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Subordination and self-decomposability
Statistics & Probability Letters 54 (2001) 317 – 324
Subordination and self-decomposability Ken-iti Sato Hachiman-yama 1101-5-103, Tenpaku-ku, Nagoya, 468-0074, Japan Received October 2000; received in revised form May 2001
Abstract Two facts are established concerning subordination and self-decomposability. (1) Any subordinated process arising from a Brownian motion with drift and a self-decomposable subordinator is self-decomposable. (2) Self-decomposable distributions of type G are not necessarily of type GL . Consequences of the 0rst fact on smoothness of the distributions c 2001 Elsevier Science B.V. All rights reserved are discussed. Keywords: Self-decomposable distribution; Subordination; Self-decomposable subordinator; Brownian motion with drift; Distribution of type G
1. Introduction and results A distribution on R is called self-decomposable if, for every b ¿ 1, there exists a distribution b on R such that their characteristic functions (z), ˆ ˆb (z) satisfy (z) ˆ = (b ˆ −1 z)ˆb (z);
z ∈ R:
(1.1)
The class L of self-decomposable distributions is a subclass of the class of in0nitely divisible distributions. Its importance in the theory of L9evy processes, processes of Ornstein–Uhlenbeck type, and selfsimilar additive processes and in applications is now getting greater. See Barndor<-Nielsen and Shephard (2001) and Sato (1999). For other aspects of self-decomposability, see Bondesson (1992) and Jurek and Mason (1993). A L9evy process whose distribution at each t is self-decomposable is called a self-decomposable process. A random variable with a self-decomposable distribution is called a self-decomposable random variable. It is an interesting problem to see whether self-decomposability is preserved under various transformations. In the case of a transformation which preserves self-decomposability, it is also interesting how big the image of the class L is. Similar problems are considered on subclasses of the class L such as the classes Lm . See, for example, Sato and Yamazato (1983) for the transformation from background driving L9evy processes to processes of Ornstein–Uhlenbeck type. In this paper we study two problems of this sort. E-mail address: [email protected] (K. Sato). c 2001 Elsevier Science B.V. All rights reserved 0167-7152/01/$ - see front matter PII: S 0 1 6 7 - 7 1 5 2 ( 0 1 ) 0 0 1 1 0 - 9
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K. Sato / Statistics & Probability Letters 54 (2001) 317 – 324
A relation of subordination and self-decomposability was found by Halgreen (1979) and Ismail and Kelker (1979). They noticed essentially the following fact. Let {Xt : t ¿ 0} and {Zt : t ¿ 0} be the Brownian motion on R and a self-decomposable subordinator, respectively. Let {Yt : t ¿ 0} be the subordinated process arising from them. By this we mean that {Xt } and {Zt } are independent
(1.2)
and that Yt = XZt ;
t ¿ 0:
(1.3)
Then {Yt } is self-decomposable. The process {Yt } is a L9evy process and the procedure above to get {Yt } is Bochner’s subordination. This fact gives the self-decomposability of normal inverse Gaussian distributions. Halgreen (1979) asked a question whether the same conclusion remains true if the Brownian motion is replaced by a Brownian motion with drift. He gave an aMrmative answer under the restriction that the distribution of Z1 is of the class of generalized gamma convolutions (GGC) (see Bondesson, 1992 for de0nition). The same result was obtained also by Shanbhag and Sreehari (1979). In this way the two papers established self-decomposability of generalized hyperbolic distributions. One of the results in this paper is the following aMrmative answer to Halgreen’s question. Theorem 1.1. Let {Xt : t¿0} be a Brownian motion with drift on R and let {Zt : t¿0} be a self-decomposable subordinator. Then the subordinated process {Yt : t ¿ 0} arising from them is self-decomposable. Using the terminology of Barndor<-Nielsen and Shephard (2001), we can express Theorem 1.1 in this way: normal variance-mean mixtures using self-decomposable mixing distributions are self-decomposable. A related paper is Barndor<-Nielsen and Halgreen (1977). The Brownian motion is strictly stable with index 2. Brownian motions with drift are stable with index 2, but not strictly stable. It is known that if the subordinand {Xt } is strictly stable and the subordinator {Zt } is self-decomposable, then the subordinated process {Yt } is self-decomposable. This is another extension of the result of Halgreen and Ismail and Kelker and a multivariate generalization of this fact is given in Theorem 6:1 of Barndor<-Nielsen et al. (2001). In Section 4 we give a simple proof of this fact. We do not know whether Theorem 1.1 can be generalized to the case where the subordinand {Xt } is a stable process which is not strictly stable. If {Xt } is a Brownian motion with drift on Rd , d ¿ 2, then Theorem 1.1 does not hold even under the additional condition that the distribution of Z1 is of GGC. This fact is proved by Takano (1989=90). A random variable Y is called of type G if there are a standard Gaussian random variable X and a nonnegative in0nitely divisible random variable Z such that X and Z are independent
(1.4)
and d
Y = Z 1=2 X;
(1.5)
see Rosinski (1991) for an account of this class. The distribution of Y is also called of type G. In other words it is a normal variance mixture using an in0nitely divisible mixing distribution. It is easy to see that a distribution on R is of type G if and only if is the distribution at time 1 of the subordinated process {Yt } arising from the Brownian motion {Xt } and a subordinator {Zt }. Thus the result of Halgreen and Ismail and Kelker is paraphrased as follows: a random variable Y of type G is self-decomposable if the Z in the de0nition of type G is self-decomposable, that is, if Y is of class GL in the terminology of Jiang (2000). The second result of this paper is that the converse assertion is not true.
