A note on a bivariate gamma distribution

Statistics and Probability Letters 80 (2010) 1991–1994

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Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro

A note on a bivariate gamma distribution Makoto Maejima ∗ , Yohei Ueda Department of Mathematics, Keio University, 3-14-1, Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan

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abstract

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Article history: Received 20 June 2010 Received in revised form 4 September 2010 Accepted 7 September 2010 Available online 21 September 2010

Vere-Jones (1967) introduced a bivariate generalization of gamma distributions and proved its infinite divisibility. Maejima and Ueda (2010) and others studied α selfdecomposability, which is a generalization of selfdecomposability. In this paper, the (−2)-selfdecomposability of bivariate gamma distributions is shown. © 2010 Elsevier B.V. All rights reserved.

MSC: 60E07 Keywords: Infinitely divisible distribution Bivariate gamma distribution α -selfdecomposable distribution

1. Introduction and preliminaries According to Vere-Jones (1967), one multivariate generalization of gamma distributions is the joint distribution of (X12 , X22 , . . . , Xd2 ), where (X1 , X2 , . . . , Xd ) is a d-dimensional Gaussian random variable whose components Xj , j = 1, 2, . . . , d, are one-dimensional standard normal random variables. Vere-Jones (1967) proved that when d = 2, W := (X12 , X22 ) is infinitely divisible. He actually gave the Lévy measure of W . Let σ be the correlation coefficient of X1 and X2 . He treated the problem in terms of moment generating functions, but if we read it in terms of characteristic functions, his result is turned out to be the following. Let Sn := (W1 + W2 + · · · + Wn )/2, where W1 , W2 , . . . , Wn are independent copies of W . (1) By (4) of Vere-Jones (1967), the density function f (x1 , x2 ) of S4 is expressed as

 



f (x1 , x2 ) = f (x1 )f (x2 ) exp{−σ 2 (x1 + x2 )/(1 − σ 2 )}(1 − σ 2 )−1 (σ 2 x1 x2 )−1/2 I1 2 σ 2 x1 x2 (1 − σ 2 )−1 ,

(1.1)

for x1 > 0 and x2 > 0, where f (x) = xe−x , x > 0. Here and in what follows, Iν ’s are modified Bessel functions. (For modified Bessel functions, see, e.g., 8.4–8.5 of Gradshteyn and Ryzhik (2007).) (2) For z ∈ R2 , C (z ) := log E ei⟨z ,S2 ⟩ =





∞

∫ 0

∫

∞



 M (dx1 dx2 )

ei⟨z ,(x1 ,x2 )⟩ − 1

0

x21 + x22

,

where

 2 2 2 −1 σ (x1 + x2 )(x1 x2 ) f (x1 , x2 )dx1 dx2 =: g (x1 , x2 )dx1 dx2 , −x1 /(1−σ 2 ) M (dx1 dx2 ) = x1 e dx1 δ0 (dx2 ),  −x2 /(1−σ 2 ) δ0 (dx1 )x2 e dx2 , ∗

for x1 > 0, x2 > 0, for x1 > 0, x2 = 0, for x1 = 0, x2 > 0.

Corresponding author. E-mail addresses: [email protected] (M. Maejima), [email protected] (Y. Ueda).

0167-7152/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2010.09.004

(1.2)

1992

M. Maejima, Y. Ueda / Statistics and Probability Letters 80 (2010) 1991–1994

(See p. 422 of Vere-Jones (1967).) We divide the integral into three parts: C (z ) =

∫∫

∫∫

∫∫

+

+

D1

D2

,

D3

where D1 = {(x1 , x2 ): x1 > 0, x2 > 0}, D2 = {(x1 , x2 ): x1 > 0, x2 = 0}, D3 = {(x1 , x2 ): x1 = 0, x2 > 0}.



For the integral

 D1

2

, let us change variables as (x1 , x2 ) = (r cos θ , r sin θ ). Write ξ = (cos θ , sin θ ) on the unit circle S in

R . Then we have π/2

∫

∫∫ =

dθ

∫

λ(dξ )

=



 1

ei⟨z ,(r cos θ,r sin θ)⟩ − 1

0

0

D1

∞

∫

∞

∫

 g (r ξ )

e i ⟨z , r ξ ⟩ − 1



r

0

S

r2

g (r cos θ , r sin θ )rdr

dr ,

(1.3)

where

λ(B) =

∫ {θ∈(0,π/2):(cos θ,sin θ)∈B}

dθ ,

B ∈ B (S).

