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On moments and tail behavior of v-stable random variables
STATISTICS& PROBABILITY LETTERS ELSEVIER
Statistics & Probability Letters 29 (1996) 307-315
On moments and tail behavior of v-stable random variables Tomasz J. Kozubowski *, Anna K. Panorska 1 Department of Mathematics, University of Tennessee at Chattanoooa, Chattanooya, TN, USA Received May 1995; revised September 1995
Abstract
In this paper a class of limiting probability distributions of normalized sums of a random number of i.i.d, random variables is considered. The representation of such distributions via stable laws and asymptotic behavior of their moments and tail probabilities are established.
AMS Classification: 60E05; 60G50 Keywords: Random summation; Tail behavior of a distribution; Stable law; Geometric stable law
I. Introduction
Let { 8 , i~> 1} be a sequence of i.i.d, random variables (r.v.'s), and let {Vp, p E (0, 1)} be an independent of X/'s family of non-negative, integer-valued r.v.'s. We consider the normalized sums vp
Sp = a ( p ) ~ (X~ + b(p)),
(1)
i=1
where a(p) > O, b(p) E N, and study the limiting behavior of Sp as p ~ 0. When Vp'S are geometric random variables with parameter p, i.e.
P(vp=k)=p(1-p)k-l,
k = 1 , 2 , 3 .....
(2)
then the class of limiting distributions in (1) consists of geometric stable (GS) laws (Mittnik and Rachev, 1989). These heavy-tail distributions, studied in Kozubowski (1994), have applications in mathematical finance (the theory of asset pricing) (Kozubowski and Rachev, 1994). In this work, we study v-stable distributions as weak limits of Sp for general classes of positive, integer-valued random variables Vp. After preliminary definitions and notation of Section 2, in Section 3 we give the representation of these distributions in terms of stable laws. In Section 4, we study the asymptotic behavior of the tail probability P(Y > 2), as 2 ~ c~, * Corresponding author. l Research partially supported by a CECA grant from the University of Tennessee at Chattanooga. 0167-7152/96/$12.00 (~) 1996 Elsevier Science B.V. All rights reserved SSDI 0 1 6 7 - 7 1 5 2 ( 9 5 ) 0 0 1 8 7 - 5
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of a v-stable r.v. Y, and show that it decays as a power of 2. We conclude the paper with Section 5, where we discuss the moments of v-stable distributions.
2. v-stable distributions Let {Vp, p E (0, 1)} be a family of non-negative, integer-valued random variables, such that when p ~ 0, P Vp --~ oo,
(3)
pVp d--~Z, as p--~O,
(4)
and
where Z is a non-negative r.v. with d.f.A. Further, for technical reasons, we make the following assumption, that we refer to as Condition (I). (I) The identity
h(q~(t)) = h(~b(t)), where ~p and ~b are infinitely divisible characteristic functions (ch.f.'s) and
h(z) =
zYdA(y),
0 < Izl < 1,
implies that ~o(t) -- q~(t). Under the above assumptions, we define v-stable distributions as follows. Definition 2.1. A r.v. Y is said to be v-stable, if there exists an independent of Vp sequence of i.i.d.r.v.'s
XI,X2 ..... and a = a(p) > O, b = b(p) E ~ such that vp
a ( p ) ~ (X,. + b(p)) ~ Y,
as p ~ O.
(5)
i=1
If b(p) - 0 then Y is called strictly v-stable. Random variables X/appearing in (5) are said to be in the domain of attraction of a v-stable r.v. Y (denoted as Xi c D(Y)). Utilizing the classical transfer theorem (Gnedenko and Fakhim, 1969) and its converse (Szasz, 1972) we obtain the following characterization of v-stable distributions.
Proposition 2.1. A r.v. Y is v-stable (strictly v-stable) if and only if its ch.f ~ has the form ~k(t) ----2 ( - log ~o(t)),
where 2(z) =
e-=dA(x)
is the Laplace transform of d.f. A, and q~ is the ch.f of an e-stable (strictly c~-stable) law.
(6)
T.J. Kozubowski, A.K. PanorskaI Statistics & Probability Letters 29 (1996) 307-315
309
Proof. Assume that Y is v-stable, that is, (5) holds together with (3) and (4), and Condition (I) is satisfied. Taking p = 1/n, we write (5) as Vln
~X,i
a_~y,
(7)
i=1
where X,i = a(1/n)(Xi + b(1/n)). Now, (7) and (4) with p = I/n, together with Condition (I), imply that there exists a r.v. X with ch.f (p such that " X . . , £ x ~ q,,
(8)
i=1
where ~o satisfies (6) (Szasz, 1972, Proposition 3). Since by (8) ¢p is a ch.f. of an a-stable law, the first part of the proposition follows. Conversely, assume that Y is a r.v. with a ch.f. q/ satisfying (6), and X is a a-stable r.v. with a ch.f. ¢p satisfying (6). Since X is a stable a-stable law, then for some ~i(n) > 0, /~(n) ¢ N, and an i.i.d, sequence X1,X2 ..... we have n
Sn = ci(n)~ (X/+/~(n)) • X.
