Characterization of a subclass of Tweedie distributions by a property of generalized stability
aa r X i v : . [ s t a t . M E ] J un Characterization of a subclass of Tweediedistributions by a property of generalized stability
Lev B. Klebanov ∗ , Grigory Temnov † December 21, 2017
Abstract
We introduce a class of distributions originating from an exponential family andhaving a property related to the strict stability property. Specifically, for two densities p θ and p linked by the relation p θ ( x ) = e θx c ( θ ) p ( x ) , we assume that their characteristic functions f θ and f satisfy f θ ( t ) = f α ( θ ) ( β ( θ ) t ) ∀ t ∈ R , θ ∈ [ a, b ] , with a and b s.t. a ≤ ≤ b . A characteristic function representation for this family is obtained and its propertiesare investigated. The proposed class relates to stable distributions and includes InverseGaussian distribution and Levy distribution as special cases.Due to its origin, the proposed distribution has a sufficient statistic. Besides, itcombines stability property at lower scales with an exponential decay of the distri-bution’s tail and has an additional flexibility due to the convenient parametrization.Apart from the basic model, certain generalizations are considered, including the onerelated to geometric stable distributions.
Key Words:
Natural exponential families, stability-under-addition, characteristic func-tions.
Random variables with the property of stability-under-addition (usually called just stabil-ity ) are of particular importance in applied probability theory and statistics. The signifi-cance of stable laws and related distributions is due to their major role in the Central limitproblem and their link with applied stochastic models used in e.g. physics and financialmathematics. ∗ Affiliation : Department of Probability and Statistics of Charles University, Prague Sokolovska 83,Prague-8, CZ 18675, Czech Republic . Email: [email protected] by the government research grant MSM 002160839 is gratefully acknowledged † Corresponding author; email: [email protected]ffiliation : School of Mathematical Sciences, University College Cork, Western Gateway Block, Cork, IE
1n practical applications, stable laws appearing naturally as limit distributions for sumsof random variables (the essence of the Central limit problem) are often used to describethe increments of stochastic processes. Stable distributions may exhibit a reliable fit ofempirical data whose nature is related to random summation. However, certain peculiaritiesof the data (e.g., lighter tails than according to stable models) motivate the search foralternatives to stable laws. Among different ways to adjust the tail behavior of the stabledistribution is the exponential (tilting), the smoothening of the tail of stable density iswidely applied, see e.g. [14].In the present work, we consider exponential families with a property that can beconsidered as a generalization of the usual strict stability-under-addition property (that is,with a real-valued function in place of the natural number of summands). The result is, asappears, related to exponentially tilted stable distributions.As the class we consider originates from natural exponential families and related tostable distributions, it is not surprising that our class appears to be a subclass of Tweediedistributions, that, for some combination of parameters, can also be characterized as ex-ponentially tilted stable distributions.Therefore, the result of our findings is a convenient representation of a certain subclassof Tweedie distributions – a characterization in terms of extended stability property. Likeit is for the Tweedie distributions, the important advantage of the proposed class is theexistence of a sufficient statistics which is due to its belonging to natural exponentialfamilies, - and the combination of the generalized stability with exponential tails.
