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A link between the Matsumoto–Yor property and an independence property on trees
ARTICLE IN PRESS
Statistics & Probability Letters 76 (2006) 1097–1101 www.elsevier.com/locate/stapro
A link between the Matsumoto–Yor property and an independence property on trees Angelo Efoe´vi Koudou Institut Elie Cartan, Laboratoire de Mathe´matiques, B.P. 239, F-54506 Vandoeuvre-le`s-Nancy, Cedex, France Received 4 October 2005 Available online 22 December 2005
Abstract We prove that an independence property established by Matsumoto and Yor [2001. An analogue of Pitman’s 2M X theorem for exponential Wiener functional, Part II: the role of the generalized inverse Gaussian laws. Nagoya Math. J. 162, 65–86] and by Letac and Wesolowski [2000. An independence property for the product of GIG and gamma laws. Ann. Probab. 28, 1371–1383] is, in a particular case, a corollary of a result by Barndorff-Nielsen and Koudou [1998. Trees with random conductivities and the (reciprocal) inverse Gaussian distribution. Adv. Appl. Probab. 30, 409–424] where, for finite trees equipped with inverse Gaussian resistances, an exact distributional and independence result was established. r 2005 Elsevier B.V. All rights reserved. Keywords: Gamma distribution; Generalized inverse Gaussian distribution; Kirchoff–Ohm laws; Matsumoto–Yor property; Tree
Introduction Let X and Y be two non-dirac, positive and independent random variables. A necessary and sufficient condition for the random variables ðX þ Y Þ1 and X 1 ðX þ Y Þ1 to be independent is that X follows a generalized inverse Gaussian distribution while Y is gamma-distributed with suitable parameters. The sufficient part of this assertion, named in the related recent literature as the Matsumoto– Yor property, is included in a work by Matsumoto and Yor (2001) in a particular case, as a proposition needed in the proof of their main result. (The latter result extends the Pitman theorem to exponential functionals of Brownian motion via generalized inverse Gaussian laws.) The general case of the Matsumoto–Yor property was given by Letac and Wesolowski (2000). (Let us note that, in spite of dates of publication, the work of Matsumoto–Yor, 2001, precedes that of Letac and Wesolowski, 2000). The origin of the present paper is the observation of the form of the random variables U ¼ ðX þ Y Þ1 and V ¼ X 1 ðX þ Y Þ1 , which look like random variables dealt with in a previous paper by Barndorff-Nielsen and Koudou (1998) (referred to in the sequel as BK98) on a finite tree equipped with inverse Gaussian random resistances. It turns out that one can prove the Matsumoto–Yor property to be, in the (non-generalized) inverse Gaussian case, a consequence of an independence property established in BK98. For this, one only has to apply the result of BK98 to a trivial tree (with only two edges). This gives a better understanding of the E-mail address: [email protected]. 0167-7152/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2005.12.006
ARTICLE IN PRESS 1098
A.E. Koudou / Statistics & Probability Letters 76 (2006) 1097–1101
independence property of Barndorff-Nielsen and Koudou (1998) which seemed a little surprising to its authors at that time. That independence property could be considered, a posteriori, as one tree-version of the Matsumoto–Yor property. Note that a nice tree-version of the Matsumoto–Yor property was recently established by Massam and Wesolowski (2004). In Section 1 we remind the definition of GIG distributions and some of their properties. Section 2 recalls the results of Matsumoto–Yor and Letac-Wesolowski. In Section 3 we explain how the result of BK98 implies a particular case of the Matsumoto–Yor property. 1. The generalized inverse Gaussian distributions We recall here the definition and some properties of the generalized inverse Gaussian (GIG) laws. Some references are Barndorff-Nielsen (1994), Matsumoto and Yor (2001), Vallois (1991). Let m 2 R, a40 and b40. The GIG distribution with parameters m, a, b is the probability measure: m b xm1 GIGðm; a; bÞðdxÞ ¼ e1=2 ða2 x1 þ b2 xÞ1ð0;1Þ ðxÞ dx, a 2K m ðabÞ where K m is the classical McDonald special function. One can have a ¼ 0 if m40 or b ¼ 0 if mo0. In the case m ¼ 12, the law GIGðm; a; bÞ is the classical inverse Gaussian distribution with density a IGða; bÞðdxÞ ¼ pffiffiffiffiffiffi eab x3=2 e1=2 ða2 x1 þ b2 xÞ1ð0;1Þ ðxÞ dx, 2p while in the case m ¼ 12 we have the reciprocal inverse Gaussian distribution with density b RIGða; bÞðdxÞ ¼ pffiffiffiffiffiffi eab x1=2 e1=2 ða2 x1 þ b2 xÞ1ð0;1Þ ðxÞ dx. 2p Note that RIGð0; bÞ is the gamma distribution with density 2 b pffiffiffiffiffiffi x1=2 eb =2x 1ð0;1Þ ðxÞ dx 2p
i.e. the gamma distribution with shape parameter Some convolution properties hold:
1 2
and scale parameter 2=b2 .