(9)
i=1
Let p, ~ 0 and k, := [1/p,] (the integer part). By (4), Vp,/k, ~ Z, where Z is a r.v. with d.f.A. Since Sk, a X, then the transfer theorem (Gnedenko and Fakhim, 1969; see also Rosinski, 1976; Rachev, 1993) gives Sv, a y. Thus, (5) holds with a(p) = 8([1/p]), b(p) =/~([1/p]), so that Y is v-stable. When Y is strictly v-stable the proof is analogous and we skip the details. [] Recall that ch.f. of a stable law has the following representation (Samorodnitsky and Taqqu, 1994): q~(t) = exp (-~r ~[tl~w(t; e, fl) + ipt),
(10)
with
w(t;e, fl) = {
1 - ifl tan(r~e/2)sign(t) if e ¢ 1, 1 + ifl(2/rt) log Itlsign(t) if e = 1,
(ll)
where 0 < ~<2,
-l~
- o o < # < oo,
a>~0.
(12)
(A r.v. with the above ch.f. is denoted as S~(a, fl, p).) Thus, (6) describes v-stable r.v.'s as a four parameter family of distributions given by their ch.f.'s. If a r.v. Y has the ch.f. as in (6) with q~ defined in (10), we say that Y is v-stable (or v~ stable) with parameters e, a, r, and # (denoted as Y ,-~ v~(cr,13,p)). Remark 1. Since any strictly v-stable r.v. corresponds to a strictly a-stable r.v. via (6), then Y ~ v~(a, fl,#) is strictly v-stable if p = 0 when e ¢ 1, or if fl = 0 for e = 1. Remark 2. Geometric stable random variables studied in Kozubowski (1994) form a special case of v-stable r.v.'s, when Vp is a geometric r.v. with mean 1/p. 3. Representation of v-stable random variables In this section we provide the representation of v-stable r.v.'s, generalizing the result for geometric stable laws (Kozubowski, 1994). Let {vp, p E (0, 1)} be a family of non-negative, integer-valued r.v.'s such that
T.J. KozubowskL A.K. Panorska/ Statistics & Probability Letters 29 (1996) 307-315
310
(3) and (4) hold, where Z is a non-negative r.v. with d.f. A (Z ~ A). Proposition 2.1 produces the following result, where a v-stable r.v. Y is written in terms of Z and an independent c~-stable r . v . X .
Proposition 3.1. Y ~ v~(a, fl, p) if and only if
~ ~=[ ~ z + z1/~x,
t
if ~ ¢ 1,
(p + X ) Z + a[3(2/rt)Z log Z,
(13)
if c~ = 1,
where Z ~ A, X ~ S~(a,~,O), and X , Z are independent.
Proof. Assume e = 1. We show that ]11 = (P + X ) Z + afl(2/rOZlogZ has ch.f. ~k satisfying (6). Indeed, first note that for any z > 0 the ch.f. ~Ox of X ~ S,(a,/?, 0) satisfies q~x(zt) = [tpx(t)]Ze -iw~(2m)z log z.
(14)
Next, utilizing (14) and conditioning on Z, we write ~k as follows: O(t) = Ee itY~ = E(E(e itY' [Z)) = =
/o
eitUz[qgx(t)]ZdA(z) =
eita#(2/n)zlog Zeit#Zq~x(zt)dA(z)
/o
[qg(t)]ZdA(z) = 2 ( - log q~(t)),
where q~(t) -- eit~tpx(t) is the ch.f. of X + p ~ S~(a, fl, p). Case ~ ~ 1 is quite similar; we proceed as above, noting that q~x(zl/~t) --- [~Ox(t)] ~. [] We conclude this section with the result that readily follows from Proposition 3.1 and Property 1.2.3 of Samorodnitsky and Taqqu (1994). It generalizes similar property of geometric stable laws (Kozubowski, 1994).
Proposition 3.2. Let Y ~ v~(a, fl, #) and let a ~ O. Then a Y ~ ~ v~(lala'sign(a)fl'a#)' L v~(la[a, sign(a)fl, a # - (2/n)aloglal~r/~),
i f ~ ~ 1,
(15)
/ f ~ = 1.
4. Tail probabilities In this section we study tail probabilities of a v-stable random variable Y : P ( Y > 2) and P ( Y < - 2 ) , as 2 ~ c~. We start with strictly v-stable Y and show that it has regularly varying tail with index ~t.
Proposition 4.1. Let Y be strictly v-stable, that is (6) (or (13)) holds with p = 0 and ~ ~ 1, or fl = 0 and ~=1. LetO<~<2. (i) The limits (as 2 ~ cxz) lira 2~P(Y > 2)
)~----~o o
and
lim 2~P(Y < - 2 )
2--*oo
(16)
are both finite if and only if q = EZ = f ~ z d A ( z ) < oo. (ii) I f both limits in (16) are finite, then lim 2~P(Y > 2 ) = C ~ - - f - - a ,t
2----~o o
(17)
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311
and lim U P ( Y < - 2 ) = Ca
a%/,
)~---*oc
(18)
where 1-~ Ca =
C(2 - c~)cos(n~/2)' 2/rt,
if ~ # 1, /f c~ = 1.
