ON A PERTURBATION OF THE STUDENT t DISTRIBUTION

  Bulletin of Mathematics Vol. 04, No. 01 (2012), pp. 25–42.

  

ON A PERTURBATION OF THE STUDENT t

DISTRIBUTION

_Abstract.

Katsuo Takano

This paper is to prove the infinite divisibility of a t distribution with

the odd degrees of freedom 2n + 1 without using of Bessel functions. The Laplace-

Stieltjes transform of an infinitely divisible probability distribution is concentrated

on the interval [0, 1). A simple expression of the Laplace-Stieltjes transform is made

by using of hypergeometric function. The L´ evy measure of the Student t distribution

obtained from the L´ evy mesure of the distribution with the density functon.

  1. INTRODUCTION A probability distribution function F (x) is called an infinitely divisi- ble probability distribution if for each integer n > 1 there is a probability distribution F (x) such that the following relation holds,

  n

  F (x) = (F )(x),

  n n

  ∗ · · · ∗ F where * denotes the convolution. If a probability distribution function F (x) is concentrated on the interval [0, ∞) and an infinitely divisible probability distribution, and if we set _{Z Z}

  ∞ ∞ −sx −sx

  η(s) = e dF (x), η n (s) = e dF n (x), Katsuo Takano – On a perturbation of the Student distributions t

  the following relation

  n

  η(s) = (η (s))

  n

  holds. It is known that the Laplace-Stieltjes transform of an infinitely di- visible probability distribution F (x) which is concentrated on the interval [0, ∞) can be written as follows: _Z

  

∞

  1

  −sx

  (e η(s) = exp{−ds + − 1) dK(x)} x

• +0

  where (c1) K(x) is nondecreasing, (c2) _R K(−0) = 0,

  ∞

  (c3) 1/x dK(x) < ∞.

  bility distribution F (x) which is concentrated on the interval [0, ∞) and if the probability distribution function F (x) has a density function f (x), the density funcion f (x) satisfies the following integral equation: _{Z x} xf (x) = f (x − t)dK(t), x > 0. If dK(t) = k(t)dt we have _Z xf (x) = f (x − t)k(t)dt, x > 0.

  (0,x)

  We will discuss about the Student t distribution. The density function of the Student t distribution with degrees of freedom r is as follows: Γ((r + 1)/2))

  1 T (t) = √

  2 (r+1)/2

  πr Γ(r/2) (1 + t /r) If r is an odd integer, r = 2n + 1 for a nonnegative integer n and if we make

  √ change of variable, t/ r = x, we have the density function Γ(n + 1)

  1 Katsuo Takano – On a perturbation of the Student distributions t

  use of the Bessel functons (cf. [3]). We will make use of the fact that if h tends to +0 the density function of the Student t distribution can be obtained by the following relation c f (x; 1, h) =

  2

  2

  2

  2

  2

  (1 + x )((1 + h) + x + x ) ) · · · ((1 + nh) c

  , →

  2 n+1

  (1 + x ) where c and c are normalised constants.

  2. THE HYPERGEOMETRIC FUNCTION Let a be a positive constant. In what follows, suppose that a = a, a =

  1

  2

  a + h, ..., a = a + nh. Let us consider the following density function

  n+1

  c f (x; a, h) = (1)

  n+1

  2

  2

  Π (a + x )

  j=1 j

  where c is a normalized constant. It holds that _X n+1 1 f (x; a, h) = c . (2)

  n+1

  2

  2

  2

  2

  Π + a )(a + x ) (−a

  j l j j=1 l=1,l6=j

  From the relation _Z

  ∞

₂

₂

  1

• x ) −t(a

  _j

  = e dt

  2

  2

  a + x

  j _Z ∞ _{2 2}

  1 √

  /v /v −a −x j −3/2

  = e π e v dv √

  πv we obtain the following equality _Z

  ∞ ₂

  1

  /v −x

  f (x; a, h) = e √

  πv

  n+1 _X √ ₂ c π /v −a _{j −3/2}

  e v dv. (3)

  n+1

  

2

  2

  Π + a ) (−a

  j l j=1 l=1,l6=j

  Let us denote the mixing density function in the integrand of (3) by g(v). The mixing density g(v) is positive on [0, ∞) and a probability density function.

