where A is a symmetric nonnegative-definite d d matrix, γ R d and ν is a measure on R d (called the Lévy measure) satisfying

Transcription

1 PROBABILITY AND MATHEMATICAL STATISTICS Vol. 29, Fasc. (29), pp NESTED SUBCLASSES OF SOME SUBCLASS OF THE CLASS OF TYPE G SELFDECOMPOSABLE DISTRIBUTIONS ON R d BY TAKAHIRO AOYA M A (NODA) Abstract. Nested subclasses, denoted by M n(r d ), n =, 2,..., of the class M(R d ), a subclass of the class of type G and selfdecomposable distributions on R d are studied. An analytic characterization in terms of Lévy measures and a probabilistic characterization by stochastic integral representations for M(R d ) are known. In this paper, analytic characterizations for M n(r d ), n =, 2,..., are given in terms of Lévy measures as well as probabilistic characterizations by stochastic integral representations are shown. A relationship with stable distributions is given. 2 AMS Mathematics Subject Classification: Primary: 6E7; Secondary: 62E. Key words and phrases: Infinitely divisible distribution on R d ; type G distribution; selfdecomposable distribution; stochastic integral representation; Lévy process.. INTRODUCTION Throughout this paper, I(R d ) (resp., I sym (R d )) stands for the class of all infinitely divisible (resp., all symmetric infinitely divisible) distributions on R d. The characteristic function µ(z), z R d, of an infinitely divisible distribution µ I(R d ) has the so-called Lévy Khintchine representation in the form: µ(z) = exp [ 2 z, Az + i γ, z + R d ( e i z,x i z, x ( + x 2 ) ) ν(dx) ], z R d, where A is a symmetric nonnegative-definite d d matrix, γ R d and ν is a measure on R d (called the Lévy measure) satisfying ν({}) = and R d ( x 2 )ν(dx) <.

2 36 T. Aoyama The triplet (A, ν, γ) is called the generating triplet of µ I(R d ). Consider a polar decomposition of ν given by (.) ν(b) = S λ(dξ) B (rξ)ν ξ (dr), where S is the unit sphere in R d, λ is a measure on S with < λ(s) and {ν ξ : ξ S} is a family of measures on (, ) such that ν ξ (B) is measurable in ξ for each B B ( (, ) ), < ν ξ ( (, ) ) for each ξ S. Here λ and {νξ } are uniquely determined by ν up to multiplication of a measurable function c(ξ) and c(ξ) with < c(ξ) <. λ is called the spherical component of ν and ν ξ the radial component. We will say that ν has the polar decomposition (λ, ν ξ ). (See, e.g., [3] and [7].) Let C µ (z) = log µ(z), z R d, be the cumulant of µ I(R d ). We can characterize five classes of infinitely divisible distributions in terms of the radical component ν ξ. (i) Class U(R d ) (Jurek class, see [5]): ν ξ (dr) = l ξ (r)dr and l ξ (r) is measurable in ξ S and nonincreasing in r for λ-a.e. ξ. (ii) Class B(R d ) (Goldie Steutel Bondesson class, see, e.g., [3]): ν ξ (dr) = l ξ (r)dr and l ξ (r) is measurable in ξ S and completely monotone in r for λ-a.e. ξ. (iii) Class L(R d ) (class of selfdecomposable distributions, see, e.g., [8]): ν ξ (dr) = k ξ (r)r dr and k ξ (r) is measurable in ξ S and nonincreasing in r for λ-a.e. ξ. (iv) Class T (R d ) (Thorin class, see, e.g., [3]): ν ξ (dr) = k ξ (r)r dr and k ξ (r) is measurable in ξ S and completely monotone in r for λ-a.e. ξ. (v) Class G(R d ) (class of type G distributions, see, e.g., [4]): µ I sym (R d ), ν ξ (dr) = g ξ (r 2 )dr and g ξ (r) is measurable in ξ S and completely monotone in r for λ-a.e. ξ. We have introduced a class named M(R d ) in the previous paper [2], which is a subclass of type G and selfdecomposable distributions on R d. Its definition is the following. DEFINITION. (Class M(R d )). µ M(R d ) if µ I sym (R d ) and (.2) ν ξ (dr) = g ξ (r 2 )r dr, where g ξ (r) is measurable in ξ S and completely monotone in r for λ-a.e. ξ. We call g ξ (r) in (.2) the g-function of ν (or µ). Denote by L(X) the law of a random variable X on R d, and for µ I(R d ) let {X (µ) t } stand for a Lévy process with L(X (µ) ) = µ. As to the definition of stochastic integrals of nonrandom functions with respect to Lévy processes {X t } on R d, we follow the definition in [9] and [], whose idea is to define integrals with respect to R d -valued independently scattered random