IGða1 ; bÞ IGða2 ; bÞ ¼ IGða1 þ a2 ; bÞ,
(1)
IGða1 ; bÞ RIGða2 ; bÞ ¼ RIGða1 þ a2 ; bÞ.
(2)
A diffusion interpretation exists for IG and RIG laws: they are, respectively, the distribution of the first and the last hitting time for a Brownian motion. Proofs can be seen in Bhattacharya and Waymire (1990) for the IG case and, for a more general proof, in Vallois (1991). The GIG density exists also as a probability measure on the set of positive definite matrices, the case a ¼ 0 defining Wishart matrices (see Letac and Wesolowski, 2000). 2. The Matsumoto–Yor independence property The following proposition comes from Matsumoto and Yor (2001), reformulated by Letac and Wesolowski (2000): Proposition 1. The Matsumoto– Yor property. Let mo0, a40 and b40. Consider two independent random variables X and Y such that X follows the law GIGðm; a; bÞ while Y follows a gamma distribution: GIGðm; 0; bÞ. Then the random variables U ¼ ðX þ Y Þ1 and V ¼ X 1 ðX þ Y Þ1 are independent. The above proposition was proved by Matsumoto and Yor in the case a ¼ b. Letac and Wesolowski (2000) noted that it is true also if aab and proved that it is in fact a characterization of GIG laws: Proposition 2. Consider two independent non-Dirac and positive random variables X and Y.
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Then the random variables U ¼ ðX þ Y Þ1 and V ¼ X 1 ðX þ Y Þ1 are independent if and only if there exist mo0, a40 and b40 such that the law of X is GIGðm; a; bÞ and Y follows the gamma distribution GIGðm; 0; bÞ. Furthermore, the laws of U and V are, respectively, GIGðm; b; aÞ and GIGðm; 0; aÞ. Note that the transformation from ð0; 1Þ2 onto ð0; 1Þ2 : ðx; yÞ7!ðu; vÞ ¼ ððx þ yÞ1 ; x1 ðx þ yÞ1 Þ is an involution and preserves, according to the above proposition, a random vector containing two independent GIG variables with suitable parameters. The expression of this transformation induces a link between the Matsumoto–Yor property and a work by Barndorff-Nielsen and Koudou (1998). This link is made explicit in the following section. 3. The Matsumoto–Yor property as a consequence of a result on finite trees equipped with random resistances Let us first recall the setting of BK98. Consider a finite tree T and denote its sets of vertices and of edges respectively by E and V. Let us select one vertex s in V, called the root, inducing thus a natural order on the tree. The set of all terminal vertices, called the boundary of T, is denoted by qT. Suppose that each edge ending at a vertex v of T has a resistance X v . Then, the total resistance R of the network is defined according to the Kirchoff–Ohm laws. For instance, for the tree illustrated by Fig. 1, with edge resistances X 1 to X 6 , the total resistance is 1 1 1 1 1 þ X 1 R ¼ X 1 þ ½fX 2 þ ðX 1 4 þ X5 þ X6 Þ g 3 .
Let us associate to each vertex v 2 V nfsg two positive real numbers av and bv fulfilling the following consistency conditions: For each v 2 V , X bv ¼ bv 0 ;
(3)
ðv;v0 Þ2E
the sum a¼
X
av
(4)
v2pnfsg
is the same along all paths p starting from the root s and ending at a terminal vertex. By Theorem 1 and Proposition 1 of BK98 we have the following:
X5 X4
X6 X2 X3 X1 s
Fig. 1. A tree-network with six edges equipped with resistances X 1 ; . . . ; X 6 .
