Weakly Stable Vectors and Magic Distribution of S. Cambanis, R. Keener and G. Simons

Transcription

1 Applied Mathematical Sciences, Vol. 1, 7, no., Weakly Stable Vectors and Magic istribution of S. Cambanis, R. Keener and G. Simons G. Mazurkiewicz Wydzia l Matematyki, Informatyki i Ekonometrii Zielona Góra, ul. prof. Z. Szafrana a, Poland G.Mazurkiewicz@wmie.uz.zgora.pl Abstract A random vector X is called weakly stable if for all random variables Θ 1 and Θ independent of X, X there eists a random variable Θ independent of X such that XΘ 1 + X Θ d XΘ. J. K. Misiewicz, K. Oleszkiewicz and K. Urbanik showed (see [6]) that for every random vector X this property is equivalent to the following condition: a, b Θ independent of X : ax + bx d XΘ. In this paper we consider weakly stable distributions and we give some elementary properties of weakly stable random vectors. The main results in the paper are connected with the distribution in n discovered by S. Cambanis, R. Keener and G. Simons in 1983, which is (up to rescaling) the only one etreme point in the family of l 1 -symmetric distributions. We give here eplicite formulas for projections of this distribution, conditional densities and plots of some of this densities. Mathematics Subject Classification: 6A1, 6B5, 6E5, 6E7, 6E1 Keywords: weakly stable distribution, symmetric stable distribution 1 Introduction Throughout the paper by L(X) we denote the distribution of the random vector X. If random vectors X and Y have the same distribution we will write X d Y. An independent copy of X is a random vector X, which is independent

2 976 G. Mazurkiewicz of X and such that X d X. A symbol µ denotes the characteristic function of the measure µ. Let E be a Banach space. By P(E) we denote the set of all probability measures on E. We will use the simplified notation P() P, P([, + )) P +. For every a and every probability measure µ we define the rescaling operator T a : P(E) P(E) by the formula: { µ( A) for a, T a µ(a) a δ (A) for a, for every Borel set A E. Equivalently T a µ is the distribution of the random vector ax if µ is the distribution of the vector X. The scale miture µ λ of a measure µ P(E) with respect to the measure λ P is defined by the formula: µ λ(a) def T s µ (A) λ(ds). It is easy to see that µ λ is the distribution of the random vector XΘ if µ L(X), λ L(Θ), X and Θ are independent. In the language of characteristic functions we obtain µ λ(t) µ(ts)λ(ds). It is known that for a symmetric random vector X independent of random variable Θ we have XΘ d X Θ. From this property we obtain that if µ is a symmetric probability distribution then µ λ(a) T s µ (A) λ (ds), + where λ L( Θ ). We say that random vector X is weakly stable if for all real numbers a, b there eists a random variable Θ independent of X, such that ax + bx d XΘ, (1) where X independent copy of X. Notice that every symmetric stable vector is weakly stable, which follows directly from the definition (see [3, 7, 8]): X (X 1, X,..., X n ) is symmetric α-stable, α (, ] (shortly SαS) if for every choice of a, b there eists a constant c ( a α + b α ) 1/α, such that ax + bx d cx,

3 Weakly stable vectors and magic distribution 977 thus it is enough to take Θ ( a α + b α ) 1/α almost everywhere. In [6] there is another eample of the symmetric weakly stable random vector. It is an n-dimensional random vector U (U 1, U,..., U n ), which is uniformly distributed on the unit sphere in n. The paper is organized as follows: In section we consider a linear transformations of weakly stable distributions. In section 3 we describe a very interesting class of l 1 -symmetric distributions introduced in the year 1983 by S. Cambanis, R. Keener and G. Simons. In section we give eplicit formulas for densities of marginal distributions and conditional distributions for l 1 -symmetric distributions on n. As a simple consequence we obtain also an interesting integral, which cannot be found in any of known for us tables of integrals. Section 5 includes the geometrical proof of K. Oleszkiewicz of the equality given by S. Cambanis, R. Keener and G. Simons. Linear transformation of weakly stable distribution The following remarks are rather trivial: Remark 1. A random vector X (X 1, X,..., X n ), n IN, is weakly stable iff for every linear operator A : n n the random vector Y AX is weakly stable. Proof. It is enough to notice that aax + bax A(aX + bx ) and A(XΘ) (AX)Θ. Remark. A random vector X (X 1, X,..., X n ), n IN, is weakly stable iff for every t (t 1,..., t n ) n a linear combination t, X n i1 t ix i is a weakly stable random variable. Proof. The necessity should be clear from the properties t, ax + bx a t, X + b t, X and t, XΘ t, X Θ. To prove the sufficiency let us notice that a characteristic function of the random variable n i1 t ix i at the point u has a form { E ep iu n } t i X i ϕ X (ut) i1 for every vector t n. Hence for the numbers a, b the condition { ( E ep iu a n t i X i + b i1 n )} t i X i i1 { E ep iuθ n } t i X i i1