K. Sato / Statistics & Probability Letters 54 (2001) 317 – 324
319
Theorem 1.2. There is a self-decomposable random variable Y of type G for which one cannot 3nd a nonnegative self-decomposable Z and a standard Gaussian X satisfying (1:4) and (1:5). In other words, a self-decomposable random variable of type G is not necessarily of type GL . This disproves a conjecture of Jiang (2000, p. 40). Proofs of Theorems 1.1 and 1.2 are given in Sections 2 and 3. 2. Proof of Theorem 1.1 The generating triplet (A; ; ) of an in0nitely divisible distribution on R, by de0nition is a triplet of nonnegative number A, a -0nite measure satisfying ({0}) = 0 and (1 ∧ x2 )(d x) ¡ ∞, and a real number such that A (z) ˆ = exp − z 2 + (eizx − 1 − izx1[−1; 1] (x))(d x) + iz : (2.1) 2 R Here A and are called the Gaussian variance and the L9evy measure of , respectively. A L9evy process {Xt : t ¿ 0} is said to have the generating triplet (A; ; ) if the distribution of X1 has the generating triplet (A; ; ). {Xt } is a subordinator if and only if A = 0, ((−∞; 0)) = 0, (0;1] x(d x) ¡ ∞, and ¿ (0;1] x(d x), that is, (eizx − 1)(d x) + i0 z (2.2) (z) ˆ = exp (0;∞)
with 0 = − (0;1] x(d x) ¿ 0, which is called the drift of the subordinator. See Sato (1999) for detailed exposition. Let {Xt } be a L9evy process on R with generating triplet (A; ; ) and {Zt } be a subordinator with L9evy measure and drift 0 . Let {Yt } be the subordinated L9evy process arising from them. Then the generating triplet (A] ; ] ; ] ) of {Yt } is given as follows: A] = 0 A;
(2.3)
] (B) = 0 (B) + ]
= 0 +
(0;∞)
(0;∞)
s (B)(ds);
(ds)
[−1;1]
xs (d x);
B ∈ B(R \ {0});
(2.4) (2.5)
where s is the distribution of Xs and B(R \ {0}) is the class of Borel subsets of R \ {0}. See Theorem 30:1 of Sato (1999). Lemma 2.1. Fix a self-decomposable process {Xt }. Suppose that; if {Zt } is a self-decomposable subordinator satisfying 0 = 0 and (ds) = s−1 1(0; a] (s) ds for some a ∈ (0; ∞); then the subordinated process {Yt } is self-decomposable. Then; for any choice of a self-decomposable subordinator {Zt }; the subordinated process {Yt } is self-decomposable. Proof. An in0nitely divisible distribution is self-decomposable if and only if its L9evy measure is of the form |x|−1 k(x) d x with k(x) increasing on (−∞; 0) and decreasing on (0; ∞) (we are using the words increase and decrease in the weak sense). See Corollary 15:11 of Sato (1999). Let {Zt } be a self-decomposable subordinator with L9evy measure and drift 0 . Then (ds) = s−1 k(s)1(0; ∞) (s) ds with k(s) decreasing on
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K. Sato / Statistics & Probability Letters 54 (2001) 317 – 324
(0; ∞). The function k(s) is the limit of an increasing sequence of functions kn (s), n = 1; 2; : : : ; of the form mn kn (s) = bn; j 1(0; an; j ] (s); j=1
where mn is a positive integer, 0 ¡ an; 1 ¡ · · · ¡ an; mn , and bn; j ¿ 0 for j = 1; : : : ; mn . Let {Zt(n) } be the subordinator with L9evy measure s−1 kn (s)1(0; ∞) (s) ds and drift 0 . Recall that convolutions of self-decomposable distributions are self-decomposable. By virtue of (2.4), the assumption implies that the subordinated process {Yt(n) } arising from {Xt } and {Zt(n) } is self-decomposable. Since the limit of a sequence of self-decomposable distributions is self-decomposable, it follows that {Yt } is self-decomposable. Proof of Theorem 1.1. Let {Xt }, {Zt }, and {Yt } be the processes in the theorem. Let (A; ; ), , 0 , and (A] ; ] ; ] ) be as above. Then A = 1, = 0, and = 0. By Lemma 2.1 it is enough to prove the theorem under the assumption that (ds) = s−1 1(0; a] (s) ds for some a ∈ (0; ∞) and 0 = 0. Thus, by (2.4), we have ] (d x) = |x|−1 k ] (x) d x with a 1 −(x−s)2 =(2s) ds ] √ e k (x) = |x| : (2.6) s 2 s 0 All we have to show is that k ] (x) is increasing on (−∞; 0) and decreasing on (0; ∞). The discussion for x ¡ 0 is reduced to that for x ¿ 0 by changing to −. So we only consider x ¿ 0. By change of variables x2 =s = u we get ∞ √ 1=2 −1=2 2 ] 2 k (x) = x e−(u −xu ) =2 (x2 =u)−3=2 (x2 =u2 ) du =
x2 =a
∞
x2 =a
e−(u
1=2
−xu−1=2 )2 =2
u−1=2 du:
(2.7)
Hence, if ¡ 0, then k ] (x) is decreasing in x on (0; ∞). Henceforth we assume that ¿ 0. Write b = a−1=2 . We have, from (2.7), ∞ √ d k ] (x) −1=2 2 1=2 −(bx−=b)2 =2 −1 2 = −e 2 (bx) 2b x + e−(u −xu ) =2 (u1=2 − xu−1=2 )(u−1=2 )u−1=2 du dx 2 2 b x ∞ 1=2 −1=2 2 = e−(u −xu ) =2 f(u) du; b2 x 2
(2.8)
where f(u) = − b(1 − 2 x2 u−2 ) + (1 − xu−1 )u−1=2 : If x ¿ b−2 , then, rewriting f(u) as f(u) = −b + 2 bx2 u−2 + u−1=2 − 2 xu−3=2 = −b(1 − b−1 u−1=2 ) − 2 xu−3=2 (1 − bxu−1=2 ) and noticing that 1 − b−1 u−1=2 ¿ 1 − b−2 x−1 ¿ 0 and 1 − bxu−1=2 ¿ 0 for u ¿ b2 x2 , we get d k ] (x)= d x ¡ 0 from (2.8). If 0 ¡ x ¡ b−2 , then b2 x2 ¡ x and we get √ d k ] (x) = I1 + I2 ; 2 dx where x ∞ 1=2 −1=2 2 1=2 −1=2 2 I1 = e−(u −xu ) =2 f(u) du; I2 = e−(u −xu ) =2 f(u) du: b2 x 2
x
K. Sato / Statistics & Probability Letters 54 (2001) 317 – 324
321
The integral I2 is, by change of variables 2 x2 u−1 = v, written as x −1=2 1=2 2 I2 = e−(xv −v ) =2 [ − b(1 − −2 x−2 v2 ) + (1 − −1 x−1 v)−1 x−1 v1=2 ]2 x2 v−2 dv 0
=−
x
0
Thus, we get
e−(v
I1 + I2 = −
1=2
−xv−1=2 )2 =2
b2 x 2
0