3 Subclasses of type G selfdecomposable distributions 37 measure induced by a Lévy process on R d. This idea was used in [] and [6] for the case d =. See also [3]. Let I log (R d ) = { µ I(R d ): log x µ(dx) < }, x > φ(x) = (2π) /2 exp( x 2 /2), m(x) = φ(u)u du, x >, and let us denote the inverse of m(x) by m (t), that is, t = m(x) if and only if x = m (t). In [2], we have shown that the stochastic integral m (t)dx (µ) t exists and is finite a.s. for any µ I log (R d ). Thus we can define the following mapping. DEFINITION.2 (M-mapping). For any µ I log (R d ), we define the mapping M by M(µ) = L ( m (t)dx (µ) ) t. One of the results in [2] was the following PROPOSITION.. We have M(R d ) = M ( I log (R d ) ). It is trivial by the definition that M(R d ) is a subclass of the class of type G and selfdecomposable distributions. However, we have more. Namely, M(R d ) is a proper subclass. Actually, in [2] we gave an example of µ which belongs to L(R d ) G(R d ) but does not belong to M(R d ). x 2. NESTED SUBCLASSES OF M(R d ) AND THEIR LÉVY MEASURES In this section, we construct nested subclasses of M(R d ) as follows. Write M (R d ) = M(R d ). We start with the following PROPOSITION 2. (Aoyama et al. [2]). Let ν and ν be the Lévy measures of µ I log (R d ) and µ := M(µ) M (R d ), respectively. Then (2.) ν (B) = ν(u B)φ(u)u du, B B (R d ). We define nested subclasses of M(R d ) in terms of their Lévy measures. DEFINITION 2. (Class M n (R d )). For any n N, define M n (R d ) = {µ M (R d ): ν in (2.) is the Lévy measure of some distribution in M n (R d )}. M (R d ) is defined by n= M n(r d ).

4 38 T. Aoyama For nonnegative integer n and x >, let η n (x) be the probability density functions of 2 (n+) Z Z... Z n, where Z i are independent standard normal random variables. (2.2) REMARK 2.. () lim x + η n (x)x = and lim x η n (x)x =. (2) η (x) = φ(x) and for n N η n (x) = φ(xu )η n (u)u du. (3) η n (x) can be written as follows: η n (x) =... φ(u )u du φ(u n )u n du n φ (x ( n ) ) u i φ(u n )u n du n. i= P r o o f. We have (2) and (3) inductively. () can be shown as follows. For < x, η n (x)x x and for any x >, η n (x)x x φ(u )u du... φ(u n )u n du n (as x +),... φ(u )u du ( ( n ) ) φ u i i= φ(u n )u n du n 2x 2( n i= = 2(2π) n/2 x (as x ). Then we have the following φ(u n )u du n n ) 2φ(un u i )u n du n THEOREM 2. (A characterization of the Lévy measures of µ n M n (R d )). Let µ n I sym (R d ), n =, 2,..., and denote its Lévy measure by ν n. Then µ n M n (R d ) if and only if (2.3) ν n (B) = ν (u B)η n (u)u du, B B (R d ), where ν is the Lévy measure of some µ M (R d ).