  Katsuo Takano – On a perturbation of the Student distributions t

  we obtain √

  η(s) = c π

  n+1 _{Z X} ∞ ₂

  1

  /v −a −sv j −3/2

  e e v dv

  n+1

  2

  

2

  Π + a ) (−a j l

  j=1 l=1,l6=j n+1 _X √

  1

  s −2a j

  = cπ e . (4)

  n+1

  2

  2

  a Π + a )

  j

  (−a

  j l j=1 l=1,l6=j

  For n = 3 we obtain

  √

  cπ

  s −2a

  η(s) = e

  3

  a3!h (2a + h)(2a + 2h)(2a + 3h)

  2

  (−3)(2m)z (−3)(−2)(2m)(2m + 1)z

• 1 + ·

  (2m + 4) (2m + 4)(2m + 5)2!

  3

  (−3)(−2)(−1)(2m)(2m + 1)(2m + 2)z . (5) +

  (2m + 4)(2m + 5)(2m + 5)3! Making use of hypergeometric function we obtain the simple expression

  2cπ

  

m

  η(s) = z (6) F (−n, 2m; 2m + n + 1; z)

  2n+1

  n!h (2m)

  n+1 √ s −2h

  where we let z = e and m = a/h. Concerning the roots of the hyperge- ometric function F (−n, 2m; 2m + n + 1; z) the author obtained the following result (cf.[11]). Theorem 2.1

If m is a positive constant and n is a natural number the hypergeometric function F (−n, 2m; 2m + n + 1; z) has roots outside the unit

  disk.

  3. THE STUDENT t DISTRIBUTIONS We show that the probability distribution with density function (2) is in- finitely divisible and obtain the L´evy measure of the Student t distribution from the L´evy mesure of the distribution with the density functon (2). Katsuo Takano – On a perturbation of the Student distributions t

  Proof 3.1 Let us show that the density function g(v) is an infinitely divisible density for every positive integer n. To show the infinite divisibility of the distribution with g(v), it suffices to show that if dK(x) = k(x)dx the following relation _Z

  ∞ ′ −sx

  (s) = η(s) e k(x)dx −η holds and k(x) is a nonnegative function and satisfies the conditions (c1),

  (c2), (c3) imposed on an infinitely divisible probability distribution. From (6) we have _X n

  2cπ (2m)

  j j m (−n) j

  η(s) = z z , (7)

  2n+1

  n!h (2m) (2m + n + 1) j!

  n+1 j j=0 √ s

  −2h

  where we set z = e and m = a/h. From this we obtain 2cπh

  ′

  η (s) = − √

  2n+1

  n!h (2m) s

  n+1 _X n

  (2m) (m + j)

  

j j

  (−n)

  m+j

  z (8) (2m + n + 1) j!

  j j=0

  and hence

  n ′ _X

  η (s) h (2m) (m + j)

  j j

  (−n) m+j = z

  − √ η(s) s (2m + n + 1) j!

  j j=0 _X n

  (2m) (m + j)

  j j

  (−n) m+j / z . (9)

  (2m + n + 1) j!

  j j=0 √

  √

  s −2h

  If we set z = e and ℜ{ s } ≥ 0, then |z| ≤ 1.

  We note that F (−n, 2m; 2m + n + 1; z) 6= 0. The denominator of (9) does not vanish in the whole complex plane except at the origin. By the contour integration of the figure after the reference we

  Katsuo Takano – On a perturbation of the Student t distributions

  re ^iθ t _hn

  j! . (A) The integral along a small circle with the center at O. From s = re

  iθ

  , √ s =

  √ r(cos θ/2 + i sin θ/2) for −π < θ < π, we see that _I e

  st

  N D ds = − ^Z

  π −π

  e

  h

  (2m + 4)

  3 _X j=0

  (−3)

  j

  (2m)

  

j

  (m + j)e

  −j2h √ re ^iθ/2

  (2m + 4)

  j

  j

  j

  Figure 1: Contour of integration (ξ > 0, t > 0, R

  z

  1 = R cos ǫ).