=−
b2 x 2
0
f(v) dv:
e−(u
1=2
−xu−1=2 )2 =2
f(u) du
e−(u
1=2
−xu−1=2 )2 =2
(xu−1 − 1)(b(xu−1 + 1) − u−1=2 ) du:
The function xu−1=2 + u1=2 is strictly decreasing for 0 ¡ u ¡ x. Since b2 x2 ¡ x, we have, for 0 ¡ u ¡ b2 x2 , xu−1=2 + u1=2 ¿ b−1 + bx ¿ b−1 : Hence I1 + I2 ¡ 0 for 0 ¡ x ¡ b−2 , which 0nishes the proof that d k ] (x)= d x ¡ 0 for x ¿ 0. 3. Proof of Theorem 1.2 We prepare two simple lemmas. Lemma 3.1. Suppose that Y satis3es (1:4) and (1:5) with a standard Gaussian random variable X and a nonnegative random variable Z. Then the distribution of Z is determined by the distribution of Y . Proof. For any real u we have E[eiuY ] = E[eiuZ
1=2
X
] = E[e−u
2
Z=2
]=
[0;∞)
e−u
2
s=2
PZ (ds);
where PZ is the distribution of Z. Thus the characteristic function of the distribution PY of Y determines the Laplace transform of PZ . Hence PZ is determined by PY . Lemma 3.2. Let a ¿ b ¿ 0 and let 1; 0 ¡ s ¡ a; f(s) = −1; a 6 s ¡ a + b; c; a + b 6 s: with c ¿ 0. Then ∞ e−xs f(s) ds ¿ 0 0
Proof. Since a e−xs ds ¿ 0
we have
a+b 0
a
e
−xs
a+b
for all x ¿ 0:
e−xs ds;
f(s) ds ¿ 0. Hence the assertion is obvious.
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K. Sato / Statistics & Probability Letters 54 (2001) 317 – 324
Proof of Theorem 1.2. By Lemma 3.1 it is enough to construct a self-decomposable subordinated process {Yt } arising from the Brownian motion {Xt } and a non-self-decomposable subordinator {Zt }. Assume, for a moment, that such {Yt }, {Xt }, and {Zt } are found. Then, as in Section 2, the generating triplet (A] ; ] ; ] ) of {Yt } is expressed by the L9evy measure and the drift 0 of {Zt } as A] = 0 ; ] (B) =
(3.1) (0;∞)
(ds)
B
√
1 −x2 =(2s) e d x: 2 s
(3.2)
By the self-decomposability the L9evy measure ] is of the form ] (d x) = |x|−1 k ] (x) d x with k ] (x) being decreasing on (0; ∞). We have k ] (−x) = k ] (x). Assume further that (ds) = s−1 k(s) ds on (0; ∞) with a nonnegative measurable function k(s). Then ∞ 1 −x2 =(2s) k(s) ] √ ds k (x) = x e s 2 s 0 ∞ 1 −1=(2r) k(x2 r) √ = dr (3.3) e r 2 r 0 for x ¿ 0. We have ∞ d k ] (x) 1 −1=(2r) 2 √ e k (x r) dr = 2x dx 2 r 0
(3.4)
for x ¿ 0, supposing that k is di
Lemma 3.2. That is, u ∈ (0; a); u ∈ (a; a + b); u ∈ (a + b; ∞):
This is equivalent to 1 −cs−3=2 ; s ∈ (0; a+b ); 1 k (s) = s−3=2 ; s ∈ ( a+b ; 1a ); −3=2 −s ; s ∈ ( 1a ; ∞): Choose k(s) such that 2cs−1=2 ; k(s) = −2s−1=2 + p; −1=2 2s ;
(3.6)
1 s ∈ (0; a+b ); 1 s ∈ [ a+b ; 1a );
s ∈ [ 1a ; ∞);
(3.7)
K. Sato / Statistics & Probability Letters 54 (2001) 317 – 324
323
1 where p is a constant satisfying −2(a + b)1=2 + p ¿ 0. Then k(s) ¿ 0 on (0; ∞), 0 k(s) ds ¡ ∞, ∞ −1 s k(s) ds ¡ ∞, and (3.6) is satis0ed. De0ne k ] (x) for x ¿ 0 by (3.3) and de0ne k ] (−x) = k ] (x). Then the 1 condition on the interchangeability of the order of integration and di
4. Remarks Let {Xt }, {Zt }, and {Yt } be the processes in Theorem 1.1. Let , , 0 , and (A] ; ] ; ] ) be as in the proof of the theorem. Let PZt and PYt be the distributions of Zt and Yt , respectively. Then, the two functions k(s) and k ] (x) which express (ds) = s−1 k(s) ds and ] (d x) = |x|−1 k ] (x) d x are connected as follows: ∞ 1 −(x−s)2 =(2s) k(s) √ k ] (x) = |x| e ds s 2 s 0 ∞ 1=2 −1=2 2 1 √ e−(u −xu ) =2 k(x2 u−1 )u−1=2 du = 2 0 x ∞ 2 2 e e−(u=2)− x =(2u) k(x2 u−1 )u−1=2 du (4.1) =√ 2 0 for x ∈ R \ {0}. It follows that
k(0+) ∞ −u=2 −1=2 k ] (0+) = k ] (0−) = √ e u du = k(0+); (4.2) 2 0 √ √ ∞ since 0 e−u=2 u−1=2 du = 2)(1=2) = 2 . Notice that the result (4.2) does not depend on . The same result holds also when {Xt } is the Brownian motion ( = 0). From the self-decomposability proved in Theorem 1.1 and from (4.2) follow many properties of PYt . First, PYt is unimodal by Yamazato’s theorem (1978). Second, the smoothness of PYt is expressed by k ] (0+) + k ] (0−) as in Sato and Yamazato (1978). Some of the results are given below; see also Sections 28 and 53 of Sato (1999). Let cZ = k(0+) and cY = k ] (0+) + k ] (0−). Then cY = 2cZ
(4.3)
by (4.2). If cZ and cY are in0nite, then PZt and PYt have C ∞ densities on R for every t ¿ 0. If 0 ¿ 0, then PYt has C ∞ density on R for every t ¿ 0. Assume that k(0+) ¡ ∞ and 0 = 0. Thus, by (2.5), {Yt } has drift 0. De0ne NZ (t) and NY (t) as the integers satisfying NZ (t) ¡ tcZ 6 NZ (t) + 1 and NY (t) ¡ tcY 6 NY (t) + 1. Then, for t ¿ 0, PYt (resp. PZt ) has C NY (t) (resp. C NZ (t) ) density on R but does not have C NY (t)+1 (resp. C NZ (t)+1 ) density on R. In this sense, the distribution of {Yt } is twice as smooth as the distribution of {Zt }. For t 6 1=cY , PYt has mode 0 and the density fY (t; x) of PYt satis0es fY (t; 0+) = ∞ and fY (t; 0−) = ∞. On the other hand, for t ¡ 1=cZ , PZt has mode 0 and fZ (t; 0+) = ∞; for t = 1=cZ , PZt has mode 0 and, in order 1 that fZ (1=cZ ; 0+) = ∞, it is necessary and suMcient that 0 (cZ − k(s))s−1 ds = ∞. In Theorem 1.1 we have treated a Bownian motion with drift, which is not strictly stable. In the case of strictly stable subordinands, more results are known. The following proposition is a special case of Theorem 6:1 of Barndor<-Nielsen et al. (2001). A simpler proof given here is essentially the same as that of Bondesson (1992, p. 19).