5 Subclasses of type G selfdecomposable distributions 39 P r o o f. (i) The only if part. Let n = and suppose µ M (R d ). Then, by the definition, ν (B) = ν (u B)φ(u)u du for some Lévy measure ν whose distribution is in M (R d ). We are going to show the assertion by induction. Suppose that the assertion is true for some n N. Namely, suppose the Lévy measure ν n of µ n M n (R d ) is given by ν n (B) = ν (u B)η n (u)u du. Suppose µ n+ M n+ (R d ) and denote its Lévy measure by ν n+. Then (2.4) ν n+ (B) = ν n (u B)φ(u)u du (by the definition of M n+ (R d )) (2.5) = φ(u)u du ν (u v B)η n (v)v dv = η n (v)v dv ν (y B)φ(yv )y dy = ν (y B)y dy η n (v)φ(yv )v dv = ν (y B)η n (y)y dy (by (2.2)). This shows that the assertion is also true for n +. (ii) The if part. The assertion is true for n =. Namely, by the definition of M (R d ), if ν (B) = ν (u B)φ(u)u du for some ν, the Lévy measure of some µ M (R d ), then µ whose Lévy measure is ν belongs to M (R d ). Suppose that the assertion is true for some n N and suppose that µ n+ I sym (R d ) have the Lévy measure ν n+ (B) = ν (u B)η n (u)u du. Then from the calculation from (2.4) to (2.5) we have ν n+ (B) = φ(u)u du ν (v B)η n (v)v dv = φ(u)u ν n (u B)du and µ n with the Lévy measure ν n belongs to M n (R d ) by the induction hypothesis. Thus µ n+ M n+ (R d ) follows from Definition 2.. This completes the proof.

6 4 T. Aoyama The following is a characterization of the Lévy measures of distributions in M n (R d ) in terms of the g-function of the Lévy measure. THEOREM 2.2. Let n N. A measure µ n I sym (R d ) belongs to M n (R d ) if and only if its Lévy measure ν n is either zero or it can be represented as ν n (B) = S where g n,ξ (r) is represented as λ(dξ) B (rξ)g n,ξ (r 2 )r dr, B B (R d ), (2.6) g n,ξ (s) = η n (s /2 y )g ξ (y 2 )y dy. Here g ξ (r) is measurable in ξ S and completely monotone in r for λ-a.e. ξ. P r o o f. Recall from (.) and (.2) that ν (B) = S λ(dξ) B (rξ)g ξ (r 2 )r dr. We see by Theorem 2. that µ n M n (R d ) if and only if ν n is represented as ν n (B) = ν (u B)η n (u)u du = η n (u)u du S λ(dξ) This completes the proof. λ(dξ) u B(yξ)g ξ (y 2 )y dy B (rξ)r dr η n (ry )g ξ (y 2 )y dy = S = λ(dξ) B (rξ)g n,ξ (r 2 )r dr. S 3. STOCHASTIC INTEGRAL CHARACTERIZATIONS OF M n(r d ), n N In this section, we characterize distributions in M n (R d ) by stochastic integral representations. Let I log n(r d ) = { µ I(R d ) : x > (log x )n µ(dx) < } and m n (x) = x η n(u)u du, x >. Since m n (x) is strictly monotone, we can define its inverse by m n(t), that is, t = m n (x) if and only if x = m n(t).

7 Subclasses of type G selfdecomposable distributions 4 LEMMA 3.. For each n N there exists C i > (i =, 2, 3) such that for every < u < η n (s)s ( (3.) ds C log(u ) n+ + ), (3.2) and (3.3) u u η n (s)ds C 2 u, u sη n (s)ds C 3 u 2. P r o o f. We have (3.2) and (3.3) by standard calculations. For n N and < u <, we have η n (s)s ds u = = φ(u )u du... φ(u )u du... C ( (log s )s ds ) + C u C ( (log u ) n+ + ), φ(u n )u n du n φ (s ( n ) u i )ds u i= ) φ (s ( n φ(u n )u ( n du n + u i= u i ) )ds where and in what follows C will denote an absolute positive constant which may be different from one to another. Thus we have (3.). This completes the proof. THEOREM 3.. For each n N the stochastic integral m n(t)dx (µ) t exists for every µ I log n+(r d ). P r o o f. For the proof, we need the following lemma, which is a special case of Proposition 5.5 of []. LEMMA 3.2. Let {X (µ) t } be a Lévy process on R d and f(t) a real-valued measurable function on [, ). Let (A, ν, γ) be the triplet of µ. Then f(t)dx (µ) t exists if the following conditions are satisfied: (3.4) f(t) 2 dt <,

