Remarks on the Stable Sα(β, γ, µ) Distribution
Tibor K. Pog´any & Saralees Nadarajah
First version: 19 December 2011
Research Report No. 11, 2011, Probability and Statistics Group School of Mathematics, The University of Manchester Remarks on the stable Sα(β, γ, µ) distribution
Tibor K. Pog´any Faculty of Maritime Studies, University of Rijeka, Rijeka, Croatia
Saralees Nadarajah School of Mathematics, University of Manchester, Manchester M13 9PL, U.K.
Abstract. Explicit closed forms are derived for the probability density function of the stable distribution Sα(β, γ, µ), α ∈ (1, 2]. Consequent asymptotic expansions are given. The expressions involve the Srivastava-Daoust generalized Kamp´e de F´eriet hypergeometric S-function, the Fox-Wright generalized hypergeometric Ψ-function, and the Gauss hypergeometric function 2F1.
2000 Mathematics subject classification. Primary 60E10; Secondary 33C60, 62G32.
Keywords and phrases. Characteristic function; Fox-Wright generalized hypergeo- metric Ψ-function; Probability density function; Srivastava-Daoust S function; Stable
distribution Sα(β, γ, µ).
1. Introduction and preliminaries A random variable (r.v.) ξ is said to have the stable distribution if its charac- teristic function (CHF) is specified by [10, p. 8, Eq. (1.6)]: exp iµt − γα|t|α 1 − iβ tan π α sgn(t) , α 6= 1, 2 φ(t) = Eeitξ = (1.1) 2 exp iµt − γ|t| 1 + iβ π ln t , α = 1, where α ∈ (0, 2], |β| ≤ 1, γ > 0 and µ ∈ R. We write ξ ∼ Sα(β, γ, µ). The stable distribution class contains the Gaussian distribution (α = 2), the Cauchy distribution (α = 1, β = 0) and the L´evy distribution (α = 1/2, β = 1) as particular cases. The Landau distribution (α = 1, β = 1) and the dirac delta distribution (α ↓ 0, γ ↓ 0) are also contained as particular cases. Stable distributions have wide-ranging applications. Some of the application areas include the lighthouse problem, distribution of masses in space, random walks, hitting times for Brownian motion, fractional diffusions, stock prices, foreign exchange rates, value at risk, multiple assets, time series, signal processing, copulas, exponential power distributions, queueing theory, geology, physics, survival analysis and reliability, network traffic, computer science, biology and medicine, discrepancies, punctuated change, central pre-limit theorems, extreme values models, behavior of the sample mean and variance, and the appropriateness of infinite variance models. For discussion on these application areas and references, we refer the readers to [10, Chapter 2]. Because of its wide applicability, it is important that the properties of ξ ∼ Sα(β, γ, µ) are known explicitly. However, even the probability density function (PDF) of ξ has not been known explicitly except for the three particular cases of the Gaussian 1 2 T.K. Pog´any and S. Nadarajah distribution (α = 2), the Cauchy distribution (α = 1, β = 0) and the L´evydistribution (α = 1/2, β = 1). There has been much development of approaches to compute the PDF numerically. We refer the readers to [19, 11, 3, 8, 9, 2, 12, 13, 14, 7, 6]. See also Luk´acs[5], Zolotarev [20], Uchaikin and Zolotarev [18] and [10, Chapter 3]. The aim of this note is to derive an explicit closed form for the PDF of ξ ∼ Sα(β, γ, µ) given by 1 Z f(x) = e−itxφ(t) dt , 2π R where φ(t) is given by (1.1). According to [10, p. 11, Theorem 1.9], stable distributions are continuous and infinitely differentiable. So, the PDF is well defined.