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K. Sato / Statistics & Probability Letters 54 (2001) 317 – 324
Proposition 4.1. A subordinated process on R arising from a strictly stable subordinand and a self-decomposable subordinator is self-decomposable. Proof. As before denote by {Xt }, {Zt }, and {Yt } the subordinand, the subordinator, and the subordinated. We assume that {Xt } is strictly stable with index , and that {Zt } is self-decomposable. Let Z = = PZ1 , the distribution of Z1 . Then, for any b ¿ 1, satis0es (1.1). The distribution b is in0nitely divisible (Sato, 1999, Proposition 15:5). Considering the representation (2.2), we can conclude that b is concentrated on [0; ∞). Thus, for each b ¿ 1, there is a subordinator {Zt(b) } independent of {Zt } such that {b−1 Zt + Zt(b) } has the same distribution as {Zt }. Let X = PX1 , the distribution of X1 . Then E[eizYt ] = E[ˆX (z)Zt ] = E[ˆX (z)b c
Since ˆX (z) = ˆX (c E[ˆX (z)b
−1
Zt
1=,
−1
Zt +Zt(b)
] = E[ˆX (z)b
−1
Zt
(b)
]E[ˆX (z)Zt ]:
(4.4)
z) for every c ¿ 0, we have
] = E[ˆX (b−1=, z)Zt ] = E[eib
−1=,
zYt
]:
(4.5)
Since b ¿ 1 is arbitrary and since the second factor of the rightmost member of (4.4) is the characteristic function of another subordinated process, (4.4) and (4.5) show the self-decomposability of {Yt }. The same proposition holds also for processes on Rd , d ¿ 2, which is proved similarly. References Barndor<-Nielsen, O., Halgreen, C., 1977. In0nite divisibility of the hyperbolic and generalized inverse Gaussian distributions. Zeit. Wahrsch. Verw. Gebiete 38, 309–311. Barndor<-Nielsen, O.E., Pedersen, J., Sato, K., 2001. Multivariate subordination, self-decomposability and stability. Adv. Appl. Probab. 33, 160–187. Barndor<-Nielsen, O.E., Shephard, N., 2001. Modelling by L9evy processes for 0nancial econometrics. In: Barndor<-Nielsen, O.E., Mikosch, T., Resnick, S.I. (Eds.), L9evy Processes—Theory and Applications, BirkhTauser, Boston, pp. 283–318. Bondesson, L., 1992. Generalized Gamma Convolutions and Related Classes of Distribution Densities. Lecture Notes in Statistics, Vol. 76. Springer, New York. Halgreen, C., 1979. Self-decomposability of the generalized inverse Gaussian and hyperbolic distributions. Zeit. Wahrsch. Verw. Gebiete 47, 13–17. Ismail, M.E.H., Kelker, D.H., 1979. Special functions, Stieltjes transforms and in0nite divisibility. SIAM J. Math. Anal. 10, 884–901. Jiang, W., 2000. Some simulation-based models towards mathematical 0nance. Ph.D. Thesis, University of Aarhus. Jurek, Z.J., Mason, J.D., 1993. Operator-Limit Distributions in Probability Theory. Wiley, New York. Rosinski, J., 1991. On a class of in0nitely divisible processes represented as mixtures of Gaussian processes. In: Cambanis, S., Samorodnitsky, G., Taqqu, M.S. (Eds.), Stable Processes and Related Topics. BirkhTauser, Boston, pp. 27 – 41. Sato, K., 1999. L9evy Processes and In0nitely Divisible Distributions. Cambridge University Press, Cambridge. Sato, K., Yamazato, M., 1978. On distribution functions of class L. Zeit. Wahrsch. Verw. Gebiete 43, 273–308. Sato, K., Yamazato, M., 1983. Stationary processes of Ornstein–Uhlenbeck type. In: Itˆo, K., Prokhorov, J.V. (Eds.), Probability Theory and Mathematical Statistics, Proceedings of the Fourth USSR–Japan Symposium 1982, Lecture Notes in Math., Vol. 1021. Springer, Berlin, pp. 541–551. Shanbhag, D.N., Sreehari, M., 1979. An extension of Goldie’s result and further results in in0nite divisibility. Zeit. Wahrsch. Verw. Gebiete 47, 19–25. Takano, K., 1989=90. On mixtures of the normal distribution by the generalized gamma convolutions. Bull. Fac. Sci. Ibaraki Univ. Ser. A 21, 29 – 41; Correction and addendum, 22, 49 –52. Yamazato, M., 1978. Unimodality of in0nitely divisible distribution functions of class L. Ann. Probab. 6, 523–531.