2. Non-selfdecomposability of W unless X1 and X2 are independent In general, we know the following, (see, e.g., Theorem 15.10 of Sato (1999)). Let µ be an infinitely divisible distribution on Rd with Lévy measure ν . Then, µ is selfdecomposable if and only if

ν(B) =

∫

λ(dξ )

∞

∫

1B (r ξ ) 0

S

kξ (r ) r

dr ,

with a finite measure λ on the unit sphere S in Rd and a nonnegative function kξ (r ) measurable in ξ ∈ S and decreasing in r > 0. Thus, for checking the selfdecomposability, it is enough to check the behavior of kξ (r ) as a function of r > 0 for each ξ . Let  µ(z ), z ∈ Rd , be the characteristic function of an infinitely divisible distribution µ. For t ≥ 0, we write µt for the distribution with characteristic function  µ(z )t and call µt the t-fold convolution of µ. Then, note that selfdecomposability is closed under multiplying selfdecomposable random variables by constants and under t-fold convolution for t > 0. Hence the selfdecomposability of W is equivalent to that of S2 . By (1.3), if ξ ̸= (1, 0) or (0, 1), equivalently if 0 < θ < π /2, then kξ (r ) = g (r ξ ), which is not nonincreasing in r > 0 unless σ = 0. Thus, if σ ̸= 0, then W is not selfdecomposable. If σ = 0, namely if the components of W are independent, then by (1.2), kξ (r ) is nonincreasing, and hence W is selfdecomposable. 3. Which class does the distribution of W belong to? In Maejima et al. (2010) and Maejima and Ueda (2010), they defined wider classes than the class of selfdecomposable distributions as follows. Let α ∈ R. An infinitely divisible distribution µ on Rd is said to be α -selfdecomposable, if any b > 1, there exists another infinitely divisible distribution ρb on Rd satisfying α

 µ(z ) =  µ(b−1 z )b  ρb (z ), z ∈ Rd . (3.1) Denote the totality of α -selfdecomposable distributions on Rd by L⟨α⟩ (Rd ). Then by the definition, L⟨0⟩ (Rd ) is the class of selfdecomposable distributions on Rd and L⟨−1⟩ (Rd ) is the so-called Jurek class of s-selfdecomposable distributions on Rd . Jurek (1988, 1989, 1992) and Jurek and Schreiber (1992) studied the classes Uβ (Q ), β ∈ R, of distributions on a real separable Banach space E, where Q is a linear operator on E with certain properties. These classes are equal to L⟨α⟩ (Rd ) if β = −α, E = Rd and Q is the identity operator. As to these classes, they studied the decomposability and stochastic integral characterizations, although some results are only for the case that Q is the identity operator. However, since, for 0 < α < 2, L⟨α⟩ (Rd ) contains all α -stable distributions and any µ ∈ L⟨α⟩ (Rd ) belongs to the normal domain of attraction of some α -stable distribution, we adopt the parametrization in (3.1). By the observations in Sections 1 and 2, the distribution of W is not only non-selfdecomposable but also not in a bigger class, the Jurek class. However, the following proposition holds. Note that, by (3.1), α -selfdecomposability is closed under multiplying α -selfdecomposable random variables by constants and under t-fold convolution for t > 0. Hence the α selfdecomposability of W is equivalent to that of S2 . In what follows, L(W ) denotes the law of W . Proposition 3.1. Let σ ̸= 0. Then

L(W )



∈ L⟨α⟩ (R2 ) ̸∈ L⟨α⟩ (R2 )

for all α ≤ −2, for all α > −2.

M. Maejima, Y. Ueda / Statistics and Probability Letters 80 (2010) 1991–1994

1993

Proof. In Maejima et al. (2010), it was shown that when α < 0, µ ∈ L⟨α⟩ (Rd ) if and only if the Lévy measure ν of µ has the form

ν(B) =

∫

λ(dξ )

∞

∫

1B (r ξ )r −α−1 ℓξ (r )dr ,

0

S

where λ is a measure on S and ℓξ (r ) is a nonnegative function measurable in ξ ∈ S and nonincreasing and right-continuous in r > 0. From the fact observed in Section 2, kξ (r )

fξ (r ) :=

r

g (r ξ )

=

r

|σ | 2 −1 (cos θ sin θ )−1/2 e−r (1−σ ) (cos θ +sin θ ) I1 = 2 1−σ



2(σ 2 cos θ sin θ )1/2 1 − σ2

 r

,

where ξ = (cos θ , sin θ ). Note that for all r > 0 and all θ ∈ (0, π /2), rfξ′ (r )/fξ (r ) = −

r 1 − σ2

 +

(cos θ + sin θ )