  Let D =

  √ s

  3 _X j=0

  (−3)

  j

  (2m)

  j

  j

  (m + j)z

  (2m + 4)

  j

  j! ,

  N = h

  3 _X j=0

  (−3)

  

j

  (2m)

  j

  j! ^o

  _π dφ

  e

  (2m + 4)

  −j2h √ Re ^iθ/2

  (m + j)e

  j

  (2m)

  j

  (−3)

  3 _X j=0

  h

  Re ^iθ t ^hn

  −ǫ

  j! ^o / ⁿ

  dθ

  iθ/2

  Re

  j! ^oi √

  j

  (2m + 4)

  −j2h √

Re

^iθ/2

  e

  j

  (2m)

  j

  3 _X j=0

  (−3)

  √ R|e

  −tR sin φ

  √ Re

  dθ = ^{Z π 2}

  tR cos θ

  √ Re

  π _{π 2}

  |dθ = ^Z

  Re ^iθ

t

  e

  iθ/2

  π _{π 2}

  (−3)

  dθ. (10) We see that _Z

  iθ/2

  Re

  j! ^oi √

  j

  (2m + 4)

  −j2h √

Re

^iθ/2

  e

  j

  (2m)

  j

  j

  3 _X j=0

  Katsuo Takano – On a perturbation of the Student t distributions

  e

  / ⁿ √

  −j2h √ Re ^{iθ/2 o}

  (m + j)e

  j

  (2m)

  j

  (−3)

  3 _X j=0

  h

  

Re

^iθ

t _hn

  π _{π 2} −ǫ

  iθ/2

  N D ds = ^Z

  st

  e

  √ R(cos θ/2 + i sin θ/2) and we see that _Z

  we have √ s =

  iθ

  N D ds → 0 as r → +0. (B) The integral along B ⌢ D. From s = Re

  st

  ) 6= 0 we have _I e

  −2 √ re ^iθ/2

  Since for every 0 < r ≤ 1 and 0 ≤ θ ≤ π F (−1, 2m; 2m + 2; e

  Re

  3 _X j=0

  j! ^o / ⁿ

  Re ^iθ t ^hn

  j

  (2m + 4)

  −j2h √ Re ^iθ/2

  (m + j)e

  j

  (2m)

  

j

  (−3)

  3 _X j=0

  h

  e

  (−3)

  π _{π 2}

  idθ = i ^Z

  iθ

  j! ^oi Re

  j

  (2m + 4)

  −j2h √ Re ^iθ/2

  e

  

j

  (2m)

  j

• i ^{Z π}
² _{π 2} Katsuo Takano – On a perturbation of the Student distributions t

  as R → +∞. We show that _{Z π 2 iθ} √

  iθ/2 Re t

  Re e (12) _π | |dθ → 0 ₂ −ǫ as R → ∞. From the fact that

  π cos θ = cos(φ + − ǫ) = sin φ, 0 ≤ φ ≤ ǫ,

  2 ξ sin ǫ =

  ≥ sin φ ≥ 0, R we see that _{Z Z 2 π π iθ 2}

  √ √

  iθ/2 Re t tR cos θ

  R|e |dθ = e Re dθ _{π π Z Z 2} −ǫ −ǫ ₂ ǫ ǫ √ √ √

  tR sin φ tRξ/R tξ

  = Re Re dφ = Re ǫ dφ ≤ √ √

  ǫ ξ ǫ

  tξ tξ

  = e ( R sin ǫ) = e ( R ) (13) → 0 sin ǫ R sin ǫ as R → ∞.

  (C) The integrals along D → G and H → E.

  √

  iπ iπ/2

  From s = ρe s = √ρe = i√ρ, we , r ≤ ρ ≤ R on D → G and from

  Katsuo Takano – On a perturbation of the Student distributions t

  see that _Z N

  st

  e ds D

  D→G √ _{iπ/2 Z}

  3 ρe _{iπ hn o X} −j2h

  (2m) (m + j)e

  j j ρe t (−3)

  = e h (2m + 4) j j!

