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Discrete stable random variables
STATISTICS & PROBABILITY LETTERS ELSEVIER
Statistics & Probability Letters 37 (1998) 243-247
Discrete stable random variables Gerd Christoph*, Karina Schreiber Fakultiitfiir Mathematik, Inst fiir Mathematische Stochastik, Otto-von-Guericke-Universit~itMagdebur9, PF 4120, 39016 Maodeburg, Germany Received 1 February 1997; received in revised form 1 May 1997
Abstract
In Steutel and van Ham (1979) a discrete analogue of stable random variables was introduced. In the present note we consider some explicit and asymptotic formulae for the probabilities of discrete stable random variables and give rates for the convergence of sequences of certain discrete stable random variables to a stable one. @ 1998 Elsevier Science B.V. All rights reserved
AMS classification: primary, 60E05; 60F05; secondary 60E07 Keywords: Strictly stable and discrete stable random variables; Explicit and asymptotic formulae for probabilities; Rate of convergence
I. Introduction and preliminaries Steutel and van H a m (1979) introduced the discrete stability for integer valued random variables (r.v.'s) via probability generating functions (p. g. f.'s), where a nonnegative lattice r.v. X with p. g. f.
gx(z)=E(zX)=exp{-2(1-z)'/},
y C(0,1],
2>0,
Izl~
(1)
is called discrete stable distributed with exponent y and parameter 2. In the mentioned paper it is proved that X is infinitely divisible, unimodal, discrete self-decomposable and normally attracted to a stable law (if 7 < 1). If y = 1 then the discrete stable r.v. X is Poisson(2) distributed. Devroye (1993) discussed methods to efficiently generate discrete stable r.v.'s. In the present note some more properties of discrete stable r.v.'s are considered. The following distribution will be useful in the representation o f discrete stable r.v.'s. To it let us consider a sequence of independent Bernoulli trials, where the kth trial successfully ends with probability 7/k. Then the random number Z of trials required to achieve the first success possesses the following probabilities:
P(Z=k)=(-1)k+'(7,_~,
k = l , 2 .... ;
7E(0,1].
* Corresponding author. Tel.: +49391 67 18652; fax: +49391 67 11172; e-mail: [email protected]. 0167-7152/98/$19.00 @ 1998 Elsevier Science B.V. All rights reserved PII SO 167-71 52(97)00123-5
(2)
G. Christoph, K. Schreiber / Statistics & Probability Letters 37 (1998) 243-247
244
A r.v. with these probabilities is called Sibuya(~) distributed. Note, that the Sibuya(1) distribution is degenerate with P(Z= 1 ) = 1. For the p.g.f, of a Sibuya(7) distributed r.v. Z we get g z ( z ) = a - (1 - z)',',
~ ~ (o, 1],
Izl ~< 1.
(3)
Combining p.g. fi's given in (1) and (3) we obtain
9x(z) =
exp{2 (gz(z) - 1)}.
(4)
Steutel and van Ham (1979) also find a canonical representation for discrete self-decomposable p. g. f.'s by a unique p.g.f. Q with Q ( 0 ) = 0 and a parameter z > 0 . Thus, for the discrete stable p.g.f. 9x the relation (4) implies that this Q is the p. g. f. of a Sibuya(7 ) distribution and z = 2/y. Following van Ham et al. (1982) or Devroye (1993) the discrete stable r. v. X may be obtained as a random sum
x
where
(5)
d=z~ + . . . + Z r ,
Z,ZbZ2....
is a sequence of independent Sibuya(y) distributed r.v.'s , Y is a Poisson(2) distributed
r.v. independent of Zi, and ~ denotes the equality in distribution. Moreover, a discrete stable r.v. X with exponent 7 and parameter 2 is distributed as a Poisson r.v. with positive random parameter S¢~ having Laplace transform ~r(z; 2 ) = e -;~z:, 0 <7~< 1, 2 > 0 . Note that S~~ with 0 < y < 1 is (strictly) stable with characteristic exponent ~, skewness parameter 1 and scale parameter 2 > 0, whereas S( has a degenerate distribution with P(S( = 2) = 1. Let us mention that the relation (4) may be obtained from (5) in a direct way.