8 42 T. Aoyama and (3.5) (3.6) dt ( f(t)x 2 ) ν(dx) <, R d f(t)γ + f(t) ( x( + f(t)x 2 ) ( + x 2 ) ) ν(dx) dt <. R d For the proof, it is enough to show that f(t) = m n(t) satisfies (3.4) (3.6) in Lemma 3.2 for every µ I log n+(r d ). Note that m n (+) = and m n ( ) =. Since m n(t) 2 dt = s 2 η n (s)s ds we have (3.4). As to (3.5), we have say. Here = 2 (n+) E( Z Z... Z n ) = (2π) (n+)/2 <, dt ( m n (t)x 2 ) ν(dx) = dm n (s) ( sx 2 )ν(dx) R d R d = η n (s)s ds ( sx 2 ν(dx) + ν(dx) ) =: I + I 2, x /s I = / x x 2 ν(dx) sη n (s)ds R d = ( x + x > ) / x x 2 ν(dx) x >/s sη n (s)ds =: I + I 2, say, and I x x 2 ν(dx) sη n (s)ds <. We have the finiteness of I 2 by (3.3) in Lemma 3.. Also, I 2 = ν(dx) η n (s)s ds = ( + ) ν(dx) η n (s)s ds R d =: I 2 + I 22, / x x x > / x

9 say. As to I 2, we have I 2 ν(dx) C x x Subclasses of type G selfdecomposable distributions 43 φ(u )u du... x 2 ν(dx) <. φ(u n )u ( n n du n i= u i ) 2 / x 2s 3 ds We have the finiteness of I 22 by (3.) in Lemma 3.. For (3.6), we have m n(t)γ + m n(t) ( x( + m n (t)x 2) ( + x 2 ) ) ν(dx) dt R d γ sdm n (s) s x ( ( + sx 2 ) ( + x 2 ) ) ν(dx) dmn (s) =: I 3 + I 4, R d say, where I 3 γ η n (s)ds <, I 4 η n (s)ds ((x x 2 s 2 ) ( ( + sx 2 )( + x 2 ) ) ) ν(dx) R d say. Here and s 2 η n (s)ds x 3 ( ( + sx 2 )( + x 2 ) ) ν(dx) R d = s 2 η n (s)ds ( =: I 4 + I 42, I 42 x + x > I 4 s 2 η n (s)ds x > = x > x ) x 3 ( ( + sx 2 )( + x 2 ) ) ν(dx) x 3 ( + x 2 ) ν(dx) <, x 3 ( + x 2 ) ν(dx) (s 2 + )( + sx 2 ) η n (s)ds x 3 ( + x 2 ) ν(dx) ( ) + (s 2 + )( + sx 2 ) η n (s)ds =: I 42 + I 422,