  D→G j=0 √ iπ/2

  3 ρe _{n oi X} −j2h

  (2m) e

  j j

  √ iπ/2 (−3) iπ / ρe e dρ

  (2m + 4) j!

  j j=0 √ _Z

  3 r ρi _{hn o X} −j2h

  (2m) (m + j)e

  j j

  (−3)

  −ρt

  e h = −

  (2m + 4) j!

  R j j=0 √

  3 _{n −j2h oi X}

ρi

  (2m) e dρ

  j j

  (−3) /

  √ (2m + 4) j! ρi

  j j=0 √ _Z

  3 R ρi _{hn o X} −j2h

  (2m) (m + j)e

  j j

  (−3)

  −ρt

  = e h (2m + 4) j!

  j r j=0 √

  3

ρi

_{n oi X} −j2h j (2m) j e dρ

  (−3) / .

  √ (2m + 4) j! ρi

  j j=0

  √ √

  −iπ

  From s = ρe ρ we see

  = −ρ, r ≤ ρ ≤ R on H → E and from s = −i that _Z N

  st

  e ds D

  H→E ₋ √ iπ/2 _Z

  3 ρe _{− hn o iπ X} −j2h j (2m) j (m + j)e ρe t (−3)

  = e h (2m + 4) j!

  j H→E j=0 √ − _iπ/2

  3 ρe _{n oi X} −j2h

  (2m) e

  j j

  √ (−3)

  

−iπ/2 −iπ

  / ρe e dρ (2m + 4) j!

  j j=0 Katsuo Takano – On a perturbation of the Student distributions t _Z

  3 r +j2h√ρi _{hn o X}

  (2m) (m + j)e

  j j

  (−3)

  −ρt

  e h = −

  (2m + 4) j!

  j R j=0

  3 _{n oi X}

+j2h√ρi

j (2m) j e dρ

  (−3) /

  √ (2m + 4) j j! ρi

  j=0 _{Z R +j2h√ρi hn o X}

  3

  (2m) (m + j)e

  j j

  (−3)

  −ρt

  = e h (2m + 4) j!

  j r j=0

  3 _{n oi X}

+j2h√ρi

  (2m) e dρ

  j j

  (−3) / . (14)

  √ (2m + 4) j! ρi

  j j=0

  Therefore we see that

  √ _Z

  3 R ρi _{hn o X} −j2h

  1 (2m) (m + j)e

  j j

  (−3)

  −ρt

  e 2πi (2m + 4) j!

  r j j=0 √

  3 _{n −j2h o X} ρi

  (2m) e

  j j

  (−3) /

  (2m + 4) j!

  

j

j=0 3 _{n o X} +j2h√ρi

  (2m) (m + j)e

  j j

  (−3)

• (2m + 4) j!

  

j

j=0 3 _{n oi X} +j2h√ρi j (2m) j e hdρ

  (−3) /

  √ (2m + 4) j! ρi

  

j

j=0 √ _Z

  3 ρi ∞ −j2h _{hn o X}

  1 (2m) (m + j)e

  j j

  (−3)

  −ρt

  e → −

  2π (2m + 4) j!

  j j=0 √

  3 ρi _{n o X}

−j2h

  (2m) e

  j j

  (−3) /

  (2m + 4) j j!

  j=0

  3 _{n o X} +j2h√ρi

  (2m) (m + j)e

  j j

  (−3)

• (2m + 4) j!