Subordination and self-decomposability Ken-iti Sato Hachiman-yama 1101-5-103, Tenpaku-ku, Nagoya, 468-0074, Japan Received October 2000; received in revised form May 2001
Abstract Two facts are established concerning subordination and self-decomposability. (1) Any subordinated process arising from a Brownian motion with drift and a self-decomposable subordinator is self-decomposable. (2) Self-decomposable distributions of type G are not necessarily of type GL . Consequences of the 0rst fact on smoothness of the distributions c 2001 Elsevier Science B.V. All rights reserved are discussed. Keywords: Self-decomposable distribution; Subordination; Self-decomposable subordinator; Brownian motion with drift; Distribution of type G
1. Introduction and results A distribution on R is called self-decomposable if, for every b ¿ 1, there exists a distribution b on R such that their characteristic functions (z), ˆ ˆb (z) satisfy (z) ˆ = (b ˆ −1 z)ˆb (z);
z ∈ R:
(1.1)
The class L of self-decomposable distributions is a subclass of the class of in0nitely divisible distributions. Its importance in the theory of L9evy processes, processes of Ornstein–Uhlenbeck type, and selfsimilar additive processes and in applications is now getting greater. See Barndor<-Nielsen and Shephard (2001) and Sato (1999). For other aspects of self-decomposability, see Bondesson (1992) and Jurek and Mason (1993). A L9evy process whose distribution at each t is self-decomposable is called a self-decomposable process. A random variable with a self-decomposable distribution is called a self-decomposable random variable. It is an interesting problem to see whether self-decomposability is preserved under various transformations. In the case of a transformation which preserves self-decomposability, it is also interesting how big the image of the class L is. Similar problems are considered on subclasses of the class L such as the classes Lm . See, for example, Sato and Yamazato (1983) for the transformation from background driving L9evy processes to processes of Ornstein–Uhlenbeck type. In this paper we study two problems of this sort. E-mail address: [email protected] (K. Sato). c 2001 Elsevier Science B.V. All rights reserved 0167-7152/01/$ - see front matter PII: S 0 1 6 7 - 7 1 5 2 ( 0 1 ) 0 0 1 1 0 - 9
318
K. Sato / Statistics & Probability Letters 54 (2001) 317 – 324
A relation of subordination and self-decomposability was found by Halgreen (1979) and Ismail and Kelker (1979). They noticed essentially the following fact. Let {Xt : t ¿ 0} and {Zt : t ¿ 0} be the Brownian motion on R and a self-decomposable subordinator, respectively. Let {Yt : t ¿ 0} be the subordinated process arising from them. By this we mean that {Xt } and {Zt } are independent
(1.2)
and that Yt = XZt ;
t ¿ 0:
(1.3)
Then {Yt } is self-decomposable. The process {Yt } is a L9evy process and the procedure above to get {Yt } is Bochner’s subordination. This fact gives the self-decomposability of normal inverse Gaussian distributions. Halgreen (1979) asked a question whether the same conclusion remains true if the Brownian motion is replaced by a Brownian motion with drift. He gave an aMrmative answer under the restriction that the distribution of Z1 is of the class of generalized gamma convolutions (GGC) (see Bondesson, 1992 for de0nition). The same result was obtained also by Shanbhag and Sreehari (1979). In this way the two papers established self-decomposability of generalized hyperbolic distributions. One of the results in this paper is the following aMrmative answer to Halgreen’s question. Theorem 1.1. Let {Xt : t¿0} be a Brownian motion with drift on R and let {Zt : t¿0} be a self-decomposable subordinator. Then the subordinated process {Yt : t ¿ 0} arising from them is self-decomposable. Using the terminology of Barndor<-Nielsen and Shephard (2001), we can express Theorem 1.1 in this way: normal variance-mean mixtures using self-decomposable mixing distributions are self-decomposable. A related paper is Barndor<-Nielsen and Halgreen (1977). The Brownian motion is strictly stable with index 2. Brownian motions with drift are stable with index 2, but not strictly stable. It is known that if the subordinand {Xt } is strictly stable and the subordinator {Zt } is self-decomposable, then the subordinated process {Yt } is self-decomposable. This is another extension of the result of Halgreen and Ismail and Kelker and a multivariate generalization of this fact is given in Theorem 6:1 of Barndor<-Nielsen et al. (2001). In Section 4 we give a simple proof of this fact. We do not know whether Theorem 1.1 can be generalized to the case where the subordinand {Xt } is a stable process which is not strictly stable. If {Xt } is a Brownian motion with drift on Rd , d ¿ 2, then Theorem 1.1 does not hold even under the additional condition that the distribution of Z1 is of GGC. This fact is proved by Takano (1989=90). A random variable Y is called of type G if there are a standard Gaussian random variable X and a nonnegative in0nitely divisible random variable Z such that X and Z are independent
(1.4)
and d
Y = Z 1=2 X;
(1.5)
see Rosinski (1991) for an account of this class. The distribution of Y is also called of type G. In other words it is a normal variance mixture using an in0nitely divisible mixing distribution. It is easy to see that a distribution on R is of type G if and only if is the distribution at time 1 of the subordinated process {Yt } arising from the Brownian motion {Xt } and a subordinator {Zt }. Thus the result of Halgreen and Ismail and Kelker is paraphrased as follows: a random variable Y of type G is self-decomposable if the Z in the de0nition of type G is self-decomposable, that is, if Y is of class GL in the terminology of Jiang (2000). The second result of this paper is that the converse assertion is not true.