2. Probabilities Expanding the p. g. f. 9x(z) given in (1) in a power series (first the exponential function and then (1 - z ) ~ Q we obtain for the probabilities of a discrete stable r.v. X
P(X=k)=(-l)
k
j!
,
k = 0 , 1 , 2 .... ;
7E(0,1],
(6)
j=0
and the sum of the right-hand side of (6) is absolutely convergent. The relation (4) leads to the formal identity
Ee(x=k)zk = e - exp 2 k=0
e(z=k)? , Izl
which connects the probabilities P(X = k ) , k = 0 , 1,2 ..... of the discrete stable distribution with the probabilities of the Sibuya(y) distribution P(Z=k), k= 1,2 . . . . . given in (2). Hence P ( X - - - 0 ) = e -;~ and using Lemma 5.6 of Petrov (1995, p. 170) we find k
P(X=k)=(-1)ke-2 Z I-I (--2)v"-----~'(m) m=l
/)m,
k=
1,2 .... ;
7E(O, 1],
(7)
Um!
where the summation is carried out over all nonnegative integer solutions (vl,v2 ..... vk) of the equation vl +2v2 + . . . +kvk =k.
G, Christoph, K. Schreiber I Statistics & Probability Letters 37 (1998) 243-247
245
Finally, Equation (4) leads us to a recursion formula. Taking the derivative in (4) we obtain g'x(Z) = 2gx(z) g~(z) for Iz[ < 1. Expanding the functions in series we find k
(k + l ) P ( X : k
+ l):2
Z
p(X:k_m)(m+
l)(_l)m
m:0
() 7 m+l
(8)
for k = 0 , 1,2 .... and P ( X = 0 ) = e - ; . Note that (8) coincides with the recursion formula (1.5) for infinitely divisible r.v.'s in Lemma 1.2 of Steutel and van Ham (1979), using again the connection between discrete stable and Sibuya r.v.'s .
3. Asymptotic behavior Now, we are interested in the asymptotic behavior of both P ( X = n) and P(X >~n) for large n. Theorem 1. Let X be discrete stable with exponent 7, 0 <7 < 1, and parameter 2. Then we have for any fixed integer m and n --+ cx~,
1 ~-~ P ( X = n ) = -~ ~= (--1) j! j+l )j sin(Tj~)B(Tj + 1,n - 7J) + O(n-''lm+l) --1)
(9)
and P(X/> n)-- 1 ~
( - 1j!) j+l 2J sin(Tjn)B(Tj, n - 7J) + O(n-~"(m+l)),
(10)
j=l
where B(x, y) = F(x) F(y)/F(x + y) is the beta function. Moreover, P(X = n ) = 1 [(~/'A (_l)j+ 1 '~JF(7j + 1)sin(TJ~) n-~'j-r + O(n-7-2) rc j!
(ll)
P(X>~n)= 1 [(~)/~'] (-1)j! j+l 2J r(7j)sin(Tj~)n-;'J + O(n-~'-l).
(12)
j=l
and
j=l
Proof. Let 7 ¢ (0, 1). To prove the asymptotic formulae we make use of the representation of X by a Poisson(S~ °) r.v. introduced in Section 1. Then
P(X = n) = fO ° S~.n e-S p~(s)ds,
(13)
where p.,X is the density of the r.v. S;'I. The relation pf~(x)= 2 -U'' p~,(x ~-U~') and the series representation of PI' which one can find e.g. in Christoph and Wolf (1993, pp. 13/14) or Feller (1971, p. 583) imply that
1 ~ (-1) j+l
p ~ ( s ) = ~ j=l
j!
,~Jr(Tj+1)sin(7jTt)s -~'j-1
+Am(s),
7 E (0,1),
(14)
G. Christoph, K. Schreiber / Statistics & Probability Letters 37 (1998) 243-247
246
for any m~> 1, where A m ( s ) = O ( s -7(m+l)-l) a s s ~ cx~. Let n be large and m be fixed with m + 1 < n . Putting (14) in (13) then all integrals involved are gamma functions. Stirling's formula leads to
F ( n - 7 j ) _ n _ ~ . j {1 + (7j)2 + 7J + O(n-2)'~ ] F(n) \ 2n
for any fixed J
as
(15)
n --+ (x3.