(19)
Proof. (*=) Assume that r / = EZ < oo. The two cases c~ = 1 and c~ # 1 are quite similar. We consider ~ # 1 and skip the details for the other case. By Proposition 3.1, Y a_ Z1/aX, X ~ Sa(a,~,0). By Property 1.2.15 in Samorodnitsky and Taqqu (1994) we have ;~--+~lim2aP(X > 2) = c a ~ l+ fl aa.
(20)
Write lim 2aP(Y > 2 ) =
lim
/0
P ( X > 2z-1/~)2~dA(z),
(21)
and note that there exists M > 0 such that for every 2 > 0
P ( X > ~-1/~)2a = P(X > 2z-l/~)(2z-l/a)~z<.Mz
(22)
(by (20), h(y) = P(X > y)y~ is hounded for all y > 0). Next, since f o M z d A ( z ) can be moved under the integral by the dominated convergence theorem lim 2aP(Y > 2 ) = )~---+~
/?
< oc, the limit in (21)
lim P ( X > )~z-1/~)(2z-l/~)azdA(z) 2--+oo
Ca---}---
'1.
Thus, the first limit in (16) is finite and is given by (17). Since 2~P(Y < - 2 ) = 2 a P ( - Y > 2) and - Y ~ va(a,-/~,0) (by Proposition 3.2), then the other limit in (16) is finite, and its form (18) easily follows. (=>) Assume that the two limits in (16) are finite. Assume ~ # 1. If/~ > - 1 , the Fatou's Lemma implies ~
> l i m i n f 2 a P ( g > 2) >~
l i m i n f P ( X > 2z-1/a)(2z-l/a)azdA(z)
=
so that EZ < oc. If fl = - 1 , consider liminf;__.~ 2~P(Y < -~.) and proceed as above. Case c~ = 1 is similar. [] Next, we give sufficient conditions for finiteness of the limits in (16) for a n0n-strictly v-stable r.v.Y. Proposition 4.2. Let Y ~., v~(a, fl,#) with 0 < ~ < 2. Let Z be as in Proposition 3.1 and
EIZlogZ I < ec if ~ = 1 or
EZ ira < oc / f ~ #
1.
(23)
T.J. Kozubowski, A.K. Panorska I Statistics & Probability Letters 29 (1996) 307-315
312
Then
l+fl lim 2~P(Y> 2 ) = C ~ r
~,. q
(24)
lim 2~P(Y < - 2 ) = C,
a~q,
(25)
2--+oo
and ,~--~ oo
where q = EZ and C~ is given by (19). Before proving Proposition 4.2, we quote a lemma from Samorodnitsky and Taqqu (1994). Lemma 4.1. Suppose that X is a r.v. with regularly varying tail, Le., there is a number ~ > 0 such that for every a > 1 P ( X > a2) lim - a -~. ;~oo P ( X > 2)
(26)
Suppose also that the tail of X dominates the tail o f a positive r.v. W in the sense that lim P ( W > 2) _ 0. ;~oo P ( X > 2)
(27)
lira P ( X + W > 2) lim P ( X - W > 2) ;~--,~ P ( X > 2) = z~ P ( X > 2) = 1.
(28)
Then
Proof. See Samorodnitsky and Taqqu (1994, Lemma 4.4.2).
[]
Proof of Proposition 4.2. Note that it is enough to show (24), as (25) would follow by Property 3.2, since P ( Y < - 2 ) = P ( - Y > 2). Case ~ < 1. Assume EZ l w = EZ < oo. By Proposition 3.1 y a= pZ + Z1/~X,
(29)
where X ~ S,(a,/3,0) and Z , X independent. By Proposition 4.1 1+/3 lim 2~p(z1/~X > 2) = C ~ a
r/.
(30)
2---*oo
Assume /3 > - 1 . Then, by (30), Z1/~X satisfies (26), and W -- ]/~IZ is a positive r.v. satisfying (27) (since EZ ~ < c~z). An application of Lemma 4.1 produces (24). If fl = - 1 , then the limit in (30) is zero. Thus, by (29), lim 2~P(Y > 2)~< lim 2~P(I~Z > 2 / 2 ) + lim 2~P(Z1/~X > 2 / 2 ) = 0,
2---*oo
2~c~
2--,oo
since 2~P(#Z > 2/2) either equals zero (for negative /~) or converges to zero (for positive /~, as EZ" < c~). Consequently, (24) holds. Case c~ = 1. Assume E[ZlogZ] < ~ , so that lira 2P(IZlogZ > 2) = 0.
