v) q=O

sm(1l"a(v

+q))f(a(v

+

q»f(l

+q)

1 ' "

(-l)q+1

sin(1l"q/2

±

Oq/a)f(q/o:)

}

+ - -

L....J xq

af(v)

1<

_{_q_ v+s+l}

« /) sin(1r(v -

q/o:))f(q)r(l - v

+

q/o:) ,

,

2a

x >

O.

This limit is uniform with respect to x on every compact subset ofR+. Let us denote by L the set of all Liouville transcendental numbers.

THEOREM 8. - Suppose that one of the fol/owing conditions is satisfied:(i) Q ¢

Q

ULand v E Q; (ii) a E Q and IJ¢ QUL; (iii) is condition (iv ) of Theorem 7. Then

( 17) _{Pa,9 ,v(±X)}

₌

_xf(v)1

{ OO

~(

_{... )}

₊

~ ~(

1

00

_{...) ,}

}

_{x> 0 ,}

where the summands in the right hand side of(17) are the same as in(16), with both series converging absolutely and uniformly with respect to x on every compact subset ofR+.

Theorem 8 fails to be true if one omits L in conditions (i) or (ii) .

THEOREM 9. - In the set of all

(Q,() ,

v) satisfying conditions (i), (ii) of Theorem 7, there is a dense subset with the power of the continuum for which both series on the right hand side of(17) diverge for all x E R+.

If (a,(J,v) does not satisfy conditions of Theorem 7, then the structure of series expansions of Pa,9,v is much more complicated than in (16) and (17). It is worth mentioning terms containing logx in the expansions.

Example. - For a

=

2/3, ()

=

1,

v

=

3/2,

we have

(18) Pa,9,v(±X)

=

cos(3/2) log - +1 0(1) , x - 4+0.

7r X

The Note is the succint version of a text on file for five years in the Academy Archives. Copy available upon request.

Note remise Ie 18 mars 1997, acceptee Ie 18 juin 1997.

References

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