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Proposition 3. Suppose that the distributions of the random variables ðX v ; v 2 V nfsgÞ are as follows: (
X v RIGðav ; bv Þ ¼ GIGð12 ; av ; bv Þ
if v 2 qT;
GIGð 12 ; av ; bv Þ
otherwise:
X v IGðav ; bv Þ ¼
(5)
Then (i) the total resistance R of the tree follows the RIGða; bÞ distribution, where a is given by (4) and X b¼ bv , v2qT
(ii) the variables W¼
X
! b2v X v
b2 R
v2V
and Z¼
X
! a2v X 1 v
a2 R1 ,
v2V
are independent, the vector ðW ; ZÞ and the variable R are independent. Furthermore, W is gamma-distributed with parameters A and 2, Z is gamma-distributed with parameters B and 2, where A ¼ ðjV j jqTjÞ=2 is half the number of internal vertices and B ¼ ðjqTj 1Þ=2. Two remarks concerning the above proposition: The idea of (i) is due to Barndorff-Nielsen (1994) and lies upon Eqs. (1) and (2). The item (ii) has not the same formulation as in BK98. In the latter paper the authors studied the conditional law of ðW ; ZÞ given R ¼ r, which turned out not to depend on r. Now we give the very simple result of the present paper: Proposition 4. The independence property established in BK98 implies the case m ¼ 12 of the Matsumoto– Yor independence property.
2
Y
1
X
s Fig. 2. A two-edge tree with resistances X and Y.
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Proof. Let a40 and b40. We recall that the inverse Gaussian law IGða; bÞ considered in BK98 is the case m ¼ 12 of the law GIGðm; a; bÞ. Consider two independent random variables X and Y such that X follows the law GIGð 12 ; a; bÞ while Y follows a gamma distribution: GIGð12 ; 0; bÞ. Let us apply BK98 to the following trivial tree: this tree consists of three vertices s, 1; 2 and two edges ðs; 1Þ and ð1; 2Þ, s being the root vertex (see Fig. 2). Suppose the edge ðs; 1Þ has resistance X while the terminal edge ð1; 2Þ has resistance Y. Then the consistency conditions (3), (4) and the assumption (5) are fulfilled, and it comes from (ii) of Proposition 3 that Z and R are independent. But we have for this trivial tree, R¼X þY and Z ¼ a2 X 1 a2 R1 ¼ a2 ðX 1 ðX þ Y Þ1 Þ. Thus ðX þ Y Þ1 and X 1 ðX þ Y Þ1 are independent, which is the Matsumoto–Yor property.
&
Acknowledgements The result of this paper was established at the end of a short stay at the Department of Mathematical Sciences of Aarhus University, Denmark. The author is grateful to Professor O.E. Barndorff-Nielsen who made possible that visit, and to Professor Letac who mentioned during a conversation the paper of Massam and Wesolowski. References Barndorff-Nielsen, O.E., 1994. A note on electrical networks and the inverse Gaussian distribution. Adv. Appl. Probab. 26, 63–67. Barndorff-Nielsen, O.E., Koudou, 1998. Trees with random conductivities and the (reciprocal) inverse Gaussian distribution. Adv. Appl. Probab. 30, 409–424. Bhattacharya, R.N., Waymire, E.C., 1990. Stochastic Processes with Applications. Wiley, New York. Letac, G., Wesolowski, J., 2000. An independence property for the product of GIG and gamma laws. Ann. Probab. 28, 1371–1383. Massam, H., Wesolowski, J., 2004. The Matsumoto–Yor property on trees. Bernoulli 10 (4), 685–700. Matsumoto, H., Yor, M., 2001. An analogue of Pitman’s 2M X theorem for exponential Wiener functional, Part II: the role of the generalized inverse Gaussian laws. Nagoya Math. J. 162, 65–86. Vallois, P., 1991. La loi Gaussienne inverse ge´ne´ralise´e comme premier ou dernier temps de passage de diffusion. Bull. Sci. Math. 2e Se´rie, 115, 301–368.