2---+o¢~
(31)
T.J. Kozubowski, A.K. PanorskalStatistics & Probability Letters 29 (1996) 307-315
313
Denoting W = a f l ( 2 / T t ) Z l o g Z we have Y a__ Z(# + X ) + W, where X ~ S l ( a , fl, O). Let fl > - 1 . Apply Lemma 4.1 with Z ( p + X ) as the r.v. with regularly varying tail (by Proposition 4.1), and IwI as the r.v. whose tail is dominated by Z(# + X), since P(IWI > 2) P(Z(u + x ) > ,t)
;~P(IWI > ,t) zP(Z(u + x ) > ,t)
---~0,
as
2---~oo,
(32)
by (31). Thus,
1 +flo.~",/, lim ZP(Z(# + X ) -1- IWl > ~) = ca-- 5 -
(33)
2---* o o
by L e m m a 4.1, and (24) follows since P(Z(I~+X)-IWl
> ~)<<.P(Z(~+x)+ w> ~)<.P(Z(~+x)+
IrVl > ~).
(34)
In case fi = - 1 we proceed in the same way as in the case c~ < 1. Case ~ > 1. The proof is the same as in the case a < 1. []
5. Moments The tail behavior (24) and (25) of a v-stable r.v. Y, implies the following result about its absolute moments.
Proposition 5.1. L e t Y ~ v~(a, f l , # ) with 0 < ~ < 2. L e t Z be as in Proposition 3.1 and ElZlogZl
< ~
i f co= 1
or
E Z w~ < co / f ~ ¢
1.
(35)
Then
EIYI p < c~ for any 0 < p < ~
and
EIYI p = ~ f o r any p > ~ .
Proof. Follows from Proposition 4.2 and the formula EIYIP = f ~ P ( l Y I p > 2 ) d 2 .
(36) []
Proposition (5.1) implies that, v~-stable r.v. does not have a finite mean for ~ < 1 . For c~ > 1 the mean exists and equals #11, where 11 = EZ. To see this, note that E Y < oo by Proposition 5.1 and by Proposition 3.1 we can write Y d= # Z + Z1/~X, where X ~ S~(a, t, 0). Since the shift parameter of an a-stable r.v. with :~ > 1 is equal to its mean (Samorodnitsky and Taqqu, 1994, Property 1.2.19), we have E X = O, and consequently E Y = E # Z + EZ1/~EX = # E Z = #11.
Thus, we established the following result.
Proposition 5.2. I f Y ~ v~(a, f l , # ) with ~ > 1, then E Y = #11, where 11 = EZ. Next, we provide an expression for absolute moments of strictly v-stable random variables.
Proposition 5.3. L e t Y ~ v~(a, fl, O) with 0 < a < 2 and fl = 0 i f o~ = 1. Then, f o r every 0 < p < ct, there is a constant C = C(~,fl, p ) such that EIYI p = CaPT,
(37)
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314
where y = EZ ply. The constant C has the f o r m C = P f-~2p-lF(1-P/~)u_----~-_i sin2-----~d u ( l + f 1 2 t a n 2 7 ) p / 2 ~ c o s ( P a r c t a n ( f l t a n T ) )
.
Proof. Let ~ ¢ 1. By Proposition 3.1, Y d Z1/~X, where X ~ &(a, fl, O). By Property 1.2.17 in Samorodnitsky and Taqqu (1994), we have E]X] p = Ct7 p, and the result follows by independence of Z and X : E ] Y I P = EZP/~EX. The proof in case ~ = 1 is similar. [] Remark. In case when Y is geometric stable, then (37) holds with 7 -- F(p/c~ + 1 ), since in this case Z has an exponential distribution with mean 1. We conclude this section with the result on the asymptotic behavior of E] YI r as r T ~, where Y ~ v~(a, r, #). It is an immediate consequence of the tail behavior of Y given in Proposition 4.2.
Proposition 5.4. Let Y ~ v~(a, r, It) with 0 < ct < 2. Let ElZlogZ I < ~
if c~ = 1
or
EZ ~v~ < c~ l f ~ # 1.
(38)
Then lim(~ - r)EIY[ r = ~C~a ~,
(39)
lim(ct - r )EY = 7C~fla ~,
(40)
rT~t
and r~
where Co is given by (19) and a (b) = lalbsign (a). Proof. The same as the proof of an analogous property of or-stable laws, see Samorodnitsky and Taqqu (1994) for details. []
Acknowledgements We thank the referee for careful review of the manuscript.
References Gertsbakh, I.B. (1984), Asymptotic methods in reliability: a review, Adv. in Appl. Probab. 16, 147-175. Gnedenko, B.V. (1982), On limit theorems for a random number of random variables, Lecture Notes in Mathematics, Vol. 1021, Fourth USSR-Japan Symp. Proc. (Springer, Berlin). Gnedenko B.V. and H. Fakhim (1969), On a transfer theorem, Dokl. Acad. Nauk USSR 187, 15-17 (in Russian). Klebanov, L.B., G.M. Maniya and I.A. Melamed (1984), A problem of Zolotarev and analogs of infinitely divisible and stable distributions in a scheme for summing a random number of random variables, Theory Probab. Appl. 29, 791-794. Kozubowski, T.J. (1994), Representation and properties of geometric stable laws, in: G. Anastassiou and S.T. Rachev, eds., Approximation, Probability, and Related Fields (Plenum, New York) pp. 321-337. Kozubowski, T.J. and S.T. Rachev (1994), The theory of geometric stable distributions and its use in modeling financial data, European J. Oper. Res. 74, 310-324. Mittnik, S. and S.T. Rachev (1993), Modeling asset returns with alternative stable distributions, Econometric Rev. 12, 261-330.