10 44 T. Aoyama say. Furthermore, I 42 = x > = x > x 3 ( + x 2 ) ν(dx) (s 2 + )( + sx 2 ) η n (s)ds x 3 ( + x 2 ) ν(dx) ( / x =: I 42 + I 422, + / x ) (s 2 + )( + sx 2 ) η n (s)ds say. We have I 42 x > by (3.2) in Lemma 3., and I 422 x > x > ν(dx) ν(dx) / x x ν(dx) η n (s)ds C / x / x by (3.) in Lemma 3.. Also I 422 = x > x > x > ν(dx) < ( sx (s 2 + ) ) ( + sx 2 ) η n (s)s ds η n (s)s ds x > ν(dx) / x η n (s)s ds < x 3 ( + x 2 ) ν(dx) (s 2 + )( + sx 2 ) η n (s)ds x 3 ( + x 2 ) 2 ν(dx) (s 2 + )η n (s)ds <. Thus we have (3.6). This completes the proof. Let M = M = M. DEFINITION 3.. Let n N. Define the mapping M n+ by M n+ (µ) = L ( m n(t)dx (µ) ) t, µ Ilog n+(r d ), and let M n+ be the (n + ) times iteration of M. That is, M n+ (µ) can be defined with M n+ (µ) = M ( M n (µ) ) if and only if M n (µ) is defined and belongs to I log (R d ).

11 THEOREM 3.2. For n N Subclasses of type G selfdecomposable distributions 45 M n (R d ) = M ( M n (R d ) I log (R d ) ). P r o o f. The proof is almost the same as that of Theorem 2.4 (i) in [2]. Let µ n M n (R d ) I log (R d ) and µ n = M(µ n ). Also, let ν n and ν n be the Lévy measures of µ n and µ n, respectively. Then, by Proposition 2., we have ν n (B) = ν n (s B)φ(s)s ds. Thus µ n M n (R d ) by Definition 2., and M ( M n (R d ) I log (R d ) ) M n (R d ). Conversely, suppose that µ n M n (R d ). Then, by the definition of M n (R d ) and Proposition 2. again, we see that µ n = L ( m (t)dx (µ) ) t for some µ M n (R d ) I log (R d ). This means that µ n M ( M n (R d ) I log (R d ) ), and M n (R d ) M ( M n (R d ) I log (R d ) ), completing the proof. COROLLARY 3.. For n N We next show THEOREM 3.3. For n N M n (R d ) = M n+( I log n+(r d ) ). M n+ ( Ilog n+(r d ) ) = M n+( I log n+(r d ) ). P r o o f. We note that µ M n+ ( Ilog n+(r d ) ) if and only if µ = L ( m n(t)dx (µ) ) t, µ Ilog n+(r d ), and that µ M n+( I log n+(r d ) ) if and only if µ = L ( We next claim that, for any µ I log n+(r d ), (3.7) m (t)dx (µ) ) t, µ Mn (R d ) I log (R d ). φ(u)u du C µ (uvz) η n (v)v dv <, z R d. If it is proved, we can exchange the order of the integrals in the calculation of cumulants below.

12 46 T. Aoyama The proof of (3.7) is as follows. The idea is from Barndorff Nielsen et al. [3]. If the generating triplet of µ is (A, ν, γ), then where Hence C µ (z) 2 (tra) z 2 + γ z + R d g(z, x) ν(dx), g(z, x) = e i z,x i z, x ( + x 2 ). C µ (uvz) 2 (tra)u 2 v 2 z 2 + γ u v z + R d g(z, uvx) ν(dx) + R d g(uvz, x) g(z, uvx) ν(dx) =: J + J 2 + J 3 + J 4, say. The finiteness of φ(u)u du (J + J 2 )η n (v)v dv is easily to be shown by the same calculation as in the proof of Theorem 3.. Noting that g(z, x) C z x 2 ( + x 2 ) with a positive constant C z depending on z, we have φ(u)u du J 3 η n (v)v dv say, and C z ν(dx) φ(u)u du uvx 2 ( + uvx 2 ) η n (v)v dv R d = C z ν(dx) sx 2 ( + sx 2 ) η n (s)s ds R d ( = C z ν(dx) + ν(dx) ) sx 2 ( + sx 2 ) η n (s)s ds x =: J 3 + J 32, J 3 C z J 32 = C z say. We have x x > =: J 32 + J 322, x > x 2 ν(dx) sη n (s)ds <, ν(dx) ( / x J 32 2 x > + / x ) sx 2 ( + sx 2 ) η n (s)s ds / x x ν(dx) η n (s)ds <, by the finiteness of I 42 in the proof of Theorem 3..