K. Sato / Statistics & Probability Letters 54 (2001) 317 – 324
319
Theorem 1.2. There is a self-decomposable random variable Y of type G for which one cannot 3nd a nonnegative self-decomposable Z and a standard Gaussian X satisfying (1:4) and (1:5). In other words, a self-decomposable random variable of type G is not necessarily of type GL . This disproves a conjecture of Jiang (2000, p. 40). Proofs of Theorems 1.1 and 1.2 are given in Sections 2 and 3. 2. Proof of Theorem 1.1 The generating triplet (A; ; ) of an in0nitely divisible distribution on R, by de0nition is a triplet of nonnegative number A, a -0nite measure satisfying ({0}) = 0 and (1 ∧ x2 )(d x) ¡ ∞, and a real number such that A (z) ˆ = exp − z 2 + (eizx − 1 − izx1[−1; 1] (x))(d x) + iz : (2.1) 2 R Here A and are called the Gaussian variance and the L9evy measure of , respectively. A L9evy process {Xt : t ¿ 0} is said to have the generating triplet (A; ; ) if the distribution of X1 has the generating triplet (A; ; ). {Xt } is a subordinator if and only if A = 0, ((−∞; 0)) = 0, (0;1] x(d x) ¡ ∞, and ¿ (0;1] x(d x), that is, (eizx − 1)(d x) + i0 z (2.2) (z) ˆ = exp (0;∞)
with 0 = − (0;1] x(d x) ¿ 0, which is called the drift of the subordinator. See Sato (1999) for detailed exposition. Let {Xt } be a L9evy process on R with generating triplet (A; ; ) and {Zt } be a subordinator with L9evy measure and drift 0 . Let {Yt } be the subordinated L9evy process arising from them. Then the generating triplet (A] ; ] ; ] ) of {Yt } is given as follows: A] = 0 A;
(2.3)
] (B) = 0 (B) + ]
= 0 +
(0;∞)
(0;∞)
s (B)(ds);
(ds)
[−1;1]
xs (d x);
B ∈ B(R \ {0});
(2.4) (2.5)
where s is the distribution of Xs and B(R \ {0}) is the class of Borel subsets of R \ {0}. See Theorem 30:1 of Sato (1999). Lemma 2.1. Fix a self-decomposable process {Xt }. Suppose that; if {Zt } is a self-decomposable subordinator satisfying 0 = 0 and (ds) = s−1 1(0; a] (s) ds for some a ∈ (0; ∞); then the subordinated process {Yt } is self-decomposable. Then; for any choice of a self-decomposable subordinator {Zt }; the subordinated process {Yt } is self-decomposable. Proof. An in0nitely divisible distribution is self-decomposable if and only if its L9evy measure is of the form |x|−1 k(x) d x with k(x) increasing on (−∞; 0) and decreasing on (0; ∞) (we are using the words increase and decrease in the weak sense). See Corollary 15:11 of Sato (1999). Let {Zt } be a self-decomposable subordinator with L9evy measure and drift 0 . Then (ds) = s−1 k(s)1(0; ∞) (s) ds with k(s) decreasing on
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K. Sato / Statistics & Probability Letters 54 (2001) 317 – 324
(0; ∞). The function k(s) is the limit of an increasing sequence of functions kn (s), n = 1; 2; : : : ; of the form mn kn (s) = bn; j 1(0; an; j ] (s); j=1
where mn is a positive integer, 0 ¡ an; 1 ¡ · · · ¡ an; mn , and bn; j ¿ 0 for j = 1; : : : ; mn . Let {Zt(n) } be the subordinator with L9evy measure s−1 kn (s)1(0; ∞) (s) ds and drift 0 . Recall that convolutions of self-decomposable distributions are self-decomposable. By virtue of (2.4), the assumption implies that the subordinated process {Yt(n) } arising from {Xt } and {Zt(n) } is self-decomposable. Since the limit of a sequence of self-decomposable distributions is self-decomposable, it follows that {Yt } is self-decomposable. Proof of Theorem 1.1. Let {Xt }, {Zt }, and {Yt } be the processes in the theorem. Let (A; ; ), , 0 , and (A] ; ] ; ] ) be as above. Then A = 1, = 0, and = 0. By Lemma 2.1 it is enough to prove the theorem under the assumption that (ds) = s−1 1(0; a] (s) ds for some a ∈ (0; ∞) and 0 = 0. Thus, by (2.4), we have ] (d x) = |x|−1 k ] (x) d x with a 1 −(x−s)2 =(2s) ds ] √ e k (x) = |x| : (2.6) s 2 s 0 All we have to show is that k ] (x) is increasing on (−∞; 0) and decreasing on (0; ∞). The discussion for x ¡ 0 is reduced to that for x ¿ 0 by changing to −. So we only consider x ¿ 0. By change of variables x2 =s = u we get ∞ √ 1=2 −1=2 2 ] 2 k (x) = x e−(u −xu ) =2 (x2 =u)−3=2 (x2 =u2 ) du =
x2 =a
∞
x2 =a
e−(u
1=2
−xu−1=2 )2 =2
u−1=2 du:
(2.7)
Hence, if ¡ 0, then k ] (x) is decreasing in x on (0; ∞). Henceforth we assume that ¿ 0. Write b = a−1=2 . We have, from (2.7), ∞ √ d k ] (x) −1=2 2 1=2 −(bx−=b)2 =2 −1 2 = −e 2 (bx) 2b x + e−(u −xu ) =2 (u1=2 − xu−1=2 )(u−1=2 )u−1=2 du dx 2 2 b x ∞ 1=2 −1=2 2 = e−(u −xu ) =2 f(u) du; b2 x 2
(2.8)
where f(u) = − b(1 − 2 x2 u−2 ) + (1 − xu−1 )u−1=2 : If x ¿ b−2 , then, rewriting f(u) as f(u) = −b + 2 bx2 u−2 + u−1=2 − 2 xu−3=2 = −b(1 − b−1 u−1=2 ) − 2 xu−3=2 (1 − bxu−1=2 ) and noticing that 1 − b−1 u−1=2 ¿ 1 − b−2 x−1 ¿ 0 and 1 − bxu−1=2 ¿ 0 for u ¿ b2 x2 , we get d k ] (x)= d x ¡ 0 from (2.8). If 0 ¡ x ¡ b−2 , then b2 x2 ¡ x and we get √ d k ] (x) = I1 + I2 ; 2 dx where x ∞ 1=2 −1=2 2 1=2 −1=2 2 I1 = e−(u −xu ) =2 f(u) du; I2 = e−(u −xu ) =2 f(u) du: b2 x 2
x
K. Sato / Statistics & Probability Letters 54 (2001) 317 – 324
321
The integral I2 is, by change of variables 2 x2 u−1 = v, written as x −1=2 1=2 2 I2 = e−(xv −v ) =2 [ − b(1 − −2 x−2 v2 ) + (1 − −1 x−1 v)−1 x−1 v1=2 ]2 x2 v−2 dv 0
=−
x
0
Thus, we get
e−(v
I1 + I2 = −
1=2
−xv−1=2 )2 =2
b2 x 2
0