2(σ 2 cos θ sin θ )1/2 1 − σ2

  r

′

I1

2(σ 2 cos θ sin θ )1/2 1 − σ2

  r

I1

2(σ 2 cos θ sin θ )1/2 1 − σ2

 r

  r 2(σ 2 cos θ sin θ )1/2 − cos θ − sin θ 1 − σ2 2 √ √ r ≤ 1− cos θ − sin θ ≤ 1, 1 − σ2 ≤ 1+

where we have used the inequality uI1′ (u)/I1 (u) ≤ u + 1,

for all u > 0,

(3.2)

which will be proved later. Then rfξ′ (r ) − fξ (r ) ≤ 0, namely, r > 0 and

ν(B) =

∫

λ(dξ )

∫

∞

1B (r ξ )fξ (r )dr =

0

S

∫

λ(dξ )

∞

∫

∂ {r −1 fξ (r )} ∂r

≤ 0. Hence ℓξ (r ) := r −1 fξ (r ) is nonincreasing in

1B (r ξ )r −(−2)−1 ℓξ (r )dr ,

0

S

which yields L(W ) ∈ L⟨−2⟩ (R2 ). Since L⟨α1 ⟩ (Rd ) ⊃ L⟨α2 ⟩ (Rd )

for α1 < α2

(Corollary 1.1(b) of Jurek (1988) or Proposition 3.1 of Maejima and Ueda (2010)), we have L(W ) ∈ L⟨α⟩ (R2 ) for all α ≤ −2. Also take into account that r α+1 fξ (r ) =

=

r α+1 |σ | 1−σ

(cos θ sin θ )−1/2 e−r (1−σ 2

r α+2 σ 2

2 −1 e−r (1−σ ) (cos θ+sin θ) 2

(1 − σ 2 )

2 )−1 (cos θ+sin θ )

 I1

2(σ 2 cos θ sin θ )1/2

(cos θ sin θ )m m!(m + 1)! m=0 ∞ −

1 − σ2



σ 1 − σ2

2m

 r

r 2m ,

where we have used the series representation of I1 given in 8.445 of Gradshteyn and Ryzhik (2007). If α > −2, then the right-hand side of the equation above tends to 0 as r ↓ 0. Hence r α+1 fξ (r ) with α > −2 is not nonincreasing and thus L(W ) ̸∈ L⟨α⟩ (R2 ). We finally prove (3.2). Since uI1′ (u) − I1 (u) = uI2 (u) (8.4864 of Gradshteyn and Ryzhik (2007)), it is enough to show I1 (u) ≥ I2 (u) for all u > 0. Also it follows from the two recurrence formulae uI1′ (u) − I1 (u) = uI2 (u) and uI2′ (u) + 2I2 (u) = uI1 (u) (8.4863 of Gradshteyn and Ryzhik (2007)) that u(I1′ (u) − I2′ (u)) + (u + 2)(I1 (u) − I2 (u)) = 3I1 (u). Then

     d  2 u u e (I1 (u) − I2 (u)) = ueu u I1′ (u) − I2′ (u) + (u + 2)(I1 (u) − I2 (u)) du = 3ueu I1 (u) ≥ 0. Hence we have u2 eu (I1 (u) − I2 (u)) ≥ 0 for all u > 0, that is, I1 (u) ≥ I2 (u) for all u > 0. Acknowledgement The authors would like to thank the referee for his/her useful suggestions.



1994

M. Maejima, Y. Ueda / Statistics and Probability Letters 80 (2010) 1991–1994

References Gradshteyn, I.S., Ryzhik, I.M., 2007. Table of Integrals, Series, and Products, Seventh ed. Elsevier. Jurek, Z.J., 1988. Random integral representations for classes of limit distributions similar to Lévy class L0 . Probab. Theory Related Fields 78, 473–490. Jurek, Z.J., 1989. Random integral representations for classes of limit distributions similar to Lévy class L0 . II. Nagoya Math. J. 114, 53–64. Jurek, Z.J., 1992. Random integral representations for classes of limit distributions similar to Lévy class L0 . III. In: Probability in Banach Spaces, 8. In: Progr. Probab., vol. 30. Birkhäuser Boston, Boston, MA, pp. 137–151. Jurek, Z.J., Schreiber, B.M., 1992. Fourier transforms of measures from the classes Uβ , −2 < β ≤ −1. J. Multivariate Anal. 41, 194–211. Maejima, M., Matsui, M., Suzuki, M., 2010. Classes of infinitely divisible distributions on Rd related to the class of selfdecomposable distributions. Tokyo J. Math. (in press) (http://www.math.keio.ac.jp/~maejima/papers/MMS.pdf). Maejima, M., Ueda, Y., 2010. α -selfdecomposable distributions and related Ornstein–Uhlenbeck type processes. Stochastic Process. Appl. doi:10.1016/j.spa.2010.08.005. Sato, K., 1999. Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge. Vere-Jones, D., 1967. The infinite divisibility of a bivariate gamma distribution. Sankhya¯ Ser. A 29, 421–422.