  

j

j=0 3 _{n oi X}

+j2h√ρi

  (2m) e hdρ

  j j

  (−3) /

  √ (2m + 4) j! ρ

  j

• j2h√ρi
• ⁿ
• j2h√ρi

  j

  (−3)

  3 _X j=0

  j! ^o / ⁿ

  j

  (2m + 4)

  −j2hyi

  (m + j)e

  j

  (2m)

  (−3)

  (2m)

  3 _X j=0

  −ty ^{2 hn}

  e

  ∞

  1 π ^Z

  √ ρ = y, we obtain k(t) =

  ρ (15) as R → ∞. By change of variable,

  j! ^oi hdρ √

  j

  (2m + 4)

  j

  j

  j

  

j

  j

  (2m + 4)

  e

  j

  (2m)

  j

  (−3)

  3 _X j=0

  j! ^o / ⁿ

  (2m + 4)

  e

  (m + j)e

  j

  (2m)

  j

  (−3)

  3 _X j=0

  j! ^o

  

j

  (2m + 4)

  −j2hyi

  e

  (2m)

  j! ^oi hdy. Katsuo Takano – On a perturbation of the Student distributions t

  N D ds

  (2m)

  j

  (−3)

  3 _X j=0

  −ρt ^hn

  e

  ∞

  1 2π ^Z

  →

  st

  (m + j)e

  e

  ξ−iR ¹

  1 2πi ^{Z ξ+iR 1}

  N D ds =

  st

  e

  A→B

  1 2πi ^Z

  as r → 0 and R → ∞. From the Cauchy theorem we see that

  Katsuo Takano – On a perturbation of the Student t distributions

  j

  −j2h √ ρi

  j

  (−3)

  (−3)

  3 _X j=0

  j! ^o / ⁿ

  

j

  (2m + 4)

  (m + j)e

  j

  (2m)

  j

  3 _X j=0

  (2m + 4) j j! ^o / ⁿ

  j! ^o

  j

  (2m + 4)

  −j2h √ ρi

  e

  j

  (2m)

  j

  (−3)

  3 _X j=0

• j2hyi
• ⁿ

• +j2hyi

  For the general case n we obtain _Z n ∞ −j2hyi _{2 hn o X} 1 (2m) (m + j)e

  j j

  (−n)

  −ty

  k(t) = e π (2m + n + 1)

  j j=0 n _{n o X} −j2hyi

  (2m) e

  j j

  (−n) /

  (2m + n + 1) j

  j=0 n _{n o X} +j2hyi j (2m) j (m + j)e

  (−n)

• (2m + n + 1) j

  j=0 n _{n oi X} +j2hyi

  (2m) e

  j j

  (−n) / hdy. (16)

  (2m + n + 1)

  j j=0

  To show the infinite divisibility it is necessary to show that the following function _{n −j2hyi o X} n (2m) (m + j)e

  

j j

  (−n) ℜ

  (2m + n + 1)

  j j=0 n _{n o X} +j2hyi

  (2m) e

  j j

  (−n) ·

  (2m + n + 1)

  j j=0

  is nonnegative for y ≥ 0. Let 2hy = θ in the above and let

  n _{n o X} −i(m+j)θ

  (2m) (m + j)e

  

j j

  (−n) A = ℜ

  (2m + n + 1) j

  j=0 n

_{n o}

_X +i(m+j)θ

j (2m) j e

  (−n) . ·

  (2m + n + 1)

  j j=0

  If n = 0 we obtain

  If n = 1 we obtain 2m(2m + 1)

  A = (1 − cos θ).

  2m + 2 Katsuo Takano – On a perturbation of the Student distributions t

  Let

  n _X i(m+j)θ

  (2m) e

  j j

  (−n)

  2 B = | |

  (2m + n + 1) j!

  j j=0 iθ

  2

  . (17) = |F (−n, 2m; 2m + n + 1; e )|

  From the fact that A is nonnegative for θ = 2hy ≥ 0 we see that the function _Z

  

∞

₂

  1

  2A

  −ty

  k(t) = e hdy (18) π B is positive for t > 0 and we obtain

  

n n

  2A 2 (2m) n+1 (1 − cos θ)

  =

  iθ

  2 B (2m + n + 1) n

  |F (−n, 2m; 2m + n + 1; e )|

  n n

  = 2 (2m)

  n+1

  (1 − cos 2hy) _{n X} n (2m) n

  j

  2n − j (2m + n + 1) j n

  n−j j=0 _o j j

  . (19) (2(n − j))!2 (1 − cos 2hy)