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315
Rachev, S.T. (1993), Rate of convergence of maxima of random arrays with applications to stock returns, Statist. Decisions 11,279-288. Rosinski, J. (1976), Weak compactness of laws of random sums of identically distributed random vectors in Banach spaces, Colloq. Math. 35, 313-325. Samorodnitsky G. and M. Taqqu (1994), Stable Non-Gaussian Random Processes (Chapman & Hall, New York). Szasz, D. (1972), On classes of limit distributions for sums of a random number of identically distributed independent random variables, Theory Probab. Appl. 17, 401-415.
Statistics & Probability Letters 29 (1996) 307-315
On moments and tail behavior of v-stable random variables Tomasz J. Kozubowski *, Anna K. Panorska 1 Department of Mathematics, University of Tennessee at Chattanoooa, Chattanooya, TN, USA Received May 1995; revised September 1995
Abstract
In this paper a class of limiting probability distributions of normalized sums of a random number of i.i.d, random variables is considered. The representation of such distributions via stable laws and asymptotic behavior of their moments and tail probabilities are established.
AMS Classification: 60E05; 60G50 Keywords: Random summation; Tail behavior of a distribution; Stable law; Geometric stable law
I. Introduction
Let { 8 , i~> 1} be a sequence of i.i.d, random variables (r.v.'s), and let {Vp, p E (0, 1)} be an independent of X/'s family of non-negative, integer-valued r.v.'s. We consider the normalized sums vp
Sp = a ( p ) ~ (X~ + b(p)),
(1)
i=1
where a(p) > O, b(p) E N, and study the limiting behavior of Sp as p ~ 0. When Vp'S are geometric random variables with parameter p, i.e.
P(vp=k)=p(1-p)k-l,
k = 1 , 2 , 3 .....
(2)
then the class of limiting distributions in (1) consists of geometric stable (GS) laws (Mittnik and Rachev, 1989). These heavy-tail distributions, studied in Kozubowski (1994), have applications in mathematical finance (the theory of asset pricing) (Kozubowski and Rachev, 1994). In this work, we study v-stable distributions as weak limits of Sp for general classes of positive, integer-valued random variables Vp. After preliminary definitions and notation of Section 2, in Section 3 we give the representation of these distributions in terms of stable laws. In Section 4, we study the asymptotic behavior of the tail probability P(Y > 2), as 2 ~ c~, * Corresponding author. l Research partially supported by a CECA grant from the University of Tennessee at Chattanooga. 0167-7152/96/$12.00 (~) 1996 Elsevier Science B.V. All rights reserved SSDI 0 1 6 7 - 7 1 5 2 ( 9 5 ) 0 0 1 8 7 - 5
T.,L Kozubowski, A.K. Panorska/Statistics & Probability Letters 29 (1996) 307-315
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of a v-stable r.v. Y, and show that it decays as a power of 2. We conclude the paper with Section 5, where we discuss the moments of v-stable distributions.
2. v-stable distributions Let {Vp, p E (0, 1)} be a family of non-negative, integer-valued random variables, such that when p ~ 0, P Vp --~ oo,
(3)
pVp d--~Z, as p--~O,
(4)
and
where Z is a non-negative r.v. with d.f.A. Further, for technical reasons, we make the following assumption, that we refer to as Condition (I). (I) The identity
h(q~(t)) = h(~b(t)), where ~p and ~b are infinitely divisible characteristic functions (ch.f.'s) and
h(z) =
zYdA(y),
0 < Izl < 1,
implies that ~o(t) -- q~(t). Under the above assumptions, we define v-stable distributions as follows. Definition 2.1. A r.v. Y is said to be v-stable, if there exists an independent of Vp sequence of i.i.d.r.v.'s
XI,X2 ..... and a = a(p) > O, b = b(p) E ~ such that vp
a ( p ) ~ (X,. + b(p)) ~ Y,
as p ~ O.
(5)
i=1
If b(p) - 0 then Y is called strictly v-stable. Random variables X/appearing in (5) are said to be in the domain of attraction of a v-stable r.v. Y (denoted as Xi c D(Y)). Utilizing the classical transfer theorem (Gnedenko and Fakhim, 1969) and its converse (Szasz, 1972) we obtain the following characterization of v-stable distributions.
Proposition 2.1. A r.v. Y is v-stable (strictly v-stable) if and only if its ch.f ~ has the form ~k(t) ----2 ( - log ~o(t)),
where 2(z) =
e-=dA(x)
is the Laplace transform of d.f. A, and q~ is the ch.f of an e-stable (strictly c~-stable) law.