Suppose X is a real random variable with cdf P and pdf p . A class of distributions associatedwith P is the natural exponential family with the pdf p θ ( x ) = e θx c ( θ ) p ( x ) , (1)with the real parameter θ ∈ [ a, b ] , where a and b are such that ∈ [ a, b ] . With f θ ( t ) denoting the characteristic function (chf) of p θ ( x ) and f ( t ) the chf of p ( x ) , the followingequality is valid according to (1): f θ ( t ) = f ( t − iθ ) f ( − iθ ) , θ ∈ [ a, b ] . (2)The problem we investigate in relation to the above chf is the following: Do real valuedfunctions α ( θ ) and β ( θ ) exist, such that f θ ( t ) = f α ( θ ) ( β ( θ ) t ) ∀ t ∈ R , θ ∈ [ a, b ] . (3)It is known (see e.g. [11]) that if a chf is analytic in the neighborhood of the origin,then it is also analytic in a horizontal strip (either this strip is the whole plane or it hasone or two horizontal boundary lines). It follows from the above that ∀ θ ∈ [ a, b ] , chf f θ ( t ) is analytic ( t ∈ C ) in the horizontal strip | Im ( t ) | < ρ ( θ ) for some ρ ( θ ) > . Then withthe notation g ( t ) := ln( f ( t )) , one gets from (2) and (3) the following balance equation g ( t − iθ ) = g ( − iθ ) + α ( θ ) g ( β ( θ ) t ) . (4)2he above functional equation can be solved using an argument based on sequential dif-ferentiating w.r.t. to θ and to t , which we postpone to Appendix. The solution w.r.t. thefunction g ( t ) has the form g ( t ) = A [1 − (1 − itc ) γ ] , (5)where γ ≤ , A, c ∈ R and the sign of A depends on the value of γ (as we shall see, A < when γ < and when γ = 2 ; A > when γ ∈ (0 , and when γ ∈ (1 , ). The expressionfor the chf f ( t ) immediately follows f ( t ) = exp { A [1 − (1 − itc ) γ ] } . Recalling (2), we can now write down the chf f θ f θ ( t ) = exp { A [(1 − cθ ) γ − (1 − itc − cθ ) γ ] } (6) = exp (cid:26) A · (1 − cθ ) γ (cid:20) − (cid:18) − itc − cθ (cid:19) γ (cid:21)(cid:27) = exp (cid:26) A · B γ (cid:20) − (cid:18) − itcB (cid:19) γ (cid:21)(cid:27) , (7)where B := 1 − cθ .Note that the functions α ( θ ) and β ( θ ) are then explicitly expressed α ( θ ) = (1 − cθ ) γ , β ( θ ) = 1 / (1 − cθ ) . γ ; role of parameters A and c The properties of introduced distributions differ with respect to various combinations oftheir parameters γ, A, c .As the role of the shape exponent γ is crucial to relations between other parameters, inthe present paragraph we analyze different cases that arise with respect to certain rangesof γ . That classification w.r.t. γ also makes it easier to clarify the relation of the introducedclass to stable laws and other important distributions.Returning to the representation for the chf ff γ ( t ) = exp { A ([1 − (1 − itc ) γ ] } , (8)note that the properties of the corresponding distribution differ depending on the rangeof γ (to mark the importance of this dependency, we shall use the notation f γ , instead ofjust f , and will point out the range of γ , such as f γ< or f γ ∈ (1 , ). Listing different casesw.r.t. γ below, we shall see that for each of the ranges of γ , f γ is a proper chf only for aparticular sign of A . In all cases, there exists a corresponding analytic function f γ ( t ) , t ∈ C ,that agrees with the characteristic function f γ ( t ) , t ∈ R . The domain of regularity forcorresponding analytic function varies depending on the ranges of γ considered below. Case a : γ < Denoting γ = − γ , for this case we have f γ< ( t ) = exp (cid:8) A (cid:2) − (1 − itc ) − γ (cid:3)(cid:9) . (9)As b γ ( t ) := (1 − itc ) − γ is the chf of a gamma distribution, we see that f γ< corresponds tothe chf of the compound Poisson r.v. with gamma-distributed summands, provided that A := − A > : f γ< ( t ) = exp (cid:8) A [ b γ ( t ) − (cid:9) . (10)3n other words, f γ< ( t ) is a chf of the random sum S = N ( t ) P i =1 X i , where N ∼ P oisson and X ∼ gamma ( γ, c ) , i.e. γ and c are the shape and the scale parameter correspondingly.Clearly, f γ< ( t ) , t ∈ C , is analytic when Im ( t ) ∈ ( − /c, ∞ ) , with the