ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXIII (1983) ROCZNIKI POLSK1EGO TOWARZYSTWA MATEMATYCZNEGO
Séria I: PRACE MATEMATYCZNE XXIII (1983)
D.
S
zynaland
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 functionsq l - p e ir
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 functions1 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
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 functionsa 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),
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.