(6)
T.J. Kozubowski, A.K. PanorskaI Statistics & Probability Letters 29 (1996) 307-315
309
Proof. Assume that Y is v-stable, that is, (5) holds together with (3) and (4), and Condition (I) is satisfied. Taking p = 1/n, we write (5) as Vln
~X,i
a_~y,
(7)
i=1
where X,i = a(1/n)(Xi + b(1/n)). Now, (7) and (4) with p = I/n, together with Condition (I), imply that there exists a r.v. X with ch.f (p such that " X . . , £ x ~ q,,
(8)
i=1
where ~o satisfies (6) (Szasz, 1972, Proposition 3). Since by (8) ¢p is a ch.f. of an a-stable law, the first part of the proposition follows. Conversely, assume that Y is a r.v. with a ch.f. q/ satisfying (6), and X is a a-stable r.v. with a ch.f. ¢p satisfying (6). Since X is a stable a-stable law, then for some ~i(n) > 0, /~(n) ¢ N, and an i.i.d, sequence X1,X2 ..... we have n
Sn = ci(n)~ (X/+/~(n)) • X.
(9)
i=1
Let p, ~ 0 and k, := [1/p,] (the integer part). By (4), Vp,/k, ~ Z, where Z is a r.v. with d.f.A. Since Sk, a X, then the transfer theorem (Gnedenko and Fakhim, 1969; see also Rosinski, 1976; Rachev, 1993) gives Sv, a y. Thus, (5) holds with a(p) = 8([1/p]), b(p) =/~([1/p]), so that Y is v-stable. When Y is strictly v-stable the proof is analogous and we skip the details. [] Recall that ch.f. of a stable law has the following representation (Samorodnitsky and Taqqu, 1994): q~(t) = exp (-~r ~[tl~w(t; e, fl) + ipt),
(10)
with
w(t;e, fl) = {
1 - ifl tan(r~e/2)sign(t) if e ¢ 1, 1 + ifl(2/rt) log Itlsign(t) if e = 1,
(ll)
where 0 < ~<2,
-l~
- o o < # < oo,
a>~0.
(12)
(A r.v. with the above ch.f. is denoted as S~(a, fl, p).) Thus, (6) describes v-stable r.v.'s as a four parameter family of distributions given by their ch.f.'s. If a r.v. Y has the ch.f. as in (6) with q~ defined in (10), we say that Y is v-stable (or v~ stable) with parameters e, a, r, and # (denoted as Y ,-~ v~(cr,13,p)). Remark 1. Since any strictly v-stable r.v. corresponds to a strictly a-stable r.v. via (6), then Y ~ v~(a, fl,#) is strictly v-stable if p = 0 when e ¢ 1, or if fl = 0 for e = 1. Remark 2. Geometric stable random variables studied in Kozubowski (1994) form a special case of v-stable r.v.'s, when Vp is a geometric r.v. with mean 1/p. 3. Representation of v-stable random variables In this section we provide the representation of v-stable r.v.'s, generalizing the result for geometric stable laws (Kozubowski, 1994). Let {vp, p E (0, 1)} be a family of non-negative, integer-valued r.v.'s such that
T.J. KozubowskL A.K. Panorska/ Statistics & Probability Letters 29 (1996) 307-315
310
(3) and (4) hold, where Z is a non-negative r.v. with d.f. A (Z ~ A). Proposition 2.1 produces the following result, where a v-stable r.v. Y is written in terms of Z and an independent c~-stable r . v . X .
Proposition 3.1. Y ~ v~(a, fl, p) if and only if
~ ~=[ ~ z + z1/~x,
t
if ~ ¢ 1,
(p + X ) Z + a[3(2/rt)Z log Z,
(13)
if c~ = 1,
where Z ~ A, X ~ S~(a,~,O), and X , Z are independent.
Proof. Assume e = 1. We show that ]11 = (P + X ) Z + afl(2/rOZlogZ has ch.f. ~k satisfying (6). Indeed, first note that for any z > 0 the ch.f. ~Ox of X ~ S,(a,/?, 0) satisfies q~x(zt) = [tpx(t)]Ze -iw~(2m)z log z.
(14)
Next, utilizing (14) and conditioning on Z, we write ~k as follows: O(t) = Ee itY~ = E(E(e itY' [Z)) = =
/o
eitUz[qgx(t)]ZdA(z) =
eita#(2/n)zlog Zeit#Zq~x(zt)dA(z)
/o
[qg(t)]ZdA(z) = 2 ( - log q~(t)),
where q~(t) -- eit~tpx(t) is the ch.f. of X + p ~ S~(a, fl, p). Case ~ ~ 1 is quite similar; we proceed as above, noting that q~x(zl/~t) --- [~Ox(t)] ~. [] We conclude this section with the result that readily follows from Proposition 3.1 and Property 1.2.3 of Samorodnitsky and Taqqu (1994). It generalizes similar property of geometric stable laws (Kozubowski, 1994).
Proposition 3.2. Let Y ~ v~(a, fl, #) and let a ~ O. Then a Y ~ ~ v~(lala'sign(a)fl'a#)' L v~(la[a, sign(a)fl, a # - (2/n)aloglal~r/~),
i f ~ ~ 1,
(15)
/ f ~ = 1.