We shall need the following mathematical tools. Let pΨq denote the Fox-Wright generalization of the hypergeometric pFq function with p numerator and q denominator parameters [17, p. 4, Eq. (2.4)] defined by p Y Γ(a + α m) " # ∞ ` ` xm (a1, α1) ,..., (ap, αp) X `=1 pΨq x = q (b1, β1) ,..., (bq, βq) m! m=0 Y Γ(b` + β`m) `=1 for suitably bounded values of |x|, where the parameters involved satisfy q p X X α` ∈ R+, ` = 1, p; βj ∈ R+, j = 1, q; ∆0 = 1 + β` − αj > 0 . `=1 j=1 We also need the Srivastava-Daoust generalization of the Kamp´ede F´eriethypergeo- metric function in two variables defined by [15, p. 199]
0 0 ! 0 0 A:B;B [(a): θ, Φ] : [(b): ψ] ; [(b ): ψ ] A:B;B SC:D;D0 0 0 x, y = SC:D;D0 (x, y) [(c): µ, ] : [(d): η] ; [(d ): η ] A B B0 Y Y Y 0 0 Γ(aj + mθj + nΦj) Γ(bj + mψj) Γ bj + nψj ∞ m n X j=1 j=1 j=1 x y := (1.2), C D D0 m! n! m,n=0 Y Y Y 0 0 Γ(cj + mµj + nj) Γ(dj + mηj) Γ dj + nηj j=1 j=1 j=1 where the parameters satisfy 0 0 θ1, . . . , θA, . . . , η1, . . . , ηD0 > 0.
For convenience, we write (a) to denote the sequence of A parameters a1, . . . , aA, with similar interpretations for (b),..., (d0). Empty products should be interpreted as unity. Srivastava and Daoust [16, p. 155] reported that the series in (1.2) converges absolutely for all x, y ∈ C when C D A B X X X X ∆1 = 1 + µj + ηj − θj − ψj > 0, j=1 j=1 j=1 j=1 Remarks on the stable Sα(β, γ, µ) distribution 3
C D0 A B0 X X 0 X X 0 ∆2 = 1 + j + ηj − Φj − ψj > 0 . j=1 j=1 j=1 j=1 A:B;B0 The complete set of conditions for convergence of the series SC:D;D0 (x, y) is given in Srivastava and Daoust [16, pp. 154-155]. Finally, we also need the logconvexity property of the gamma function (a direct consequence of the H¨olderinequality): Γ(λx + (1 − λ)y) ≤ Γ(x)λ Γ(y)1−λ, min(x, y) > 0, λ ∈ [0, 1] . (1.3) This inequality is sharp.
2. Main results
Theorem 1 derives an explicit closed form for the PDF of ξ ∼ Sα(β, γ, µ). Some corresponding asymptotic expansions are given in Theorem 2.
α −1 Theorem 1. Let ξ ∼ Sα(β, γ, µ). Then for all α ∈ (1, 2] and |β tan(π 2 )| < 1 − α the PDF of ξ can be expressed as ( !) 1 [1/α : 1/α; 1] : − ; − i(µ − x) f(x) = < S1:0;0 , iβ tan π α . (2.1) 0:0;0 2 παγ − : − ; − γ Proof. Consider 1 Z f(x) = exp {i(µ − x)t − γα|t|α [1 − iβΦ sgn(t)]} dt, 2π R α α where Φ = tan(π 2 ), i(µ − x) =: X and iβγ Φ =: Y . The Maclaurin series of the two unimodular terms in the integrand gives us ∞ Z m n 1 X n X Y f(x) = tm (sgn(t)|t|α) exp {−γα|t|α} dt · 2π m! n! m,n=0 R ∞ ( Z 0 1 X n = tm (−(−t)α) exp {−γα(−t)α} dt 2π m,n=0 −∞ ) Z ∞ Xm Y n + tm+αn exp {−γαtα} dt · 0 m! n! ∞ Z ∞ m n 1 X α X Y = 1 + (−1)m+n tm+αne−(γt) dt · 2π m! n! m,n=0 0 ∞ 1 X Z ∞ = 1 + (−1)m+n t1/α+m/α+n−1 e−t dt 2παγ m,n=0 0 m n Xγ−1 Y γ−α · · m! n! ∞ −1m −αn 1 X 1 + m Xγ Y γ = 1 + (−1)m+n Γ + n · (2.2). 2παγ α m! n! m,n=0 4 T.K. Pog´any and S. Nadarajah