=−
b2 x 2
0
f(v) dv:
e−(u
1=2
−xu−1=2 )2 =2
f(u) du
e−(u
1=2
−xu−1=2 )2 =2
(xu−1 − 1)(b(xu−1 + 1) − u−1=2 ) du:
The function xu−1=2 + u1=2 is strictly decreasing for 0 ¡ u ¡ x. Since b2 x2 ¡ x, we have, for 0 ¡ u ¡ b2 x2 , xu−1=2 + u1=2 ¿ b−1 + bx ¿ b−1 : Hence I1 + I2 ¡ 0 for 0 ¡ x ¡ b−2 , which 0nishes the proof that d k ] (x)= d x ¡ 0 for x ¿ 0. 3. Proof of Theorem 1.2 We prepare two simple lemmas. Lemma 3.1. Suppose that Y satis3es (1:4) and (1:5) with a standard Gaussian random variable X and a nonnegative random variable Z. Then the distribution of Z is determined by the distribution of Y . Proof. For any real u we have E[eiuY ] = E[eiuZ
1=2
X
] = E[e−u
2
Z=2
]=
[0;∞)
e−u
2
s=2
PZ (ds);
where PZ is the distribution of Z. Thus the characteristic function of the distribution PY of Y determines the Laplace transform of PZ . Hence PZ is determined by PY . Lemma 3.2. Let a ¿ b ¿ 0 and let 1; 0 ¡ s ¡ a; f(s) = −1; a 6 s ¡ a + b; c; a + b 6 s: with c ¿ 0. Then ∞ e−xs f(s) ds ¿ 0 0
Proof. Since a e−xs ds ¿ 0
we have
a+b 0
a
e
−xs
a+b
for all x ¿ 0:
e−xs ds;
f(s) ds ¿ 0. Hence the assertion is obvious.
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K. Sato / Statistics & Probability Letters 54 (2001) 317 – 324
Proof of Theorem 1.2. By Lemma 3.1 it is enough to construct a self-decomposable subordinated process {Yt } arising from the Brownian motion {Xt } and a non-self-decomposable subordinator {Zt }. Assume, for a moment, that such {Yt }, {Xt }, and {Zt } are found. Then, as in Section 2, the generating triplet (A] ; ] ; ] ) of {Yt } is expressed by the L9evy measure and the drift 0 of {Zt } as A] = 0 ; ] (B) =
(3.1) (0;∞)
(ds)
B
√
1 −x2 =(2s) e d x: 2 s
(3.2)
By the self-decomposability the L9evy measure ] is of the form ] (d x) = |x|−1 k ] (x) d x with k ] (x) being decreasing on (0; ∞). We have k ] (−x) = k ] (x). Assume further that (ds) = s−1 k(s) ds on (0; ∞) with a nonnegative measurable function k(s). Then ∞ 1 −x2 =(2s) k(s) ] √ ds k (x) = x e s 2 s 0 ∞ 1 −1=(2r) k(x2 r) √ = dr (3.3) e r 2 r 0 for x ¿ 0. We have ∞ d k ] (x) 1 −1=(2r) 2 √ e k (x r) dr = 2x dx 2 r 0
(3.4)
for x ¿ 0, supposing that k is di
Lemma 3.2. That is, u ∈ (0; a); u ∈ (a; a + b); u ∈ (a + b; ∞):
This is equivalent to 1 −cs−3=2 ; s ∈ (0; a+b ); 1 k (s) = s−3=2 ; s ∈ ( a+b ; 1a ); −3=2 −s ; s ∈ ( 1a ; ∞): Choose k(s) such that 2cs−1=2 ; k(s) = −2s−1=2 + p; −1=2 2s ;
(3.6)
1 s ∈ (0; a+b ); 1 s ∈ [ a+b ; 1a );
s ∈ [ 1a ; ∞);
(3.7)
K. Sato / Statistics & Probability Letters 54 (2001) 317 – 324
323
1 where p is a constant satisfying −2(a + b)1=2 + p ¿ 0. Then k(s) ¿ 0 on (0; ∞), 0 k(s) ds ¡ ∞, ∞ −1 s k(s) ds ¡ ∞, and (3.6) is satis0ed. De0ne k ] (x) for x ¿ 0 by (3.3) and de0ne k ] (−x) = k ] (x). Then the 1 condition on the interchangeability of the order of integration and di
4. Remarks Let {Xt }, {Zt }, and {Yt } be the processes in Theorem 1.1. Let , , 0 , and (A] ; ] ; ] ) be as in the proof of the theorem. Let PZt and PYt be the distributions of Zt and Yt , respectively. Then, the two functions k(s) and k ] (x) which express (ds) = s−1 k(s) ds and ] (d x) = |x|−1 k ] (x) d x are connected as follows: ∞ 1 −(x−s)2 =(2s) k(s) √ k ] (x) = |x| e ds s 2 s 0 ∞ 1=2 −1=2 2 1 √ e−(u −xu ) =2 k(x2 u−1 )u−1=2 du = 2 0 x ∞ 2 2 e e−(u=2)− x =(2u) k(x2 u−1 )u−1=2 du (4.1) =√ 2 0 for x ∈ R \ {0}. It follows that
k(0+) ∞ −u=2 −1=2 k ] (0+) = k ] (0−) = √ e u du = k(0+); (4.2) 2 0 √ √ ∞ since 0 e−u=2 u−1=2 du = 2)(1=2) = 2 . Notice that the result (4.2) does not depend on . The same result holds also when {Xt } is the Brownian motion ( = 0). From the self-decomposability proved in Theorem 1.1 and from (4.2) follow many properties of PYt . First, PYt is unimodal by Yamazato’s theorem (1978). Second, the smoothness of PYt is expressed by k ] (0+) + k ] (0−) as in Sato and Yamazato (1978). Some of the results are given below; see also Sections 28 and 53 of Sato (1999). Let cZ = k(0+) and cY = k ] (0+) + k ] (0−). Then cY = 2cZ
(4.3)
by (4.2). If cZ and cY are in0nite, then PZt and PYt have C ∞ densities on R for every t ¿ 0. If 0 ¿ 0, then PYt has C ∞ density on R for every t ¿ 0. Assume that k(0+) ¡ ∞ and 0 = 0. Thus, by (2.5), {Yt } has drift 0. De0ne NZ (t) and NY (t) as the integers satisfying NZ (t) ¡ tcZ 6 NZ (t) + 1 and NY (t) ¡ tcY 6 NY (t) + 1. Then, for t ¿ 0, PYt (resp. PZt ) has C NY (t) (resp. C NZ (t) ) density on R but does not have C NY (t)+1 (resp. C NZ (t)+1 ) density on R. In this sense, the distribution of {Yt } is twice as smooth as the distribution of {Zt }. For t 6 1=cY , PYt has mode 0 and the density fY (t; x) of PYt satis0es fY (t; 0+) = ∞ and fY (t; 0−) = ∞. On the other hand, for t ¡ 1=cZ , PZt has mode 0 and fZ (t; 0+) = ∞; for t = 1=cZ , PZt has mode 0 and, in order 1 that fZ (1=cZ ; 0+) = ∞, it is necessary and suMcient that 0 (cZ − k(s))s−1 ds = ∞. In Theorem 1.1 we have treated a Bownian motion with drift, which is not strictly stable. In the case of strictly stable subordinands, more results are known. The following proposition is a special case of Theorem 6:1 of Barndor<-Nielsen et al. (2001). A simpler proof given here is essentially the same as that of Bondesson (1992, p. 19).