4 978 G. Mazurkiewicz is equivalent with ϕ ax (ut)ϕ bx (ut) Eϕ XΘ (ut), which holds for every u, t n and X (X 1,..., X n) - an independent copy of X. Now it is enough to take u 1. However it is not true that if AX is weakly stable for some linear operator A, then vector X is weakly stable. Theorem 1 gives the counter-eample for A being a projection operator on any of coordinates. Theorem 1. If X is SαS, Y is SβS, α, β (, ] and X, Y are independent random variables then random vector (X, Y ) is weakly stable iff α β. Proof. Let L(X) µ is SαS and L(Y ) ν is SβS, X and Y independent. Then the vector (X, Y ) has a characteristic function Φ(t, s) µ(t) ν(s) and it satisfies (1) if and only if for all real numbers a, b there eists a probability distribution λ P such that Φ(at, as)φ(bt, bs) Φ(ut, us)λ(du). Symmetry of distribution µ and ν implies that µ(at) ν(as) µ(bt) ν(bs) µ(ut) ν(us) λ (du). () + Assume that s. Since ν() 1 then the equation () has a form µ(at) µ(bt) µ(ut) λ (du). (3) + A random variable X is symmetric α-stable, and λ is concentrated on [, ). It follows from theorem 1 in [5] that λ is uniquely determined and λ δ ( a α + b α ) In the same way, if in the equation () we assume t we obtain 1/α λ δ ( a β + b β ) 1/β. This means that the equality () holds iff a α + b α a β + b β for every a, b, so α β. Corollary 1. If a random vector X (X 1,..., X n ) is SαS and a random vector Y (Y 1,..., Y k ) is SβS, n, k IN, α, β (, ], and vectors X and Y are independent, then the random vector (X 1,..., X n, Y 1,..., Y k ) is weakly stable iff α β. 3 l 1 -symmetric distributions S. Cambanis, R. Keener and G. Simons studied l α -symmetric distributions, i.e. distributions µ on n for which characteristic function depends on the l α -

5 Weakly stable vectors and magic distribution 979 norm on n, i.e.: µ() ϕ( 1 α + α n α ), ( 1,,..., n ) n for some function ϕ on [, ) (notation ϕ Φ n (α)). In the case α 1 they have obtained a full characterization of such measures and the corresponding functions ϕ Φ n (1) (see []). Their result is based on the following surprisingly general definite integral identity: π ( s f sin θ + ) π t dθ f cos θ ( (s + t) sin θ ) dθ, which holds for all s, t, and all functions f for which the integrals make sense. They gave the following. Theorem. ([]) (a) ϕ Φ n (1), n, if and only if ϕ() ϕ n (r)λ(dr), where λ P + and ϕ n Φ n (1) is given by Γ ( ) n ϕ n () ( πγ n 1 ) Ω n (u )u n n 3 (u 1) du. 1 (b) Equivalently, X (X 1, X,..., X n ) is l 1 -symmetric if and only if ( ) X d U1 U Θ U,,..., n, 1 n where Θ is a nonnegative random variable with distribution λ P +, U (U 1, U,..., U n ) denotes an n-dimensional random vector which is uniformly distributed on the unit sphere in n, ( 1,,..., n ) denotes an n- dimensional vector with irichlet distribution and parameters ( 1, 1,..., 1 ), and Ω n (t 1 + t t n) is the characteristic function of U, and ( n )( ) n Ω n (r ) Γ 1 J n r 1 (r), r >, where J n 1 is a Bessel function of the first kind; Θ, U and are independent. In the two-dimensional case we obtain that ϕ Φ (1) if and only if ϕ() π sin r λ(dr), >, r