13 Then Subclasses of type G selfdecomposable distributions 47 Also, we have the finiteness of J 322 by (3.) in Lemma 3.. As to J 4, note that for a > g(az, x) g(z, ax) = az, x x 2 a 2 ( + x 2 ) ( + a x 2 ) φ(u)u du z x 3 a( + a 2 )( + x 2 ) ( + a x 2 ). J 4 η n (v)v dv z ν(dx) φ(u)u du R d x 3 uv( + u 2 v 2 )( + x 2 ) ( + u 2 v 2 x 2 ) η n (v)v dv = z ν(dx) x 3 s( + s 2 )( + x 2 ) ( + sx 2 ) η n (s)s ds R d = z ( x =: J 4 + J 42, say. Here and J 4 z J 42 = z + x > x 2 z 2 z x > ) ν(dx) x 3 ( + s 2 )( + x 2 ) ( + sx 2 ) η n (s)ds x 2 ν(dx) x ( + s 2 )( + x 2 ) ( + sx 2 ) η n (s)ds x 2 ν(dx) ( + s 2 )( + sx 2 ) η n (s)ds x x x 2 ν(dx) ( + s 2 )η n (s)ds <, x 3 ( + x 2 ) ν(dx) ( + s 2 )( + sx 2 ) η n (s)ds <. The finiteness of J 42 follows from I 42 in the proof of Theorem 3.. This completes the proof of (3.7). If we calculate the necessary cumulants, we have ( C Mn+ (µ)(z) = C µ m n (t)z ) dt = C µ (uz)dm n (u) = C µ (uz)η n (u)u du,

14 48 T. Aoyama C M n+ (µ)(z) = C M (µ)( n m (t)z ) dt = = dm(u) C µ (uvz)dm n (v) = φ(u)u du ( dt C µ m (t)m n (s)z ) ds C µ (uvz)η n (v)v dv = C µ (yz)y dy φ(yv )η n (v)v dv = C µ (yz)η n (y)y dy = C Mn+ (µ)(z). This completes the proof of Theorem 3.3. The following is a goal of this section and an M n -version of Proposition.. Namely, any µ M n (R d ) has the stochastic integral representation defined in Definition 3.. THEOREM 3.4. We have M n (R d ) = M n+ ( Ilog n+(r d ) ). P r o o f. The statement is an immediate consequence of Corollary 3. and Theorem 3.3. THEOREM 4.. We have 4. THE CLASS M (R d ) M (R d ) S sym (R d ), where S sym (R d ) is the class of all symmetric stable distributions on R d. P r o o f. Let n. When µ A is Gaussian with zero mean and covariance matrix A, suppose {X t } is a Gaussian Lévy process such that the covariance matrix of X is c n A, where c n = m n(t) 2 dt. Then we have µ A = L ( m ) n(t)dx t Mn (R d ) for any n. Hence µ M (R d ). When µ is non-gaussian α-stable with the Lévy measure ν, we have ν(b) = λ(dξ) B (rξ)r (+α) dr = λ n (dξ) B (rξ)c n r (+α) dr, S S

15 Subclasses of type G selfdecomposable distributions 49 where c n = m n (t) α dt and λ n (dξ) = c n λ(dξ). We also have c n r (+α) = r (+α) u α dm n (u) = r (ur ) α η n (u)u dt where = r η n (ry )y (+α) dy = r η n (ry )g(y 2 )y dy, g(s) = s α/2, which is completely monotone. Thus, by Theorem 2.2, c n r (+α) can be regarded as g n,ξ (r)r, implying that ν is the Lévy measure of a distribution in M n (R d ). This is true for all n, and thus µ M (R d ). 5. MORE ABOUT THE CLASSES M n(r d ) WHEN d = When d =, it is known that µ is of type G if and only if µ = L(V /2 Z) for some infinitely divisible nonnegative random variable V independent of the standard normal random variable Z. That is, µ is a variance mixture of normal distributions. And in [2], we showed the following PROPOSITION 5.. µ M(R) if and only if µ = L(V /2 Z), where L(V ) I(R + ) has an absolutely continuous Lévy measure ν V of the form (5.) ν V (dr) = l(r)r dr, r >, and the function l is given by (5.2) l(r) = (x r) /2 ρ(dx), where ρ is a measure on (, ) satisfying the integrability condition (5.3) r x /2 ρ(dx) + ( + log x)x /2 ρ(dx) <. We characterize the distribution of the random variance V in the case of µ M n (R).