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K. Sato / Statistics & Probability Letters 54 (2001) 317 – 324
Proposition 4.1. A subordinated process on R arising from a strictly stable subordinand and a self-decomposable subordinator is self-decomposable. Proof. As before denote by {Xt }, {Zt }, and {Yt } the subordinand, the subordinator, and the subordinated. We assume that {Xt } is strictly stable with index , and that {Zt } is self-decomposable. Let Z = = PZ1 , the distribution of Z1 . Then, for any b ¿ 1, satis0es (1.1). The distribution b is in0nitely divisible (Sato, 1999, Proposition 15:5). Considering the representation (2.2), we can conclude that b is concentrated on [0; ∞). Thus, for each b ¿ 1, there is a subordinator {Zt(b) } independent of {Zt } such that {b−1 Zt + Zt(b) } has the same distribution as {Zt }. Let X = PX1 , the distribution of X1 . Then E[eizYt ] = E[ˆX (z)Zt ] = E[ˆX (z)b c
Since ˆX (z) = ˆX (c E[ˆX (z)b
−1
Zt
1=,
−1
Zt +Zt(b)
] = E[ˆX (z)b
−1
Zt
(b)
]E[ˆX (z)Zt ]:
(4.4)
z) for every c ¿ 0, we have
] = E[ˆX (b−1=, z)Zt ] = E[eib
−1=,
zYt
]:
(4.5)
Since b ¿ 1 is arbitrary and since the second factor of the rightmost member of (4.4) is the characteristic function of another subordinated process, (4.4) and (4.5) show the self-decomposability of {Yt }. The same proposition holds also for processes on Rd , d ¿ 2, which is proved similarly. References Barndor<-Nielsen, O., Halgreen, C., 1977. In0nite divisibility of the hyperbolic and generalized inverse Gaussian distributions. Zeit. Wahrsch. Verw. Gebiete 38, 309–311. Barndor<-Nielsen, O.E., Pedersen, J., Sato, K., 2001. Multivariate subordination, self-decomposability and stability. Adv. Appl. Probab. 33, 160–187. Barndor<-Nielsen, O.E., Shephard, N., 2001. Modelling by L9evy processes for 0nancial econometrics. In: Barndor<-Nielsen, O.E., Mikosch, T., Resnick, S.I. (Eds.), L9evy Processes—Theory and Applications, BirkhTauser, Boston, pp. 283–318. Bondesson, L., 1992. Generalized Gamma Convolutions and Related Classes of Distribution Densities. Lecture Notes in Statistics, Vol. 76. Springer, New York. Halgreen, C., 1979. Self-decomposability of the generalized inverse Gaussian and hyperbolic distributions. Zeit. Wahrsch. Verw. Gebiete 47, 13–17. Ismail, M.E.H., Kelker, D.H., 1979. Special functions, Stieltjes transforms and in0nite divisibility. SIAM J. Math. Anal. 10, 884–901. Jiang, W., 2000. Some simulation-based models towards mathematical 0nance. Ph.D. Thesis, University of Aarhus. Jurek, Z.J., Mason, J.D., 1993. Operator-Limit Distributions in Probability Theory. Wiley, New York. Rosinski, J., 1991. On a class of in0nitely divisible processes represented as mixtures of Gaussian processes. In: Cambanis, S., Samorodnitsky, G., Taqqu, M.S. (Eds.), Stable Processes and Related Topics. BirkhTauser, Boston, pp. 27 – 41. Sato, K., 1999. L9evy Processes and In0nitely Divisible Distributions. Cambridge University Press, Cambridge. Sato, K., Yamazato, M., 1978. On distribution functions of class L. Zeit. Wahrsch. Verw. Gebiete 43, 273–308. Sato, K., Yamazato, M., 1983. Stationary processes of Ornstein–Uhlenbeck type. In: Itˆo, K., Prokhorov, J.V. (Eds.), Probability Theory and Mathematical Statistics, Proceedings of the Fourth USSR–Japan Symposium 1982, Lecture Notes in Math., Vol. 1021. Springer, Berlin, pp. 541–551. Shanbhag, D.N., Sreehari, M., 1979. An extension of Goldie’s result and further results in in0nite divisibility. Zeit. Wahrsch. Verw. Gebiete 47, 19–25. Takano, K., 1989=90. On mixtures of the normal distribution by the generalized gamma convolutions. Bull. Fac. Sci. Ibaraki Univ. Ser. A 21, 29 – 41; Correction and addendum, 22, 49 –52. Yamazato, M., 1978. Unimodality of in0nitely divisible distribution functions of class L. Ann. Probab. 6, 523–531.