6 98 G. Mazurkiewicz or equivalently, the characteristic function of the random vector (X, Y ) has the form E ep i(tx + sy ) ϕ( t + s ) if and only if ( ) (X, Y ) d U1 U, Θ, B 1 B where Θ has distribution λ P +, U (U 1, U ) is uniformly distributed on the unit sphere in, B has Beta( 1, 1 ) distribution; Θ, U and B are independent. The paper of Cambanis et al. contains also the eplicit formulas for the density of l 1 -symmetric distributions with given distribution λ, namely in n : ( ) f n () r n g n λ(dr), ( 1,,..., n ) n, + r where g n () [ Γ ( n )] (n )!π n n k1 ( k 1)n + n j1,j k ( k j ), k j for k j. In order to simplify the notation we will write ( ) ( ) n U1 U U, n U,...,. 1 n ( ) n U The random vector has the characteristic function ϕn and the density function g n, moreover it is l 1 -symmetric, since in this case λ δ 1. ( ) n U Theorem 3. Random vector is weakly stable. ( ) n ( ) n U Proof. Let and be independent copy of U and let a, b. Then ( ) n ( ) n U the characteristic function ϕ of the random vector a + b U has the following form ϕ(t) ϕ n ( at l1 )ϕ n ( bt l1 ) ϕ n ( a t l1 )ϕ n ( b t l1 ), for t n, t l1 t 1 + t t n. This means that for every fied a, b the function ϕ(t) depends only on the norm t l1, thus theorem implies that there eists a distribution λ a,b P +, such that ϕ(t) This is equivalent with the representation ( ) n U a + b ϕ n (r t l1 )λ a,b (dr). ( U ) n d ( U ) n Θ, ( ) U n. where λ a,b L(Θ), Θ independent of This ends the proof.

7 Weakly stable vectors and magic distribution 981 More about the magic distribution of S. Cambanis, R. Keener and G. Simons ( ) n U Let us consider a weakly stable vector X with density function g n and assume that linear operator A : n n, n IN is defined by a nonsingular matri A [a ij ] n i,j1 with an inverse matri A 1 [b ij ] n i,j1. Hence random vector AX has a density function given by the formula: f n () [ Γ ( n )] (n )!π n A n (( n i1 b ki i ) 1) n + n j1,j k ( n i1 (b ki b ji ) n i i1 (b ki + b ji ) i ), () k1 where n i1 b ki i n i1 b ji i, for k j. According to remark 1, vector AX is weakly stable. Let Π k (X 1,..., X n ) (X 1,... X k ). The density function for the random vector Π k (X) cannot be obtained from the formula () since the corresponding matri A, such that Π k (X) AX, is singular. Thus we have to calculate the following: Π n k ( 1,..., n k )... g n ( 1,..., n )d n k+1 d n k+... d n. According to remark 1, each (n k)-dimensional projection of the distribution given by density g n is weakly stable. We can also obtain a conditional density of the vector (X 1, X,..., X k ) given that (X k+1, X k+,..., X n ) ( k+1, k+,..., n ): f(( 1,,..., k ) ( k+1, k+,..., n )) g n ( 1,,..., n ) Π n k ( k+1, k+,..., n ). The following theorem is based on straightforward but very laborious calculations: Theorem. For every n, α IN, n α, the marginal densities Π n k of the density g n are given by the following: For k α Π n α ( 1,..., n α ) [ ( Γ n )] (n )!π n α [ n α k1 ] ( k 1)n + n α j1,j k ( k j ) n α α α 1. k

8 98 G. Mazurkiewicz For k α + 1 Π n α 1 ( 1,..., n α 1 ) [ ( Γ n )] + (n )!π n α α i1 For k α + [ n α 1 k1 (αc n α 1 i i 1 Π n α ( 1,..., n α ) [ ( Γ n )] For k α + 3 (n )!π n α [ n α Π n α 3 ( 1,..., n α 3 ) [ ( Γ n )] [ (n )!π n α α+1 i1 k1 n α 3 k1 (α+c n α 3 i i 1 ( k 1)n n α 1 j1,j k ( k j ) k ) ] n α 1 α i. α+1 ln 1 + k 1 k ( k ] 1)n n α j1,j k ( k j ) k + α+1c n α α+ 1. ( k 1)n n α 3 j1,j k ( k j ) k ) ] n α 3 α+1 i. In these formulas we use the following notations: Π n ( 1,,..., n ) g n ( 1,,..., n ), α+3 ln 1 if j, j m ( 1) j( ) n j j 1 i ( 1)j i Sj i m i m if j 1,,..., t, if j <, where w n m, t w [ γ 1 ], here γ k, m, w, IN, { Si m ( m i ) (k 1,...,k i ) ( m i ) k l if i {1,..., m}, if i or i > m, 1 + k 1 k where (k 1,..., k i ) denotes any choice of i elements from the set {1,,..., m}, { S j,m i ( m 1 i ) (k 1,...,k i ) j ( m i ) k l if i {1,..., m}, if i or i > m,