16 5 T. Aoyama THEOREM 5.. Let n =, 2,... A necessary and sufficient condition for that µ M (R) belongs to a smaller class M n (R) is the following: (5.4) { ρ(dx) = 2 (2πx) /2 φ(u )u du... φ(u n )u n 2 du n 2 ( φ(u n )u n g x ( n ) 2 ) u i du n }dx, i= where g( ) is completely monotone. The proof is almost the same as that of Theorem 5.2 in [2]. P r o o f. (i) The only if part. Suppose µ M n (R). Since M n (R) G(R d ), we have µ = L(V /2 Z) for some V I(R + ). Thus, we get for z R E[exp(izV /2 Z)] = E[exp( V z 2 /2)] = exp { 2 Az 2 ( + exp( vz 2 /2) ) ν V (dv) } + = exp { 2 Az 2 + ν V (dv) = exp { 2 Az ( exp(izx) ) dx ( exp(izv /2 u) ) φ(u) du } where A. Therefore, the Lévy measure ν of µ is of the form (5.5) ν(dx) = ( + + φ(v /2 x)v /2 ν V (dv) ) dx. φ(v /2 x)v /2 ν V (dv) }, By Theorem 2.2, µ M n (R) if and only if ν(dx) = x g n (x 2 )dx, where g n is given by (2.6). Since µ M (R d ), g n is completely monotone. It can be written as g n (r) = e ry/2 Q(dy), r >, for a measure Q on (, ) given by Q(dy) = (2π) /2 (2y) { ( φ(u n )u n g where g( ) is completely monotone. φ(u )u du... φ(u n )u n 2 du n 2 y ( n i= u i ) 2 ) du n }dy,

17 Subclasses of type G selfdecomposable distributions 5 By (5.5), we get (5.6) φ(v /2 x)v /2 ν V (dv) = x g n (x 2 ). Since we obtain + r /2 = (2π) /2 e rw/2 w /2 dw, r >, r /2 g(r) = (2π) /2 e r(w+y)/2 w /2 dwq(dy) Taking x = r /2 > in (5.6), we get (5.7) (2π) /2 Let (5.8) + = (2π) /2 Q(dy) e ru/2 (u y) /2 du y = (2π) /2 u e ru/2 du (u y) /2 Q(dy). e r/2v v /2 ν V (dv) ρ(dx) = x /2 Q ( d(x ) ) = 2 (2πx) /2 { ( φ(u n )u n g Then l(r) in (5.2) becomes = (2π) /2 u e ru/2 du (u y) /2 Q(dy). φ(u )u du... φ(u n )u n 2 du n 2 ) ) 2 u i du n }dx. x ( n i= l(r) = (x r) /2 x /2 Q ( d(x ) ) r = (y r) /2 y /2 Q(dy) r r = ( yr) /2 r /2 Q(dy) = r (r y) /2 Q(dy). Thus, by (5.7), e r/2v v /2 ν V (dv) = e ru/2 u /2 l(u ) du +

