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ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXIII (1983) ROCZNIKI POLSK1EGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE XXIII (1983)

D.

S

zynal

and

A.

W

olinska

(Lublin)

On classes of couples of characteristic functions satisfying the condition of Dugué

D. Dugué in [1] was interested in finding couples (q>1, q>2) of characte­ ristic functions satisfying the condition

(D) --- --- =

(pi(t)<p2(t)-He has remarked that the characteristic functions

<Pi(t) 1

1 + if ’ <P 2 ( t ) = 1

1 —

it satisfy the condition (D).

L. Rubik has given in [2] two classes of couples (<pl5 (p2) of characteris­ tic functions for which a more general condition

(K) P(Pi(t) + q(p2(t) = (pi(t)<p2(t), P + q =? 1, P > 0, q > 0 holds. The first class is created by characteristic functions

<Pi (0 - a

a + i t ’ q>2(t) =

pa

p a — i t q

a >

0,

while the second one by Kcharacteristic functions

<Pi (t) = q + p cos bt — i p sin bt, <p2(t) = p + q cos bt + i q sin bt, beR, where p + q — 1, p > 0, q >

0.-In this note we give some new classes of couples {(px, q>2) of characteris­ tic functions satisfying (K) or (D).

E

xample 1. It can easily be stated that (K) is satisfied by the functions

q l - p e ir

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326 D. S z y n a l and A. W o lin s k a

being the characteristic functions of random variables and X2, respec­ tively, which have the discrete distributions

^ [ * 1 = k = 0 , 1 , 2 . . . , P [ X 2 = - 1 ] = 1.

The random variable X = X x + X 2 (and Y = pXt + q X 2) has the characteris­ tic function

cp(t) = q>i(t)(p2(t) = 4 ( 1 — pea) e" and the discrete distribution

P [ X = - l ] = q , P [X = k ] = q p k+1, k = 0, 1 ,2 ,...

E

xample 2. Condition (K) is satisfied by functions <Pi(r) = qe"

(1 +q)e" — 1 (p2(t) = pe“

1 — qelt

being the characteristic functions of random variables X { and X 2 respec­ tively of geometric distributions

PCXi = - * ] = «( j— J . * = 0,1,2,...,

P [ X 2 = fc] = pqk~1, k = 1, 2, . . .

The random variable X = Х г + Х 2 has the characteristic function

pqe2it

<P(t) = (pi{t)<p2(*) = (1 - q e u)[( 1 +q)eu- 1]

and the discrete distribution

P [X = - * ] = w ( Щ J , * = 0 , 1 , 2 ...

P[ X = *] = w ‘, * = 1 , 2 , . . .

E

xample 3. Condition (D) is satisfied by functions

1 1

<Pi(t) =

TJ" ' [(4/ 2 - l ) + 0 ] [ ( v /2+ 1 ) - й ] ’

being the characteristic functions of absolutely continuous distributions given by the density functions

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Classes o f couples o f characteristic functions 327

respectively. The random variable X = X l + X 2 has the characteristic function

<p(t) = ( P i ( t ) ( p 2 (t) =

and the density function

1

( 1 - if ) 2 [ ( ^ 2 —1) +

i t ] [ ( ^ / 2 + 1 ) - i t ] ,( 'J2- 1) f i x ) = 4 ч/2 \xe x-f 1 4 y / l v'l+l) if x ^ 0, if x > 0.

E

xample 4. Condition (K) is satisfied by functions

a 1 = Я + Р (p2(t) = p + q IP a t + 'Щ —= = J a 2 - f t 2 y j a 2 - f t 2 a >

0,

being, so-called, generalized characteristic functions (see, i.e., [4]) Moreover, one can verify that and

q

>2 are the generalized characteristic functions of the generalized distributions .which can be written respectively as follows:

F t (*) = F ld(x) + F lc(x), F 2 (x) = F 2d{x) + F 2c(x), where F id and F ic, i = 1, 2, are given by the formulas:

f 0, if x ^ 0, ( 0, if x ^ 0, F u W = T " F 2d(x) = \[ q, if x > 0, I p , if x > 0, F lc(x) = pa 00 0 } * cos x u yju2- fa : F 2c (*) q a cos yu , \ , q cos xu K J V J J u 2 - fa 2 J 71

J

J u 2-fa* со 0 0 d u , d u .

Rem ark. It is not difficult to note that functions of the form

<PiU) = я+рФ(г)-1Рхи), (p2{t) = p+q'l'(t)+iqx(t),

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328 D. S z y n a l and A. W o l in s k a

References

[1 ] D. D u g ué, Arithmétique des lois de probabilités, Mémor. Sci. Math. 137, Paris ^1957. [2 ] L. K u b ik , Sur un problèm e de M. D. Dugué, Pràce Mat. (Comment. Math.) 13 (1969),

p. 1-2.

[3 ] E. L u k a c s , C haracteristic functions, London 1960.

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