This is equivalent to (2.1). Since ∆1 = 1 − 1/α and ∆2 = 0, a case not covered by Srivastava and Daoust [16], the series convergence needs further study. Substituting 1 1 α 1 λ := , x := + m, y := + n α 2 α − 1 2α into (1.3), we obtain the estimate 1 1 1− 1 1 1 α 1 + nα α Γ + m + n ≤ Γ + m Γ 2 (2.3) α α 2 α − 1 for all m, n ∈ N0 := {0, 1, 2,...}. Applying (2.3) to (2.2), we deduce 1 1 ∞ −1m ∞ 1 1− −αn 1 X 1 α |X|γ X + nα α |Y |γ |f(x)| ≤ Γ + m · Γ 2 . παγ 2 m! α − 1 n! m=0 n=0 −1 The resulting power series’ radii or convergence are ρX = ∞ and ρY = 1 − α respectively. Therefore, the first one converges for all x ∈ R, while the second one converges for |Y |γ−α = |βΦ| < 1 − α−1. The proof is complete. α Theorem 2. Assume α ∈ (1, 2] and |β tan(π 2 )| → 0. Then we have " # 1 x − µ2 (1/α, 1/α) f(x) = √ 1Ψ1 − . (2.4) παγ (1/2, 1) 4γ Moreover, 1 Γ 1 + " 1 1 1 # α , + 1 2 f(µ) = F 2α 2 α − β tan π α (2.5) 2 1 1 2 πγ 2 π −1 for all α ∈ (1, 2] and | β tan( 2 α) |< 1 − α . Proof. Note that " # 0 A:B;B ((a), θ) , ((b), ψ) lim SC:D;D0 (x, y) = A+BΨC+D x . y→0 ((c), µ) , ((d), η) α Now, letting |β tan(π 2 )| → 0 in (2.1), we see that ( ∞ m ) 1 X 1 + m [i(µ − x)/γ] f(x) = < Γ παγ α m! m=0 ∞ 2 2m 1 X 1 + 2m −(x − µ) /γ = Γ παγ α (2m)! m=0 ∞ 1 2 2m 1 X Γ + m −(x − µ)/ 4γ = α α , 3/2 Γ 1 + m m! π αγ m=0 2 Remarks on the stable Sα(β, γ, µ) distribution 5
m 1 √ where we have applied (2m)! = 2mΓ(2m) = 4 Γ(m + 1)Γ(m + 2 )/ π. So, we have (2.4) since the confluent Fox-Wright series, 1Ψ1, converges for all 2α > 1. Now, letting x → µ in (2.1) and calculating the real parts of the resulting special functions, we arrive at ( ∞ n ) 1 X 1 [iβΦ] f(µ) = < Γ + n παγ α n! n=0 ∞ 2n 1 X 1 −(βΦ) = Γ + 2n παγ α (2n)! n=0 1/α−1 ∞ 1 1 1 2n 2 X Γ + n Γ + + n −(βΦ) = 2α 2α 2 , παγ Γ 1 + n n! n=0 2 where we have applied the Legendre duplication formula, Γ(2z) = 22z−1Γ(z)Γ(z + 1 √ 2 )/ π. To reduce the sum to a Gauss hypergeometric function, 2F1, we need the Pochhammer symbols (z)n = Γ(z + n)/Γ(z): 1/α−1 ∞ 1 1 1 2n 2 1 1 1 X + −(βΦ) f(µ) = Γ Γ + 2α n 2α 2 n . 3/2 2α 2α 2 1 n! π αγ n=0 2 n
But, this is equivalent to (2.5) since 2F1 converges inside the open unit disc – note −1 from Theorem 1 that |βΦ| < 1 − α . Zolotarev [20, p. 87, §2.4] reported on the following inequality for the kth derivative of the PDF of Sα(β, 1, 0): 1 + k Γ α −(1+k)/α f (k)(x) ≤ cos π βK(α) , k ∈ (2.6) πα 2 N for some constant K(α). Zolotarev did not discuss the background or the sharpness of (2.6). The following theorem generalizes this inequality giving a functional upper bound for the modulus of the derivatives of f(x).