9 Weakly stable vectors and magic distribution 983 where (k 1,..., k i ) j denotes any choice of i elements from the set {1,,..., j 1, j + 1,..., m}, ( j 1)n γ+1a m j m if j 1,,..., m, i1,i j ( j i ) j γ+1 ( 1 1)n 1 if j m 1. γ+1 The value of γ C m j is given by the recursive formula γc m j [ j 1 ] ( 1) i+1 Si m γcj i m + ( 1) [ j+1 i1 ( j+1 t+[ n + ( 1) ] t + [ j+1 ] ) t ( 1) i ] m i1 t+[ j+1 γ+1a m i S i,m [ j 1 ] G j( i ) ] i S m t+[ j+1 ] im i for G j (r) r if j is even, and G j (r) 1 if j is odd, j 1,,..., γ 1, and γc m γ ( 1)w m. i1 i Proof. We give here only a sketch of the proof. We will use the method of the mathematical induction with respect to α IN. At the beginning notice that the formulas are true for α, 1,, 3, since Π n ( 1,,..., n ) g n ( 1,,..., n ) [ ( Γ n )] n ( k 1)n + (n )!π n n k1 j1,j k ( k j ), Π n 1 ( 1,,..., n 1 ) g n ( 1,,..., n )d n [ Γ ( n )] (n )!π n n 1 k1 Π n ( 1,,..., n ) ( k 1)n n 1 j1,j k ( k j ) k ln 1 + k 1 k, Π n 1 ( 1,,..., n 1 )d n 1 [ ( Γ n )] [ n ( k 1)n (n )!π n n k1 j1,j k ( k j ) k + 1 n Π n 3 ( 1,,..., n 3 ) Π n ( 1,,..., n )d n k1 k1 k [ ( Γ n )] [ n 3 ( k 1)n (n )!π n n 3 j1,j m ( k j ) k ln 1 + k 3 1 k ],

10 98 G. Mazurkiewicz ( ] 1 + n 3 + 1), k1 k where k j for k j. Assume now that these formulas holds for every l α, l, α IN, n α 5. Particularly, we can use these formulas for Π n α i ( 1,..., n α i ), where i, 1,, 3. Now our aim is to show that the corresponding formulas stay true also for Π n α i ( 1,..., n α i ) Π n α i+1 ( 1,..., n α i+1 )d n α i+1, where i, 5, 6, 7 with the assumption n α 1 + i. We start with integrating Π n α 3 in order to obtain Π n α, namely Π n α ( 1,..., n α ) Π n α 3 ( 1,..., n α 3 )d n α 3. In the formula of Π n α 3 we assume n α 3 and separate the last epression of the sum from k 1 to k n α 3. In order to integrate this epression with respect to d we need the following identity, crucial for further calculations: ( 1) m+w m 1 j1 ( j ) γ + γ j1 t j γc m 1 (1 + ( 1) γ+j ) j + j where w + γ, constants t, m 1 j m 1 j w+ γ j m 1 j1 ( γ+1a m 1 j ) j + γ+1 B m 1 j, (5) + j, γ C m 1, γ+1 A m 1 are as in theorem and γ+1 B m 1 j ( 1) γ+1 γ+1a m 1 j. It is not difficult to notice that if γ is even, then γ Cj m γ+1 Cj+1 m for j 1,..., [ ] γ 1 and m IN. Another very important for further calculations formulas are the following relations: γc m jd m γ γc m 1 i ij j j (i j) m d m for even γ (6) and m j d m j i m 1 i (j i) m d m. (7)

11 Weakly stable vectors and magic distribution 985 We can also simplify some calculations if we notice that for n IN we have n 1 d du and n ln 1 + un+ 1 d 1 ln 1 + u un+ 1 u du, where u 1. Moreover it is easy to notice that d y 1 y ln 1 + y 1 y and This implies that Moreover <1 >1 d y 1 y ln 1 + y 1 y for y, for y. d for y. (8) y ( 1 y + 1 ) ln y 1 d π (y 1) for y, where (a) for a and (a) a for a < (see [9]). Notice also that 1 (y 1) (y 1) +, where (a) + for a and (a) + a for a >. Using all these facts we obtain that Π n α is equal to the sum of the epression given in theorem 1 for k (α 1) and some integral formula I. This formula has a form: [ (1 ) ( ) I q + α+3cα+3 m 1 α + 1 α+1 ln α j1 α j1 ( m 1 j + α+3 Cα j+3) m 1 α j+1 ln ( α+cα j+ m j m α j + 1 where m n α 3, q [Γ( n )] (n )!π n α. Then for m the relations (6) and (7) imply the following: ( α+c m α j+ m j ) d j i ( m 1 i The remark given after (5) allows us to write ( α+c m α j+ m j ) d j i ( m 1 i ) ] d, (9) + α+ C m 1 α i+) (j i) d. + α+3 C m 1 α i+3) (j i) d. (1)