  √ After all, by change of variable, y = w, we obtain _Z

  ∞

  √

  n −tw n−1

  k(t) = e 2 (2m) w) h dw

  n+1

  (1 − cos 2h

  n

  √ _{. X} j (2m) 2 w n

  j

  2n − j π

  (2m + n + 1) j n

  

n−j

j=0

  √

  j

  w) (2(n − j))!(1 − cos 2h and therefore _Z

  ∞

  √

  2n −tw 2n−1

  k(t) = e 2 (2m) (sin h w) h dw

  n+1 n

  √ _{. X} 2j (2m) 2 w n

  j

  2n − j π

  (2m + n + 1) j n

  

n−j

j=0

  √

  2j Katsuo Takano – On a perturbation of the Student distributions t

  distribution with the density function (2) is infinitely divisible since it is a mixture density of the normal distributions.

  Let us denote the characteristic fuction of the probability distribution with the density function (2) in the following form _{h i Z} itx l(x)

  

itx

φ(t) = exp e dx .

  − 1 −

  2

  1 + x x

  R−{0}

  In what follows we will obtain the measure l(x)dx/x. We have _{Z Z}

• ∞ ∞
₂

  1

  itx /v −x

  φ(t) = e e g(v)dv dx √

  πv _Z −∞ ∞ ₂ /4

  −vt

  = e g(v)dv and _Z

• ∞

  k(x)

  

−sx

  log φ(t) = (e dx − 1) x _{Z Z} +0 ∞ ∞

  1

  

−sx −xw

  = (e e U (w)dw dx − 1) x _Z +0

  

2

∞

  t log(1 + )U (w)dw (21) = −

  4w

  2

  where we set s = t /4 and √

  2n 2n−1

  U (w) = 2 (2m) (sin h w) h

  n+1 n

  √ _{. X}

2j

(2m) 2 w n

  j

  2n − j π

  (2m + n + 1) j n

  n−j j=0

  √

  2j w) .

  (2(n − j))!(sin h Using the following equality _Z

  2

√

  t itu du

  itu −2 w|u|

  ) = e e , − log (1 + − 1 −

  2

  4w 1 + u |u|

  R−{0}

  we obtain φ(t) _{Z Z h n √ o i} ∞ itu du

  itu Katsuo Takano – On a perturbation of the Student distributions t

  We see that the function l(x) can be given in the following form _Z

  

∞

√ −2 w |x|

  l(x) = (sgn x) e U (w)dw _Z

  ∞ −|x| v 2n−1 = (sgn x) e _h

  2

  2n

  (2m) h(sin(hv/2)) /

  n+1 n _X 2j

  n (2m) j

  2 2n − j

  π (2m + n + 1) j n

  n−j j=0 _i 2j

  dv. (22) (2(n − j))!(sin(hv/2))

  Let us denote the characteristic function of the Student t distribution with odd degrees of freedom in the following form _{h i Z} itx l (x)

  st itx

  φ(t) = exp e dx .

  − 1 −

  2

  1 + x x

  R−{0}

  Theorem 3.2 The function l st (x) can be given in the explicit form _Z

  ∞ −|x|v

  l (x) = (sgn x) e

  st _X n

  n

  2n+1 2n 2j 2n − j 2j

  (2a) v dv/ 2π (2a) (2(n − j))!v }. (23) j n

  j=0 We take a = 1 for the Student t distribution.

  Proof 3.2 By (22) and hm = a we see that _Z

  ∞

−|x| v

_h l(x) = (sgn x) e 2n 2n−1

  2 (2m) h (sin(hv/2))/(hv/2)

  n+1 n _{n X} 2j

  (2m) 2 n

  j

  2n − j Katsuo Takano – On a perturbation of the Student distributions t

  From the above we see that _Z

  ∞ _{h 2 n+1} 2n −|x| v

  l st (x) = (sgn x) e (2a) v / _X n _i n

  2j 2n − j 2j

  2π (2a) dv (2(n − j))!v j n

  j=0 as h tends +0 and we obtain (23).