Statistics & Probability Letters 76 (2006) 1097–1101 www.elsevier.com/locate/stapro
A link between the Matsumoto–Yor property and an independence property on trees Angelo Efoe´vi Koudou Institut Elie Cartan, Laboratoire de Mathe´matiques, B.P. 239, F-54506 Vandoeuvre-le`s-Nancy, Cedex, France Received 4 October 2005 Available online 22 December 2005
Abstract We prove that an independence property established by Matsumoto and Yor [2001. An analogue of Pitman’s 2M X theorem for exponential Wiener functional, Part II: the role of the generalized inverse Gaussian laws. Nagoya Math. J. 162, 65–86] and by Letac and Wesolowski [2000. An independence property for the product of GIG and gamma laws. Ann. Probab. 28, 1371–1383] is, in a particular case, a corollary of a result by Barndorff-Nielsen and Koudou [1998. Trees with random conductivities and the (reciprocal) inverse Gaussian distribution. Adv. Appl. Probab. 30, 409–424] where, for finite trees equipped with inverse Gaussian resistances, an exact distributional and independence result was established. r 2005 Elsevier B.V. All rights reserved. Keywords: Gamma distribution; Generalized inverse Gaussian distribution; Kirchoff–Ohm laws; Matsumoto–Yor property; Tree
Introduction Let X and Y be two non-dirac, positive and independent random variables. A necessary and sufficient condition for the random variables ðX þ Y Þ1 and X 1 ðX þ Y Þ1 to be independent is that X follows a generalized inverse Gaussian distribution while Y is gamma-distributed with suitable parameters. The sufficient part of this assertion, named in the related recent literature as the Matsumoto– Yor property, is included in a work by Matsumoto and Yor (2001) in a particular case, as a proposition needed in the proof of their main result. (The latter result extends the Pitman theorem to exponential functionals of Brownian motion via generalized inverse Gaussian laws.) The general case of the Matsumoto–Yor property was given by Letac and Wesolowski (2000). (Let us note that, in spite of dates of publication, the work of Matsumoto–Yor, 2001, precedes that of Letac and Wesolowski, 2000). The origin of the present paper is the observation of the form of the random variables U ¼ ðX þ Y Þ1 and V ¼ X 1 ðX þ Y Þ1 , which look like random variables dealt with in a previous paper by Barndorff-Nielsen and Koudou (1998) (referred to in the sequel as BK98) on a finite tree equipped with inverse Gaussian random resistances. It turns out that one can prove the Matsumoto–Yor property to be, in the (non-generalized) inverse Gaussian case, a consequence of an independence property established in BK98. For this, one only has to apply the result of BK98 to a trivial tree (with only two edges). This gives a better understanding of the E-mail address: [email protected]. 0167-7152/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2005.12.006
ARTICLE IN PRESS 1098
A.E. Koudou / Statistics & Probability Letters 76 (2006) 1097–1101
independence property of Barndorff-Nielsen and Koudou (1998) which seemed a little surprising to its authors at that time. That independence property could be considered, a posteriori, as one tree-version of the Matsumoto–Yor property. Note that a nice tree-version of the Matsumoto–Yor property was recently established by Massam and Wesolowski (2004). In Section 1 we remind the definition of GIG distributions and some of their properties. Section 2 recalls the results of Matsumoto–Yor and Letac-Wesolowski. In Section 3 we explain how the result of BK98 implies a particular case of the Matsumoto–Yor property. 1. The generalized inverse Gaussian distributions We recall here the definition and some properties of the generalized inverse Gaussian (GIG) laws. Some references are Barndorff-Nielsen (1994), Matsumoto and Yor (2001), Vallois (1991). Let m 2 R, a40 and b40. The GIG distribution with parameters m, a, b is the probability measure: m b xm1 GIGðm; a; bÞðdxÞ ¼ e1=2 ða2 x1 þ b2 xÞ1ð0;1Þ ðxÞ dx, a 2K m ðabÞ where K m is the classical McDonald special function. One can have a ¼ 0 if m40 or b ¼ 0 if mo0. In the case m ¼ 12, the law GIGðm; a; bÞ is the classical inverse Gaussian distribution with density a IGða; bÞðdxÞ ¼ pffiffiffiffiffiffi eab x3=2 e1=2 ða2 x1 þ b2 xÞ1ð0;1Þ ðxÞ dx, 2p while in the case m ¼ 12 we have the reciprocal inverse Gaussian distribution with density b RIGða; bÞðdxÞ ¼ pffiffiffiffiffiffi eab x1=2 e1=2 ða2 x1 þ b2 xÞ1ð0;1Þ ðxÞ dx. 2p Note that RIGð0; bÞ is the gamma distribution with density 2 b pffiffiffiffiffiffi x1=2 eb =2x 1ð0;1Þ ðxÞ dx 2p
i.e. the gamma distribution with shape parameter Some convolution properties hold:
1 2
and scale parameter 2=b2 .