18 52 T. Aoyama or Therefore, + e r/2v v /2 ν V (dv) = e r/2v v 3/2 l n (v) dv, r >. v /2 ν V (dv) = v 3/2 l(v) dv, v >, which yields (5.). The integrability condition (5.3) for Q is obtained from the fact that > (x 2 ) ν(dx) = x R R( x )g n (x 2 )dx. For, this yields that and and hence xdx exp( x 2 y/2) Q(dy) < x dx exp( x 2 y/2) Q(dy) <, [ y ( exp( y/2) ) + 2 u exp( u/2) du ] Q(dy) <. It is obvious that the above condition is equivalent to y (5.9) ( + log y )Q(dy) + y Q(dy) <. On the other hand, and x /2 ρ(dx) = xq ( d(x ) ) = y Q(dy) ( + log x)x /2 ρ(dx) = ( + log x)q ( d(x ) ) = ( + log y )Q(dy). Thus, we get (5.3) from (5.9) and (5.4) by (5.8). The only if part is thus proved. (ii) The if part. Suppose µ = L(V /2 Z) and the Lévy measure ν V of V satisfies (5.) (5.3).

19 Subclasses of type G selfdecomposable distributions 53 We first claim that the integrability condition (5.3) implies that ν V is really a Lévy measure on (, ) of a positive infinitely divisible random variable, namely it satisfies (5.) (r )ν V (dr) <. We have (r )ν V (dr) = As to the first integral, we have rν V (dr) = l(r)dr = = rν V (dr) + ν V (dr). dr (x r) /2 ρ(dx) r x ρ(dx) (x r) /2 dr + ρ(dx) (x r) /2 dr = 2 x /2 ( ρ(dx) + 2 x /2 (x ) /2) ρ(dx) 2 x /2 ρ(dx) + const x /2 ρ(dx) = 2 xq ( d(x ) ) const Q ( d(x ) ) = 2 x Q(dx) + const Next, as to the second integral, we obtain ν V (dr) = r l(r)dr = Q(dx). r dr (x r) /2 ρ(dx) r x = ρ(dx) r (x r) /2 dr = (log x + const)x /2 ρ(dx) = (log x + const)q ( d(x ) ) = (log x + const)q(dx). Therefore, (5.3) implies (5.). Furthermore, as we have already seen, ν µ is expressed as in (5.5). So, to complete the proof, it is enough to show that when we put g n (x 2 ) = x φ(v /2 x)v /2 ν V (dv),

20 54 T. Aoyama then g n (r) is as (2.6) in Theorem 2.2. However, for that, it is enough to follow the proof of the only if part from bottom to top. This completes the proof. Acknowledgements. The author would like to express his sincere appreciation to Makoto Maejima for his valuable comments during the work on this paper. He is also grateful to Jan Rosiński, Ken-iti Sato, Toshiro Watanabe and a referee for their many helpful comments. REFERENCES [] T. Aoyama and M. Maejima, Characterizations of subclasses of type G distributions on R d by stochastic integral representations, Bernoulli 3 (27), pp [2] T. Aoyama, M. Maejima and J. Rosiński, A subclass of type G selfdecomposable distributions, J. Theoret. Probab. 2 (28), pp [3] O. E. Barndorff- Nielsen, M. Maejima and K. Sato, Some classes of multivariate infinitely divisible distributions admitting stochastic integral representation, Bernoulli 2 (26), pp. 33. [4] W. Feller, An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edition, Wiley, 966. [5] Z. J. Jurek, Relations between the s-selfdecomposable and selfdecomposable measures, Ann. Probab. 3 (985), pp [6] B. Rajput and J. Rosiński, Spectral representations of infinitely divisible processes, Probab. Theory Related Fields 82 (989), pp [7] J. Rosiński, On series representations of infinitely divisible random vectors, Ann. Probab. 8 (99), pp [8] K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, 999. [9] K. Sato, Stochastic integrals in additive processes and application to semi-lévy processes, Osaka J. Math. 4 (24), pp [] K. Sato, Additive processes and stochastic integrals, Illinois J. Math. 5 (Doob Volume) (26), pp [] K. Urbanik and W. A. Woyczyński, Random integrals and Orlicz spaces, Bull. Acad. Polon. Sci. 5 (967), pp Department of Mathematics Faculty of Science and Technology Tokyo University of Science 264, Yamazaki, Noda , Japan aoyama_takahiro@ma.noda.tus.ac.jp Received on ; revised version on