Using (15) with j = m + 1 for the remaining term, we obtain (9). To derive (11) we have to use both (14) and (15) with m = [ ( 7 + 1)/7] and apply (15) for each term of the sum, too. The convergence rate of the remaining term follows from the first term of the sum and (15). The relations (10) and (12) concerning the tail of the distribution follow from (9) and (11) considering
2
k - 7 ) - 1 =--n-1
k=n
7J Jr- O ( n - T j - 1 ) ,
n ---+ c¢
B(k - z, z + 1 ) = B ( n - %'0,
and
7J
k=,
where the latter equation follows from B(k - r, ~ + 1 ) = B(k - r, v) + B(k + 1 - z, z).
[]
Let us mention that the first term of the sum in (9) for the probabilities of a discrete stable r.v. X with parameter 0 < 7 < 1 and 2 = 1 coincides with the probabilities of a Sibuya(7) r . v . Z . With (2) and some calculus we find
P(Z=n)=(-1)n+'
( ~ ) - F ( n - 7)Tff-~-7~-ff' 7zl s i n ( T r t ) B ( n - j ' l + 7 )
1 F( 7 + 1) sin(Trr)n - ~ ' - I + O ( n - ~ - 2 )
as n ---+cx~,
7"[
where the equations in the first line are exact for all n/> 1. As a completion we consider now the well-known case of 7 = I, where X is Poisson(2) distributed. Then integrating by parts we find 1
[;
2n
P(X>>.n)= _(n _ 1)! a0 tn-l e-tdt=-~, e-;.(1 + O ( n - l ) ) = P ( X
=n),
as n ~ c~. Hence in both cases we have the same exponential convergence rate.
4. Convergence to a stable law
Let {X(")}n denote a sequence of discrete stable r.v.'s with exponent 7 and parameter 2n. Because of the infinite divisibility it follows that X {n) has the same distribution as XI -4- .-. + An, where X1 . . . . . X, are independent copies of the discrete stable r.v. X with exponent 7 and parameter 2. If 7 = 1 then {X(m}, is Poisson(2n) distributed and the sequence {n-IX{")}, tends to E ( X ) = 2. If 0 < 7 < 1 then X belongs to the domain of normal attraction of the r. v. S~', hence the limit (in distribution) of the sequence {n-l/';X(")}, is S~~. For the latter case we, finally, consider the rate of convergence. Let ~,~(z; 7, 2) be the Laplace transform of n-1/rX {~). Then some calculus leads to
~n(z; 7, 2) = 9n-, : x,,,~(e - z ) = e x p { - 2 n ( 1 - e - " - ' :'z)~,} =e-;~z;'(l+~Tz'+~n-'/~;+O(n-2/~'))
as n ---~ cx~.
(16)
G. Christoph, K. Schreiber / Statistics & Probabilio, Letters 37 (1998) 243-247
247
With Esseen's smoothness lemma we have proved
Theorem 2. Suppose 0 < 7 < 1. Then P { n - I/:' X I" ) <~x } - P { S : ! ~<~x} : O(n -1/:') as n --* vo, where the rate o f converqence m a y not be improved.
Acknowledgements The authors would like to thank the referee for his helpful suggestions and comments, which led to several improvements. References Christoph, G., Wolf, W., 1993. Convergence Theorems with a Stable Limit Law, Mathematical Research, vol. 70. Akademie-Verlag, Berlin. Devroye, L,, 1993. A triptych of discrete distributions related to the stable law. Statist. Probab. Lett. 18, 349 351. Feller, W., 1971. An Introduction to Probability Theory and its Applications, vol. 2. Wiley Series in Probability, Wiley, New York. Petrov, V.V., 1995. Limit Theorems of Probability Theory: Sequences of Independent Random Variables. Clarendon Press, Oxford. Steutel, F.W., van Ham, K., 1979. Discrete analogues of self-decomposability and stability. Ann. Probab. 7, 893-899. van Ham, K., Steutel, F.W., Vervaat, W., 1982. Self-decomposable discrete distributions and branching processes, Z. Wahrsch. verw. Gebiete 61, 97-118.