  If a = 1 we have _Z

  

∞

_h 2n −|x|y

  l st (x) = (sgn x) e 2(2y) /

_{n oi}

_X n n

  2n − j 2j 2π dy.

  (2(n − j))!(2y) j n

  j=0

  (25) In order to show that the results here coincide with those formulae of which have been already obtained we write down the several cases (cf. [1]).

  If n = 1 _{Z ∞}

  2

  y

  −|x|y l (x) = (sgn x) e dy. st

  2

  π(1 + y ) If n = 2 _{Z ∞}

  4

  y

  

−|x|y

l st (x) = (sgn x) e dy.

  2

  2

  4

  π(3 + 3y + y ) If n = 3 _{Z ∞}

  6

  y

  −|x|y l (x) = (sgn x) e dy. st

  2

  4

  6

  π(225 + 45y + 6y + y ) If n = 4 _{Z ∞}

  −|x|y

  l st (x) = (sgn x) e

  

8

  y dy. (26)

  2

  4

  6

  8

  π(11025 + 1575y + 135y + 10y + y ) From the above we see that the function l (x) can be decomposed to the Katsuo Takano – On a perturbation of the Student distributions t

  If n = 1 _{Z ∞}

  1

  1

  −|x|y

  l st (x) = e dy − (sgn x)

  2

  πx π(1 + y ) _{Z Z h} →∞ →∞ _i 1 sgn x sin y cos y

  = dy ,

  − cos |x| dy − sin |x| πx π y y

  |x| |x|

  (27) (x 6= 0). If n = 2 _{Z ∞}

  2

  2

  1 3 + 3y

  −|x|y l st (x) = e dy.

  − (sgn x)

  2

  2

  4

  πx π(3 + 3y + y ) If n = 3

  1 l st (x) = πx _{Z ∞}

  2

  4

  225 + 45y + 6y

  −|x|y e dy.

  −(sgn x)

  2

  4

  6

  π(225 + 45y + 6y + y )

  References

  [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Func- tions, New York, Dover, 1970. [2] L. Bondesson, Generalized Gamma Convolutions and Related Classes of Distributions and Densities, Lecture Note in Statistics, 76, Springer-

  Verlag, 1992 [3] E. Grosswald, The Student t-distribution of any degree of freedom is infinitely divisible, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 36

  (1976), 103–109. [4] C. Halgreen, Self-decomposability of the generalized inverse Gaussian distribution and hyperbolic distributions, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 47 (1979), 13–17. [5] D. H. Kelker, Infinite divisibility and variance mixtures of the normal distribution, Ann. Math. Statist., 42 (1971), 802–808.

  Katsuo Takano – On a perturbation of the Student distributions t [7] L. J. Slater, Generalized hypergeometric functions, Cambridge Univ.

  Press, Cambridge, 1966. [8] F. W. Steutel, Preservation of infinite divisibility under mixing and related topics, Math. Centre Tracts, Math. Centre, Amsterdam, 33.

  1970 [9] F.W.Steutel, K.van Harn, Infinite divisibility of probability distribu- tions on the real line

  , Marcel Dekker, 2004 [10] K. Takano, On infinite divisibility of normed product of Cauchy densi- ties, J. Comput. Applied Math., 150(2003), 253-263.

  [11] K. Takano & H. Okazaki, The Gauss hypergeometric series with roots outside of the unit disk, Proceedings of the 8-th international conference on difference equations and applications, Edited by S. Elaydi, G. Ladas et all, (Held at Czech) (2005), 265-272.

  [12] K. Takano, Hypergeometric functions and infinite divisibility of prob- ability distributions consisting of Gamma functions, International J.

  Pure and Applied Math., 20 no.3 (2005), 379-404. [13] O. Thorin, On the infinite divisibility of the Pareto distribution, Scand.

  Acturial. J., (1977), 31–40. [14] G. N. Watson, A Treatise on the Theory of Bessel Functions, Second _{Katsuo Takano} Edition, Cambridge University Press, 1966

  : Department Mathematics, Faculty of Science, Ibaraki University, Mito-city, Ibaraki 310 Japan

  E-mail: ktaka@mx.ibaraki.ac.jp