4. Tail probabilities In this section we study tail probabilities of a v-stable random variable Y : P ( Y > 2) and P ( Y < - 2 ) , as 2 ~ c~. We start with strictly v-stable Y and show that it has regularly varying tail with index ~t.
Proposition 4.1. Let Y be strictly v-stable, that is (6) (or (13)) holds with p = 0 and ~ ~ 1, or fl = 0 and ~=1. LetO<~<2. (i) The limits (as 2 ~ cxz) lira 2~P(Y > 2)
)~----~o o
and
lim 2~P(Y < - 2 )
2--*oo
(16)
are both finite if and only if q = EZ = f ~ z d A ( z ) < oo. (ii) I f both limits in (16) are finite, then lim 2~P(Y > 2 ) = C ~ - - f - - a ,t
2----~o o
(17)
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311
and lim U P ( Y < - 2 ) = Ca
a%/,
)~---*oc
(18)
where 1-~ Ca =
C(2 - c~)cos(n~/2)' 2/rt,
if ~ # 1, /f c~ = 1.
(19)
Proof. (*=) Assume that r / = EZ < oo. The two cases c~ = 1 and c~ # 1 are quite similar. We consider ~ # 1 and skip the details for the other case. By Proposition 3.1, Y a_ Z1/aX, X ~ Sa(a,~,0). By Property 1.2.15 in Samorodnitsky and Taqqu (1994) we have ;~--+~lim2aP(X > 2) = c a ~ l+ fl aa.
(20)
Write lim 2aP(Y > 2 ) =
lim
/0
P ( X > 2z-1/~)2~dA(z),
(21)
and note that there exists M > 0 such that for every 2 > 0
P ( X > ~-1/~)2a = P(X > 2z-l/~)(2z-l/a)~z<.Mz
(22)
(by (20), h(y) = P(X > y)y~ is hounded for all y > 0). Next, since f o M z d A ( z ) can be moved under the integral by the dominated convergence theorem lim 2aP(Y > 2 ) = )~---+~
/?
< oc, the limit in (21)
lim P ( X > )~z-1/~)(2z-l/~)azdA(z) 2--+oo
Ca---}---
'1.
Thus, the first limit in (16) is finite and is given by (17). Since 2~P(Y < - 2 ) = 2 a P ( - Y > 2) and - Y ~ va(a,-/~,0) (by Proposition 3.2), then the other limit in (16) is finite, and its form (18) easily follows. (=>) Assume that the two limits in (16) are finite. Assume ~ # 1. If/~ > - 1 , the Fatou's Lemma implies ~
> l i m i n f 2 a P ( g > 2) >~
l i m i n f P ( X > 2z-1/a)(2z-l/a)azdA(z)
=
so that EZ < oc. If fl = - 1 , consider liminf;__.~ 2~P(Y < -~.) and proceed as above. Case c~ = 1 is similar. [] Next, we give sufficient conditions for finiteness of the limits in (16) for a n0n-strictly v-stable r.v.Y. Proposition 4.2. Let Y ~., v~(a, fl,#) with 0 < ~ < 2. Let Z be as in Proposition 3.1 and
EIZlogZ I < ec if ~ = 1 or
EZ ira < oc / f ~ #
1.
(23)
T.J. Kozubowski, A.K. Panorska I Statistics & Probability Letters 29 (1996) 307-315
312
Then
l+fl lim 2~P(Y> 2 ) = C ~ r
~,. q
(24)
lim 2~P(Y < - 2 ) = C,
a~q,
(25)
2--+oo
and ,~--~ oo
where q = EZ and C~ is given by (19). Before proving Proposition 4.2, we quote a lemma from Samorodnitsky and Taqqu (1994). Lemma 4.1. Suppose that X is a r.v. with regularly varying tail, Le., there is a number ~ > 0 such that for every a > 1 P ( X > a2) lim - a -~. ;~oo P ( X > 2)
(26)
Suppose also that the tail of X dominates the tail o f a positive r.v. W in the sense that lim P ( W > 2) _ 0. ;~oo P ( X > 2)
(27)
lira P ( X + W > 2) lim P ( X - W > 2) ;~--,~ P ( X > 2) = z~ P ( X > 2) = 1.
(28)
Then
Proof. See Samorodnitsky and Taqqu (1994, Lemma 4.4.2).
[]
Proof of Proposition 4.2. Note that it is enough to show (24), as (25) would follow by Property 3.2, since P ( Y < - 2 ) = P ( - Y > 2). Case ~ < 1. Assume EZ l w = EZ < oo. By Proposition 3.1 y a= pZ + Z1/~X,
(29)
where X ~ S,(a,/3,0) and Z , X independent. By Proposition 4.1 1+/3 lim 2~p(z1/~X > 2) = C ~ a
r/.