IGða1 ; bÞ IGða2 ; bÞ ¼ IGða1 þ a2 ; bÞ,
(1)
IGða1 ; bÞ RIGða2 ; bÞ ¼ RIGða1 þ a2 ; bÞ.
(2)
A diffusion interpretation exists for IG and RIG laws: they are, respectively, the distribution of the first and the last hitting time for a Brownian motion. Proofs can be seen in Bhattacharya and Waymire (1990) for the IG case and, for a more general proof, in Vallois (1991). The GIG density exists also as a probability measure on the set of positive definite matrices, the case a ¼ 0 defining Wishart matrices (see Letac and Wesolowski, 2000). 2. The Matsumoto–Yor independence property The following proposition comes from Matsumoto and Yor (2001), reformulated by Letac and Wesolowski (2000): Proposition 1. The Matsumoto– Yor property. Let mo0, a40 and b40. Consider two independent random variables X and Y such that X follows the law GIGðm; a; bÞ while Y follows a gamma distribution: GIGðm; 0; bÞ. Then the random variables U ¼ ðX þ Y Þ1 and V ¼ X 1 ðX þ Y Þ1 are independent. The above proposition was proved by Matsumoto and Yor in the case a ¼ b. Letac and Wesolowski (2000) noted that it is true also if aab and proved that it is in fact a characterization of GIG laws: Proposition 2. Consider two independent non-Dirac and positive random variables X and Y.
ARTICLE IN PRESS A.E. Koudou / Statistics & Probability Letters 76 (2006) 1097–1101
1099
Then the random variables U ¼ ðX þ Y Þ1 and V ¼ X 1 ðX þ Y Þ1 are independent if and only if there exist mo0, a40 and b40 such that the law of X is GIGðm; a; bÞ and Y follows the gamma distribution GIGðm; 0; bÞ. Furthermore, the laws of U and V are, respectively, GIGðm; b; aÞ and GIGðm; 0; aÞ. Note that the transformation from ð0; 1Þ2 onto ð0; 1Þ2 : ðx; yÞ7!ðu; vÞ ¼ ððx þ yÞ1 ; x1 ðx þ yÞ1 Þ is an involution and preserves, according to the above proposition, a random vector containing two independent GIG variables with suitable parameters. The expression of this transformation induces a link between the Matsumoto–Yor property and a work by Barndorff-Nielsen and Koudou (1998). This link is made explicit in the following section. 3. The Matsumoto–Yor property as a consequence of a result on finite trees equipped with random resistances Let us first recall the setting of BK98. Consider a finite tree T and denote its sets of vertices and of edges respectively by E and V. Let us select one vertex s in V, called the root, inducing thus a natural order on the tree. The set of all terminal vertices, called the boundary of T, is denoted by qT. Suppose that each edge ending at a vertex v of T has a resistance X v . Then, the total resistance R of the network is defined according to the Kirchoff–Ohm laws. For instance, for the tree illustrated by Fig. 1, with edge resistances X 1 to X 6 , the total resistance is 1 1 1 1 1 þ X 1 R ¼ X 1 þ ½fX 2 þ ðX 1 4 þ X5 þ X6 Þ g 3 .
Let us associate to each vertex v 2 V nfsg two positive real numbers av and bv fulfilling the following consistency conditions: For each v 2 V , X bv ¼ bv 0 ;
(3)
ðv;v0 Þ2E
the sum a¼
X
av
(4)
v2pnfsg
is the same along all paths p starting from the root s and ending at a terminal vertex. By Theorem 1 and Proposition 1 of BK98 we have the following:
X5 X4
X6 X2 X3 X1 s
Fig. 1. A tree-network with six edges equipped with resistances X 1 ; . . . ; X 6 .