12 986 G. Mazurkiewicz Another crucial relation is the following identity: α j1 1 α j + 1 α j1 ( m 1 j j i ( m 1 i ) + α+3 C m 1 α i+3 (j i) ) α j + α+3 C m 1 (α j i) α j+3 i + 1 i + ( ) α α+3 Cα+3 m 1 (α i) i + 1. (11) i Now we must apply (1) and (11) in (9) and make order to observe that [ (1 I q + α+3cα+3) ( ) α m 1 (α i) i + 1 (α)+1 ln i α ( + m 1 j + α+3 Cα j+3) ( α j ] m 1 (α j i) (α j)+1 ln 1 + i ) d. j1 i In order to calculate I we tried to find the proper integral formulas in [1] or [9], but we failed. It occurs that we should use some trick to prove that I. First of all we must recall that we know all formulas of Π n l 3, Π n l 1 for l α and these are densities functions. Hence if we take n l + and n l + respectively, and notice that Π (l+) l 3d 1 and Π (l+) l 1d 1, then we will obtain the whole series of new integration formulas, which have the form: ( ) l (l i) i + 1 l+1 ln d for l α, l IN. i All of them must be used to show that I. In this way we proved that Π n α has an appropriate form. Now we will discuss the proof of formula of Π n α 5 ( 1,,..., n α 5 ) Π n α ( 1,,..., n α )d n α. As before we must separate the last epression of the sum from k 1 to k n α. Assume that n α. uring integrating this epression with respect to d observe that <1 ( 1) + h()d for every function h:. The property (8) allows us to simplify some terms of the sum from k 1 to k n α 5. This time we must also apply proper versions of (5) and use (7). We should also notice that ( 1 y 1 ) d ln 1 + y + y 1 y for y 1

13 Weakly stable vectors and magic distribution 987 (this is why in formulas of the form Π n l 1 appear logarithms). In this way we obtain the appropriate formula of Π n α 5. The procedure of calculations of Π n α 6 is very similar to the procedure of calculations of Π n α. This time we must add one more integration formula - this, which follows from the equation Π (α+6) α 5d 1. We can also observe that the way of calculations of Π n α 7 is very similar to the one of Π n α 5. Every density Π n k 1 was used in calculations of Π n k. Hence we obtained a new integral formula, which cannot be found in I. S. Gradshteyn, I. M. Ryzhik ([1]) or A. P. Prudnikow, J. A. Bryczkow, O. I. Mariczew ([9]) tables, namely Corollary. ( ) n (n i) i + 1 n+1 ln d for n IN. i The formulas of Π n k are very long and very complicated and it can be difficult to make them useful, ecept that now calculations can be done by the computer. This way we can obtain the plots of the marginal densities in and, which are very interesting and beautiful - see the pictures below. Especially for Π 3 1 it is hard to believe that all one dimensional projections of this measure are the same up to a scale parameter, which means that every distribution of random variable t, X, t n, is the same as distribution of c(t)x 1, where a random variable X 1 has a density function Π n (n 1). And this is just the most useful and interesting property of l 1 -symmetric distributions. It is easy to find a dependency between these plots. Plot of density Π n (n 1) in is analogous to the plot of density Π (n+1) (n 1) in. This property was a great help in verifying correctness of calculations.