(30)
2---*oo
Assume /3 > - 1 . Then, by (30), Z1/~X satisfies (26), and W -- ]/~IZ is a positive r.v. satisfying (27) (since EZ ~ < c~z). An application of Lemma 4.1 produces (24). If fl = - 1 , then the limit in (30) is zero. Thus, by (29), lim 2~P(Y > 2)~< lim 2~P(I~Z > 2 / 2 ) + lim 2~P(Z1/~X > 2 / 2 ) = 0,
2---*oo
2~c~
2--,oo
since 2~P(#Z > 2/2) either equals zero (for negative /~) or converges to zero (for positive /~, as EZ" < c~). Consequently, (24) holds. Case c~ = 1. Assume E[ZlogZ] < ~ , so that lira 2P(IZlogZ > 2) = 0.
2---+o¢~
(31)
T.J. Kozubowski, A.K. PanorskalStatistics & Probability Letters 29 (1996) 307-315
313
Denoting W = a f l ( 2 / T t ) Z l o g Z we have Y a__ Z(# + X ) + W, where X ~ S l ( a , fl, O). Let fl > - 1 . Apply Lemma 4.1 with Z ( p + X ) as the r.v. with regularly varying tail (by Proposition 4.1), and IwI as the r.v. whose tail is dominated by Z(# + X), since P(IWI > 2) P(Z(u + x ) > ,t)
;~P(IWI > ,t) zP(Z(u + x ) > ,t)
---~0,
as
2---~oo,
(32)
by (31). Thus,
1 +flo.~",/, lim ZP(Z(# + X ) -1- IWl > ~) = ca-- 5 -
(33)
2---* o o
by L e m m a 4.1, and (24) follows since P(Z(I~+X)-IWl
> ~)<<.P(Z(~+x)+ w> ~)<.P(Z(~+x)+
IrVl > ~).
(34)
In case fi = - 1 we proceed in the same way as in the case c~ < 1. Case ~ > 1. The proof is the same as in the case a < 1. []
5. Moments The tail behavior (24) and (25) of a v-stable r.v. Y, implies the following result about its absolute moments.
Proposition 5.1. L e t Y ~ v~(a, f l , # ) with 0 < ~ < 2. L e t Z be as in Proposition 3.1 and ElZlogZl
< ~
i f co= 1
or
E Z w~ < co / f ~ ¢
1.
(35)
Then
EIYI p < c~ for any 0 < p < ~
and
EIYI p = ~ f o r any p > ~ .
Proof. Follows from Proposition 4.2 and the formula EIYIP = f ~ P ( l Y I p > 2 ) d 2 .
(36) []
Proposition (5.1) implies that, v~-stable r.v. does not have a finite mean for ~ < 1 . For c~ > 1 the mean exists and equals #11, where 11 = EZ. To see this, note that E Y < oo by Proposition 5.1 and by Proposition 3.1 we can write Y d= # Z + Z1/~X, where X ~ S~(a, t, 0). Since the shift parameter of an a-stable r.v. with :~ > 1 is equal to its mean (Samorodnitsky and Taqqu, 1994, Property 1.2.19), we have E X = O, and consequently E Y = E # Z + EZ1/~EX = # E Z = #11.
Thus, we established the following result.
Proposition 5.2. I f Y ~ v~(a, f l , # ) with ~ > 1, then E Y = #11, where 11 = EZ. Next, we provide an expression for absolute moments of strictly v-stable random variables.
Proposition 5.3. L e t Y ~ v~(a, fl, O) with 0 < a < 2 and fl = 0 i f o~ = 1. Then, f o r every 0 < p < ct, there is a constant C = C(~,fl, p ) such that EIYI p = CaPT,
(37)
T.J. Kozubowski, A.K. Panorska/ Statistics & Probability Letters 29 (1996) 307-315
314
where y = EZ ply. The constant C has the f o r m C = P f-~2p-lF(1-P/~)u_----~-_i sin2-----~d u ( l + f 1 2 t a n 2 7 ) p / 2 ~ c o s ( P a r c t a n ( f l t a n T ) )
.
Proof. Let ~ ¢ 1. By Proposition 3.1, Y d Z1/~X, where X ~ &(a, fl, O). By Property 1.2.17 in Samorodnitsky and Taqqu (1994), we have E]X] p = Ct7 p, and the result follows by independence of Z and X : E ] Y I P = EZP/~EX. The proof in case ~ = 1 is similar. [] Remark. In case when Y is geometric stable, then (37) holds with 7 -- F(p/c~ + 1 ), since in this case Z has an exponential distribution with mean 1. We conclude this section with the result on the asymptotic behavior of E] YI r as r T ~, where Y ~ v~(a, r, #). It is an immediate consequence of the tail behavior of Y given in Proposition 4.2.
Proposition 5.4. Let Y ~ v~(a, r, It) with 0 < ct < 2. Let ElZlogZ I < ~
if c~ = 1
or
EZ ~v~ < c~ l f ~ # 1.
(38)
Then lim(~ - r)EIY[ r = ~C~a ~,
(39)
lim(ct - r )EY
(40)
rT~t
and r~
where Co is given by (19) and a (b) = lalbsign (a). Proof. The same as the proof of an analogous property of or-stable laws, see Samorodnitsky and Taqqu (1994) for details. []
Acknowledgements We thank the referee for careful review of the manuscript.
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