Statistics & Probability Letters 37 (1998) 243-247
Discrete stable random variables Gerd Christoph*, Karina Schreiber Fakultiitfiir Mathematik, Inst fiir Mathematische Stochastik, Otto-von-Guericke-Universit~itMagdebur9, PF 4120, 39016 Maodeburg, Germany Received 1 February 1997; received in revised form 1 May 1997
Abstract
In Steutel and van Ham (1979) a discrete analogue of stable random variables was introduced. In the present note we consider some explicit and asymptotic formulae for the probabilities of discrete stable random variables and give rates for the convergence of sequences of certain discrete stable random variables to a stable one. @ 1998 Elsevier Science B.V. All rights reserved
AMS classification: primary, 60E05; 60F05; secondary 60E07 Keywords: Strictly stable and discrete stable random variables; Explicit and asymptotic formulae for probabilities; Rate of convergence
I. Introduction and preliminaries Steutel and van H a m (1979) introduced the discrete stability for integer valued random variables (r.v.'s) via probability generating functions (p. g. f.'s), where a nonnegative lattice r.v. X with p. g. f.
gx(z)=E(zX)=exp{-2(1-z)'/},
y C(0,1],
2>0,
Izl~
(1)
is called discrete stable distributed with exponent y and parameter 2. In the mentioned paper it is proved that X is infinitely divisible, unimodal, discrete self-decomposable and normally attracted to a stable law (if 7 < 1). If y = 1 then the discrete stable r.v. X is Poisson(2) distributed. Devroye (1993) discussed methods to efficiently generate discrete stable r.v.'s. In the present note some more properties of discrete stable r.v.'s are considered. The following distribution will be useful in the representation o f discrete stable r.v.'s. To it let us consider a sequence of independent Bernoulli trials, where the kth trial successfully ends with probability 7/k. Then the random number Z of trials required to achieve the first success possesses the following probabilities:
P(Z=k)=(-1)k+'(7,_~,
k = l , 2 .... ;
7E(0,1].
* Corresponding author. Tel.: +49391 67 18652; fax: +49391 67 11172; e-mail: [email protected]. 0167-7152/98/$19.00 @ 1998 Elsevier Science B.V. All rights reserved PII SO 167-71 52(97)00123-5
(2)
G. Christoph, K. Schreiber / Statistics & Probability Letters 37 (1998) 243-247
244
A r.v. with these probabilities is called Sibuya(~) distributed. Note, that the Sibuya(1) distribution is degenerate with P(Z= 1 ) = 1. For the p.g.f, of a Sibuya(7) distributed r.v. Z we get g z ( z ) = a - (1 - z)',',
~ ~ (o, 1],
Izl ~< 1.
(3)
Combining p.g. fi's given in (1) and (3) we obtain
9x(z) =
exp{2 (gz(z) - 1)}.
(4)
Steutel and van Ham (1979) also find a canonical representation for discrete self-decomposable p. g. f.'s by a unique p.g.f. Q with Q ( 0 ) = 0 and a parameter z > 0 . Thus, for the discrete stable p.g.f. 9x the relation (4) implies that this Q is the p. g. f. of a Sibuya(7 ) distribution and z = 2/y. Following van Ham et al. (1982) or Devroye (1993) the discrete stable r. v. X may be obtained as a random sum
x
where
(5)
d=z~ + . . . + Z r ,
Z,ZbZ2....
is a sequence of independent Sibuya(y) distributed r.v.'s , Y is a Poisson(2) distributed
r.v. independent of Zi, and ~ denotes the equality in distribution. Moreover, a discrete stable r.v. X with exponent 7 and parameter 2 is distributed as a Poisson r.v. with positive random parameter S¢~ having Laplace transform ~r(z; 2 ) = e -;~z:, 0 <7~< 1, 2 > 0 . Note that S~~ with 0 < y < 1 is (strictly) stable with characteristic exponent ~, skewness parameter 1 and scale parameter 2 > 0, whereas S( has a degenerate distribution with P(S( = 2) = 1. Let us mention that the relation (4) may be obtained from (5) in a direct way.