ARTICLE IN PRESS A.E. Koudou / Statistics & Probability Letters 76 (2006) 1097–1101
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Proposition 3. Suppose that the distributions of the random variables ðX v ; v 2 V nfsgÞ are as follows: (
X v RIGðav ; bv Þ ¼ GIGð12 ; av ; bv Þ
if v 2 qT;
GIGð 12 ; av ; bv Þ
otherwise:
X v IGðav ; bv Þ ¼
(5)
Then (i) the total resistance R of the tree follows the RIGða; bÞ distribution, where a is given by (4) and X b¼ bv , v2qT
(ii) the variables W¼
X
! b2v X v
b2 R
v2V
and Z¼
X
! a2v X 1 v
a2 R1 ,
v2V
are independent, the vector ðW ; ZÞ and the variable R are independent. Furthermore, W is gamma-distributed with parameters A and 2, Z is gamma-distributed with parameters B and 2, where A ¼ ðjV j jqTjÞ=2 is half the number of internal vertices and B ¼ ðjqTj 1Þ=2. Two remarks concerning the above proposition: The idea of (i) is due to Barndorff-Nielsen (1994) and lies upon Eqs. (1) and (2). The item (ii) has not the same formulation as in BK98. In the latter paper the authors studied the conditional law of ðW ; ZÞ given R ¼ r, which turned out not to depend on r. Now we give the very simple result of the present paper: Proposition 4. The independence property established in BK98 implies the case m ¼ 12 of the Matsumoto– Yor independence property.
2
Y
1
X
s Fig. 2. A two-edge tree with resistances X and Y.
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Proof. Let a40 and b40. We recall that the inverse Gaussian law IGða; bÞ considered in BK98 is the case m ¼ 12 of the law GIGðm; a; bÞ. Consider two independent random variables X and Y such that X follows the law GIGð 12 ; a; bÞ while Y follows a gamma distribution: GIGð12 ; 0; bÞ. Let us apply BK98 to the following trivial tree: this tree consists of three vertices s, 1; 2 and two edges ðs; 1Þ and ð1; 2Þ, s being the root vertex (see Fig. 2). Suppose the edge ðs; 1Þ has resistance X while the terminal edge ð1; 2Þ has resistance Y. Then the consistency conditions (3), (4) and the assumption (5) are fulfilled, and it comes from (ii) of Proposition 3 that Z and R are independent. But we have for this trivial tree, R¼X þY and Z ¼ a2 X 1 a2 R1 ¼ a2 ðX 1 ðX þ Y Þ1 Þ. Thus ðX þ Y Þ1 and X 1 ðX þ Y Þ1 are independent, which is the Matsumoto–Yor property.
&
Acknowledgements The result of this paper was established at the end of a short stay at the Department of Mathematical Sciences of Aarhus University, Denmark. The author is grateful to Professor O.E. Barndorff-Nielsen who made possible that visit, and to Professor Letac who mentioned during a conversation the paper of Massam and Wesolowski. References Barndorff-Nielsen, O.E., 1994. A note on electrical networks and the inverse Gaussian distribution. Adv. Appl. Probab. 26, 63–67. Barndorff-Nielsen, O.E., Koudou, 1998. Trees with random conductivities and the (reciprocal) inverse Gaussian distribution. Adv. Appl. Probab. 30, 409–424. Bhattacharya, R.N., Waymire, E.C., 1990. Stochastic Processes with Applications. Wiley, New York. Letac, G., Wesolowski, J., 2000. An independence property for the product of GIG and gamma laws. Ann. Probab. 28, 1371–1383. Massam, H., Wesolowski, J., 2004. The Matsumoto–Yor property on trees. Bernoulli 10 (4), 685–700. Matsumoto, H., Yor, M., 2001. An analogue of Pitman’s 2M X theorem for exponential Wiener functional, Part II: the role of the generalized inverse Gaussian laws. Nagoya Math. J. 162, 65–86. Vallois, P., 1991. La loi Gaussienne inverse ge´ne´ralise´e comme premier ou dernier temps de passage de diffusion. Bull. Sci. Math. 2e Se´rie, 115, 301–368.