On some representation of the density function gamma type

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A C T A U N I V E R S I T A T I S L O D Z I E N S I S FOLIA MATHEMATICA 8, 1996

Włodzimierz Krysicki

O N T H E R E P R E SE N T A T IO N O F T H E D E N S IT Y F U N C T IO N S G A M M A T Y P E

To Professor Lech Włodarski on His 80th birthday

In this paper we present the random variable X of garnm a type as infinite (or finite) p rod uct of independent random variables X k ,

k € N, in the following sense:

OO

* = n x *-

k = 1

Let the random variable X have three-param eters gam m a distri-bution with the density given by the formula

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f ( x \ e , A, p) = jrRj * p -1 exp , x > 0, p, A, 0 > 0, where 0 ' / A is the scale param eter and p, A are the shape-param eter. In the present note it will be used the Mellin transform (2) of the function (1)

M ( s ) = M [ f ) x \ Q , A, p), s\ = E X °

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where s denote a complex variable: [1], [2]: recall th a t Sneddon [3], Dytkin and Prudników [4] have defined the Mellin transform M(s) as E X ' - 1.

By applying the Knar formula [5.8, p. 324]

00 ^ n ,

r(* + i) = 2” n w n ' Rel>- ł’

k= 1 V 2 / into (2) we obtain 00 T ( l + P-*t i ) M ( s) = (4 0 )s/ a TT —^_{A l} _{/ i . u—A \}R e > - p . fc=l 7

The last form ula can be written as

( l + f i x )

~ (40)*/A2‘ r ( i + L = ^ )

(3) M ( s ) =

[ T ---

2---

L,

R es > - p . r (* + « * )

fc=l

Let us denote fc-factor occuring in the right hand side of (3) by (s). Next, we apply the inverse Mellin transform for every factor fifjt(s); the result of this transform is denoted by /jt(x |0 , A, p):

c + i oo

(4) /fc(ar|0 , A, p) = —

J

x 3 1 gk(s) ds, R e s > - p .

P utting c = 0 in view Re s > —p, we get (5) /* (x |0 , A, p)

2 a.(2*-1-l)A+ p-l exp i I , a; > 0. r ( i + S x ) ( 4 0 ) i + i i r ' V 2 0

It is easy to prove that the condition J0°°/jt(a:|0 , A, p)dx — 1 is satisfied for any k 6 N. In view of the inequality /jt(x |0 , A, p) > 0

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ON TH E DEN SITY FU NC TIO N S GAMMA T Y P E 21

this implies th a t the form ula (5) defines the density of probability of the random variable X k in the interval [0, oo). In consequaence every factor <7it(s), k € N of the infinity product (3) is the Mellin transform of the random variable X * with the density given by (5).

Now we shall show th a t </*(.$) is the Mellin transform of n r A, p). To do this we find an estim ate of an absolute value of any factor of (3). In fact setting R e s = c = 0 according to Re s = c = 0 we get

( 4 0 ) a / 2 * A r

₍

î

+

ï

- \ + A_J ( 4 0 ) R e " / 2 * Ar ( 1 i r ) - A + R e . i \ V, 2 2* A J

r (è + Sx)

r k

₂

+

₂

*x)

Let us denote M n « - ) = m u gk(s )- Besides, for any n € N |A/r,(js)| < |il/,(.s)| and M i ( s ) is an absolute integrable function. From the above and in view of the Lebesgue dom inated convergence theorem [6. Th. 1.34] and (3) we obtain successively

I OO

lim [ x ~ a~ A (a/(.s) - lim M n(s) \ ds = 0.

n —*oo J \ n—»oo /

— i oo

T hus, we have proved the following

T h e o r e m . The density of the three-parameter gamma di stnbut ui n o f the form (1) is equal to the density o f the infinite product X k , where the density /* (:r|0 , A, p) o f X k are independent and defined by the formula (5), the scale parameter being (40 ) 1/ 2*A respectively, what can be formally denoted as

OO x = n x k.

k= 1

Re f e r e n c e s

[1] B. Epstein, Som e application o f llie M ellin transform in statistics, A.M.S. 19 (1948), 370-379.

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[2] B.M. Zolotariew, Prieobrazowania M ellina-Stieltiesa w teorii wierojatnostiej, Teoria w ierojatnostiej i jejo prim ienienia 2 no. 4 (1957), 444-469.

[3] J. Sneddon, Fourier Transforms, New York, 1951.

[4] B.A. Dytkin and A.P. Prudników, Integralnyje Prcobrazowania, Moskwa, 1961.

[5] J.S. G radztein and J M Ryzhik, Tables o f Integrals, Sums, S en e s and Prod-ucts, National Publishing Company of Physical and M athem atical Literature, Moskwa, 1962.

[6] W. R udin, Real and Complex Analysis, Me Graw-Hill, New York, 1974.

Włodzimierz Krysicki

O P E W N E J R E P R E Z E N T A C J I F U N K C J I G Ę S T O Ś C I T Y P U G A M M A

Przed ca 30 laty ukazało się wiele prac dotyczących wyznacza-nia gęstości skończonego iloczynu zmiennych losowych niezależnych o rozkładach Beta, Gamma, Normalnym, Bessela, w tym prace pra-cowników naukowych Politechniki Łódzkiej: Środki, Podolskiego i Krysickiego.

Prezentowana praca stawia sobie jako cel: cel przeciwny, którym jest przedstawienie zmiennej losowej X typu Gam m a jako

nieskończo-nego (bądź skończonieskończo-nego) iloczynu zmiennych losowych niezależnych k € N, s sensie stichastycznym tzn. w sensie równości gęstości t r st och t TOC \ r

X = l l j x k.

In stitu te of M athem atics

Łódź Technical University al. Politechniki 11, 90-924 Łódź, Poland