2. Probabilities Expanding the p. g. f. 9x(z) given in (1) in a power series (first the exponential function and then (1 - z ) ~ Q we obtain for the probabilities of a discrete stable r.v. X
P(X=k)=(-l)
k
j!
,
k = 0 , 1 , 2 .... ;
7E(0,1],
(6)
j=0
and the sum of the right-hand side of (6) is absolutely convergent. The relation (4) leads to the formal identity
Ee(x=k)zk = e - exp 2 k=0
e(z=k)? , Izl
which connects the probabilities P(X = k ) , k = 0 , 1,2 ..... of the discrete stable distribution with the probabilities of the Sibuya(y) distribution P(Z=k), k= 1,2 . . . . . given in (2). Hence P ( X - - - 0 ) = e -;~ and using Lemma 5.6 of Petrov (1995, p. 170) we find k
P(X=k)=(-1)ke-2 Z I-I (--2)v"-----~'(m) m=l
/)m,
k=
1,2 .... ;
7E(O, 1],
(7)
Um!
where the summation is carried out over all nonnegative integer solutions (vl,v2 ..... vk) of the equation vl +2v2 + . . . +kvk =k.
G, Christoph, K. Schreiber I Statistics & Probability Letters 37 (1998) 243-247
245
Finally, Equation (4) leads us to a recursion formula. Taking the derivative in (4) we obtain g'x(Z) = 2gx(z) g~(z) for Iz[ < 1. Expanding the functions in series we find k
(k + l ) P ( X : k
+ l):2
Z
p(X:k_m)(m+
l)(_l)m
m:0
() 7 m+l
(8)
for k = 0 , 1,2 .... and P ( X = 0 ) = e - ; . Note that (8) coincides with the recursion formula (1.5) for infinitely divisible r.v.'s in Lemma 1.2 of Steutel and van Ham (1979), using again the connection between discrete stable and Sibuya r.v.'s .
3. Asymptotic behavior Now, we are interested in the asymptotic behavior of both P ( X = n) and P(X >~n) for large n. Theorem 1. Let X be discrete stable with exponent 7, 0 <7 < 1, and parameter 2. Then we have for any fixed integer m and n --+ cx~,
1 ~-~ P ( X = n ) = -~ ~= (--1) j! j+l )j sin(Tj~)B(Tj + 1,n - 7J) + O(n-''lm+l) --1)
(9)
and P(X/> n)-- 1 ~
( - 1j!) j+l 2J sin(Tjn)B(Tj, n - 7J) + O(n-~"(m+l)),
(10)
j=l
where B(x, y) = F(x) F(y)/F(x + y) is the beta function. Moreover, P(X = n ) = 1 [(~/'A (_l)j+ 1 '~JF(7j + 1)sin(TJ~) n-~'j-r + O(n-7-2) rc j!
(ll)
P(X>~n)= 1 [(~)/~'] (-1)j! j+l 2J r(7j)sin(Tj~)n-;'J + O(n-~'-l).
(12)
j=l
and
j=l
Proof. Let 7 ¢ (0, 1). To prove the asymptotic formulae we make use of the representation of X by a Poisson(S~ °) r.v. introduced in Section 1. Then
P(X = n) = fO ° S~.n e-S p~(s)ds,
(13)
where p.,X is the density of the r.v. S;'I. The relation pf~(x)= 2 -U'' p~,(x ~-U~') and the series representation of PI' which one can find e.g. in Christoph and Wolf (1993, pp. 13/14) or Feller (1971, p. 583) imply that
1 ~ (-1) j+l
p ~ ( s ) = ~ j=l
j!
,~Jr(Tj+1)sin(7jTt)s -~'j-1
+Am(s),
7 E (0,1),
(14)
G. Christoph, K. Schreiber / Statistics & Probability Letters 37 (1998) 243-247
246
for any m~> 1, where A m ( s ) = O ( s -7(m+l)-l) a s s ~ cx~. Let n be large and m be fixed with m + 1 < n . Putting (14) in (13) then all integrals involved are gamma functions. Stirling's formula leads to
F ( n - 7 j ) _ n _ ~ . j {1 + (7j)2 + 7J + O(n-2)'~ ] F(n) \ 2n
for any fixed J
as
(15)
n --+ (x3.