α −1 Theorem 3. Let ξ ∼ Sα(β, γ, µ). Then, for α ∈ (1, 2], | β tan(π 2 ) |< 1 − α and for any κ ∈ N, we have ! 1 1+κ : 1 ; 1 : − ; − |x − µ| f (κ)(x) ≤ S1:0;0 α α , β tan π α . (2.7) κ+1 0:0;0 2 παγ − ; − ; − γ Proof. Following the proof of Theorem 1, we obtain: κ ∞ Z ∞ m n (−i) X α α X Y f (κ)(x) = tκ+m+αne−γ t dt 1 + (−1)κ+m+n · . 2π m! n! m,n=0 0 Thus, ∞ −1m n 1 X 1 + κ m |x − µ|γ [|βΦ|] f (κ)(x) ≤ Γ + + n · , (2.8) παγκ+1 α α m! n! m,n=0 6 T.K. Pog´any and S. Nadarajah and (2.7) is proved, because the double series converges for the stated range of para- meters. Remark 1. Alternatively, separate the gamma-term with the aid of (1.3) into 1 1 1− 1 + κ m 1 + κ α 1+κ + αn α Γ + + n ≤ Γ + m · Γ 2 α α 2 α − 1 by choosing 1 1 + κ α 1 + κ λ := , x := + m, y := + n . α 2 α − 1 2α Applying this estimate to the right hand double sum in (2.8), we conclude 1 ∞ −1m 1 X 1 + κ α |x − µ|γ f (κ)(x) ≤ Γ + m παγκ+1 2 m! m=0 1 ∞ 1− n X 1+κ + αn α [|βΦ|] · Γ 2 . (2.9) α − 1 n! n=0
The involved power series’ have the same radii of convergence for all κ ∈ N0. This is not the case for the ones in the proof of Theorem 1. Unfortunately, upper bounds like [Γ(x)]λ ≤ Γ(x), λ ∈ [0, 1], x ≥ 2 can not be used to estimate (2.9) since the resulting confluent hypergeometric function, 1F0, is defined only in the interval x ∈ (µ − γ, µ + γ) and the resulting Fox-Wright series, 1Ψ0, is formal, converges only at βΦ = 0.
3. Discussion and final remarks
The following asymptotic expansion for the PDF of Sα(0, 1, 0) ∞ 2m 1 1 X 1 + 2m −x Γ f(x) = (−1)m Γ ∼ α πα α (2m)! πα m=0 for x → 0 has been obtained by Bergstr¨om[1] and mentioned in Zolotarev [20, p. 94, Theorem 2.5.1, Corollary 2]. An exhaustive account of the asymptotics of the PDF is given in [20, Sections 2.4 and 2.5]. See also [18, Eq. (4.1.3)]. Our (2.5) in Theorem 2 generalizes these asymptotics of the PDF f(x) for the α −1 α nonstandard Sα(β, γ, µ) case when | β tan(π 2 ) |< 1 − α . When | β tan(π 2 ) |≥ 1 − α−1, we can use the analytical continuation principle. For this, consult the series expansion by Uchaikin and Zolotarev [18, p. 108, Eq. (4.2.10)]. Hoffmann-Jørgensen [4, Theorem, Eq. (2.2)] has also expressed the PDF of ξ ∼ Sα(β, γ, µ) in terms of known special functions. Using the Mellin-transform technique and introducing the generalized incomplete hypergeometric function, pGq, Hoffmann- Jørgensen [4] obtained expressions for (2.4) and (2.5) when α ∈ (1, 2]. For this method, consult [18, pp. 179-180, §6.7.]. Remarks on the stable Sα(β, γ, µ) distribution 7
Let us mention that Hoffmann-Jørgensen’s function pGq [4, Eq. (6)] is, in fact, Fox-Wright generalized pΨq-function with equal parameters 0 < γ < 1, that is p Y Γ(aj + γm) ∞ m Γ(a ) X z j=1 j G (a , . . . , a ; b , . . . , b ; γ, z) = p q 1 p 1 q m! q m=0 Y Γ(bk + γm) Γ(bk) k=1 q Y Γ(b ) j " #
j=1 (ap, γ) = p pΨq z . Y (bq, γ) Γ(aj) j=1
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