14 988 G. Mazurkiewicz For n y - - The graphic illustration of the density function g (, y) Π (, y) given by S. Cambanis, R. Keener and G. Simons The marginal density Π 1 () Π (, y)dy Level curves of function Π

15 Weakly stable vectors and magic distribution 989 For n y - The density function Π 3 1 (, y) - g 3 (, y, z)dz The marginal density Π 3 () Π 3 1 (, y)dy Level curves of Π 3 1

16 99 G. Mazurkiewicz For n y - The density function Π (, y) - g (, y, z 1, z )dz 1 dz The marginal density Π 3 () Π (, y)dy Level curves of Π

17 Weakly stable vectors and magic distribution 991 For n y - The density function Π 5 3 (, y) - 3 g 5 (, y, z 1, z, z 3 )dz 1 dz dz The marginal density Π 5 () Π 5 3 (, y)dy Level curves of Π 5 3

18 99 G. Mazurkiewicz For n y - The density function Π 6 (, y) - g 6 (, y, z 1,..., z )dz 1... dz The marginal density Π 6 5 () Π 6 (, y)dy Level curves Π 6

19 Weakly stable vectors and magic distribution 993 For n y - - The density function Π n (n ) (, y) lim n g (, y, z 1,..., z n )dz 1... dz n given by Bretagnolle et al. [1]. This is the density function of the random vector (X, Y ) with independent components and identically distributed symmetric Cauchy with parameter α 1. n The marginal density Π n (n 1) () Π n (n ) (, y)dy Level curves Π n (n )

20 99 G. Mazurkiewicz 5 Appendi The proof of S. Cambanis, R. Keener and G. Simons of the equality: s B + t 1 B d ( s + t ), B where B is Beta( 1, 1 ) and s, t was not very difficult but comple. They constructed a special random variable T (θ) cos(θ), with θ having uniform sin(θ) distribution on the interval [, π ]. Then they showed, calculating directly, that the distribution function of T (θ) does not depend on the value [ 1, 1]. Few years ago K. Oleszkiewicz gave a very simple geometrical proof of this equality. With the permission of K. Oleszkiewicz we want to present this proof here since otherwise it would be lost and the beauty of geometrical construction is worth our attention. Consider the unit sphere in, and assume that the chords EF and G are parallel, the points A, B, C, O, P are situated as on the Figure 1. B C X A O P E A r O O P H E G F Figure 1 Figure Let CP AE, OP and P C CP B α. Notice that the length of the chord EF determines the length of the curve EF and it depends only on P EF π α. We obtain B BE E EG F G EF, which means that B depends only on α and do not depend on. Let s, t and consider the sphere S with the radius r s + t. We assume that the random variable θ is uniformly distributed on the interval [, π] and the point H is the projection of X on the diameter AE - see the Figure. Hence the corresponding random vector X is uniformly distributed on S and P ( ctg XOH < u) P ( ctg XP H < u) for all u.

21 Weakly stable vectors and magic distribution 995 This equality follows from the property that the points of the sphere S, for which the inequality ctg XOH < u holds, and the points of the sphere S, for which the inequality ctg XP H < u holds, form subsets of S with the same Lebesgue measure. This implies that cos θ sin θ d r cos θ r sin θ. The same argument can be used to show that r cos θ d r sin θ r cos(θ) r sin(θ) and in consequence we have ( ) r cos(θ) + 1 d r sin(θ) ( ) cos θ + 1. sin θ It is easily seen that right hand side of this equation is equal to 1 (r+) sin θ and the left hand side is equal + (r ). In order to obtain the formula given r cos θ r sin θ by S. Cambanis, R. Keener and G. Simons it is enough to multiply the last equation by r and substitute r s + t, s t. References [1] J. Bretagnolle,. acunha Castelle, J. - L. Krivine, Lois stables et espaces L p, Ann. Inst. H. Poincaré Sect., B (1966), [] S. Cambanis, R. Keener and G. Simons, On α-symmetric Multivariate istributions, J. Multivariate Anal. 13 (1983), [3] W. Feller, An Introduction to Probability Theory and its Applications, vol., John Wiley, New York, [] E. Lukács, Characteristic Functions, Griffin, London, 196. [5] G. Mazurkiewicz, When a scale miture is equivalent with rescaling, Journal of Mathematical Sciences, (), [6] J. K. Misiewicz, K. Oleszkiewicz, K. Urbanik, Classes of measures closed under miing and convolution. Weak stability, Studia Mathematica (5), [7] G. Samorodnitsky, M.S. Taqqu, Stable Non-Gaussian Random Processes, Chapman & Hall, New York, 199. [8] V. M. Zolotarev, One-dimensional Stable istributions, Transl. Math. Monographs 65, Amer. Math. Soc., Providence.

22 996 G. Mazurkiewicz [9] A. P. Prudnikov, J. A. Brychkov, O. I. Marichev, Integrals and Series (in Russian), Nauka, Moscow, [1] I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products, Academic Press, New York, 198. Received: November 3, 6