Using (15) with j = m + 1 for the remaining term, we obtain (9). To derive (11) we have to use both (14) and (15) with m = [ ( 7 + 1)/7] and apply (15) for each term of the sum, too. The convergence rate of the remaining term follows from the first term of the sum and (15). The relations (10) and (12) concerning the tail of the distribution follow from (9) and (11) considering
2
k - 7 ) - 1 =--n-1
k=n
7J Jr- O ( n - T j - 1 ) ,
n ---+ c¢
B(k - z, z + 1 ) = B ( n - %'0,
and
7J
k=,
where the latter equation follows from B(k - r, ~ + 1 ) = B(k - r, v) + B(k + 1 - z, z).
[]
Let us mention that the first term of the sum in (9) for the probabilities of a discrete stable r.v. X with parameter 0 < 7 < 1 and 2 = 1 coincides with the probabilities of a Sibuya(7) r . v . Z . With (2) and some calculus we find
P(Z=n)=(-1)n+'
( ~ ) - F ( n - 7)Tff-~-7~-ff' 7zl s i n ( T r t ) B ( n - j ' l + 7 )
1 F( 7 + 1) sin(Trr)n - ~ ' - I + O ( n - ~ - 2 )
as n ---+cx~,
7"[
where the equations in the first line are exact for all n/> 1. As a completion we consider now the well-known case of 7 = I, where X is Poisson(2) distributed. Then integrating by parts we find 1
[;
2n
P(X>>.n)= _(n _ 1)! a0 tn-l e-tdt=-~, e-;.(1 + O ( n - l ) ) = P ( X
=n),
as n ~ c~. Hence in both cases we have the same exponential convergence rate.
4. Convergence to a stable law
Let {X(")}n denote a sequence of discrete stable r.v.'s with exponent 7 and parameter 2n. Because of the infinite divisibility it follows that X {n) has the same distribution as XI -4- .-. + An, where X1 . . . . . X, are independent copies of the discrete stable r.v. X with exponent 7 and parameter 2. If 7 = 1 then {X(m}, is Poisson(2n) distributed and the sequence {n-IX{")}, tends to E ( X ) = 2. If 0 < 7 < 1 then X belongs to the domain of normal attraction of the r. v. S~', hence the limit (in distribution) of the sequence {n-l/';X(")}, is S~~. For the latter case we, finally, consider the rate of convergence. Let ~,~(z; 7, 2) be the Laplace transform of n-1/rX {~). Then some calculus leads to
~n(z; 7, 2) = 9n-, : x,,,~(e - z ) = e x p { - 2 n ( 1 - e - " - ' :'z)~,} =e-;~z;'(l+~Tz'+~n-'/~;+O(n-2/~'))
as n ---~ cx~.
(16)
G. Christoph, K. Schreiber / Statistics & Probabilio, Letters 37 (1998) 243-247
247
With Esseen's smoothness lemma we have proved
Theorem 2. Suppose 0 < 7 < 1. Then P { n - I/:' X I" ) <~x } - P { S : ! ~<~x} : O(n -1/:') as n --* vo, where the rate o f converqence m a y not be improved.
Acknowledgements The authors would like to thank the referee for his helpful suggestions and comments, which led to several improvements. References Christoph, G., Wolf, W., 1993. Convergence Theorems with a Stable Limit Law, Mathematical Research, vol. 70. Akademie-Verlag, Berlin. Devroye, L,, 1993. A triptych of discrete distributions related to the stable law. Statist. Probab. Lett. 18, 349 351. Feller, W., 1971. An Introduction to Probability Theory and its Applications, vol. 2. Wiley Series in Probability, Wiley, New York. Petrov, V.V., 1995. Limit Theorems of Probability Theory: Sequences of Independent Random Variables. Clarendon Press, Oxford. Steutel, F.W., van Ham, K., 1979. Discrete analogues of self-decomposability and stability. Ann. Probab. 7, 893-899. van Ham, K., Steutel, F.W., Vervaat, W., 1982. Self-decomposable discrete distributions and branching processes, Z. Wahrsch. verw. Gebiete 61, 97-118.