entire line t = − i/c being the region of singularity for f ( t ) . Case b : γ ∈ (0 , This case, after a complex shift of the variable, corresponds to acertain subclass of stable distributions. Specifically, recall that if e X = X + x is a shiftedone-sided Levy-stable variable then its chf can be represented as e φ ( z ) = exp { izx − ( − iz ) α e a } , e a > . (11)Clearly, (11) with x = 0 brings us close to our case (8), differing by just a proper changeof the variable. Specifically, with cz := ct + i we have f γ ∈ (0 , ( t ) = e A exp {− (1 − itc ) γ A } ⇐⇒ f γ ∈ (0 , ( z − i/c ) = e A exp {− ( − izc ) γ A } , (12)which is close to (11) up to the normalization factor e A (provided that A > ).Furthermore, note that the re-parametrization cz = ct + i relates to the transformationof the density, similar to the one used in the introduction of f θ , according to (1) and (2).When t = z − i/c then in order for f γ ∈ (0 , ( t ) to correspond to a proper chf we needa normalization factor f γ ∈ (0 , ( − i/c ) . From (12), we see that f γ ∈ (0 , ( z − i/c ) f γ ∈ (0 , ( − i/c ) =: f stabγ ( z ) corresponds to one-sided stable chf and that f γ ∈ (0 , ( − i/c ) = e A .Hence, one-sided stable density corresponding to chf f stabγ is the result of an exponentialtransformation (take the exponent parameter /c in (1) instead of θ ) of the density corre-sponding to chf f γ ∈ (0 , . That is, the r.v. with chf f γ ∈ (0 , in turn relates to the exponentialtransformation of the one-sided stable density with the exponent parameter being − /c .Note that both f stabγ and f γ ∈ (0 , are proper characteristic functions only when A > .Like in Case a , the function f γ ∈ (0 , ( t ) , t ∈ C , is analytic in Im ( t ) ∈ ( − /c, ∞ ) , butcontrary to the previous case, there exists a limit for f γ ∈ (0 , ( t ) w.r.t t approaching the line t = − i/c (with the only point of non-regularity being (0 , − i/c ) ). This limit corresponds to f stabγ introduced above which is the chf of one-sided stable law.Note that the above is in correspondence with one of the classical results on char-acteristic functions that says that a necessary condition for a function analytic in someneighborhood of the origin to be a characteristic function is that in either half-plane thesingularity nearest to the real axis is located on the imaginary axis (see e.g. [11] for details). Remark 1
Check that the well-known Inverse Gaussian distribution as a special case with γ = 1 / . Case c : γ = 1 This degenerate case corresponds to the constant r.v. whose cdf is theHeaviside step function, since the distribution corresponding to the chf e iAct has a singleunit jump at point Ac . Case d : γ ∈ (1 , Consider a re-parametrization e γ = γ/ allowing to write down thechf f ( t ) as e f γ ∈ (1 , ( t ) = exp n A [1 − ((1 − itc ) ) e γ ] o = exp n A [1 − (1 − c e t ) e γ ] o = f e γ ∈ (0 , ( − i e t ) , (13)4here e t := 2 it + ct .While f e γ ∈ (0 , ( t ) is a chf of the type considered above (Case b ), f γ ∈ (1 , ( t ) is also achf. Indeed, while φ ( µ,σ ) ( t ) = exp (cid:8) µit − σ t / (cid:9) is the chf of the Gaussian r.v. (withexpectation µ and variance σ ), the function f e γ ∈ (0 , ( − i e t ) = Z e e tx p e γ ( x ) dx = Z (cid:2) φ (2 , c ) ( t ) (cid:3) x p e γ ( x ) dx (14)which is a chf of some probabilistic distribution, as a continuous mixture of the Gaussianchf φ (2 , c ) ( t ) = exp (cid:8) it − ct (cid:9) (provided that c := − c > ) with the pdf p e γ whose chf f e γ ∈ (0 , is the one from the above Case b .Since we already know from Case b that it should be that A > for f γ ∈ (0 , to be achf, in the present case we also have A > as a necessary condition for f γ ∈ (1 , given by(14) to be a chf. Like in Case b , f ( t ) , t ∈ C , is analytic only in the strip restricted bythe horizontal line t = − i/c , i.e. when Im ( t ) ∈ ( − /c, ∞ ) , but additionally we have thecondition that c < . Case e : γ = 2 Check that f γ =2 is the chf of the normal r.v. f γ =2 ( t ) = exp (cid:8) A (cid:0) − cit − c t (cid:1)(cid:9) = φ ( − Ac, Ac ) ( t ) , (15)where A = − A . As the variance is σ = 2 Ac , it should be that A < . The sign of c Discussing the cases with respect to the parameter γ , we noted that thesign of the parameter A is crucial in each of the cases. Commenting on the sign of the otherparameter, c , we noted that its sign is only crucial in Case d . In other cases, c can take anysign when f γ is a proper chf due to the property of characteristic functions providing that f ( − t ) is a chf as soon as f ( t ) is a chf. However, we should keep in mind that the function f γ ( t ) ( t ∈ C ), in all relevant cases, is analytic only in a strip depending on c . Limit behavior of f regarding mutual limit w.r.t. c and A Note that the one-sided stable distributions appearing in Case b and obtained via the change of the complexvariable cz := ct + i could also be viewed as a limiting case w.r.t c → ∞ and A → .Additionally, c and A should be linked so that A ∼ e c/c γ ( e c ∈ R ), hence the limit w.r.t A → and c → ∞ (keeping e c constant) leads to f γ ∈ (0 , ( t ) = e A exp (cid:26) − (cid:18) c − it (cid:19) γ ( c γ A ) (cid:27) −−−→ c →∞ A → exp {− ( − it ) γ e c } =: f ( lim ) γ ∈ (0 , ( t ) , which corresponds to one-sided stable distributions appearing in Case b above. Recallingthe notation of Case b , we see that f ( lim ) ( t ) = f stabγ ( t ) = exp {− ( − itc ) γ A } . θ Let us turn again to the representation for f θ . Rewrite (6) noting that an additional re-parametrization with e A := A · B γ and c θ = c − cθ leads to the form (8) f γ,θ ( t ) = exp n e A · [1 − (1 − itc θ ) γ ] o , (16)5here we used the notation f γ,θ instead of just f θ , indicating the significance of both ofthe parameters.Clearly, Cases considered above in relation to chf f γ can be revisited for f θ , but apartfrom the ranges of the shape exponent γ , the role the natural exponent θ is important.Obviously, the borders of the analyticity strips depending on c , as well as the conditionson A , will change in correspondence with θ , i.e. one can view introducing f γ,θ in place of f γ as passing from θ = 0 to non-zero θ , so that the "new" c and A will depend on θ .In particular, replacements through Case a – Case e are the following: • the strips of analyticity: ( − i/c, ∞ ) = ⇒ ( − i (1 /c − θ ) , ∞ ) in Cases a , b and d ; • the signs of coefficients: recall that Case a and Case e imply A < , while in both Case b and Case d we have A > ; for f γ,θ the same holds with A = ⇒ A · B γ = A · (1 − cθ ) γ .Revisit, for instance, Case d where we have a representation analogous to the oneexpressed by (13) and (14): e f γ ∈ (1 , ,θ ( t ) = f e γ ∈ (0 , ,θ (2 t − ict ) = Z (cid:2) φ (2 , c θ ) ( t ) (cid:3) x p e γ,θ ( x ) dx , (17)where pdf p e γ,θ corresponds to the chf f e γ ∈ (0 , ,θ = exp n e A (cid:2) − (1 − itc θ ) e γ (cid:3)o and φ (2 , c θ ) is a chf of the normal r.v., i.e. φ (2 , c θ ) ( t ) = exp (cid:8) it − c θ t (cid:9) (with c θ := − c θ ). So that inorder for e f γ ∈ (1 , ,θ to be a chf, it should be that c θ = c/ (1 − cθ ) < and e A = A · B γ > .All other cases w.r.t γ can be revisited for f γ,θ analogously. Additionally, certain valuesand ranges of θ correspond to particular special cases which we consider below. Range of θ , singularities and corresponding limits. While c can be positive ornegative, the point when θ = 1 /c is the case of singularity and/or the limiting case.Specifically, Case a with θ = 1 /c is the case of singularity of the chf f γ,θ . In Case b ,the limit /c ← θ exists with limiting chf’s f ( lim ) γ ( t ) = exp {− ( − izc ) γ A } .In a way, approaching the point /c with respect to θ means approaching correspond-ing stable (or mixed normal-stable) distributions, while increasing the exponent parameter θ corresponds to lightening the tails of distributions P θ to exponential ones. Limit case θ −→ ∞ . The question in interest is: what distributions appear in thelimit case θ −→ ∞ ? In [1], limit laws for the whole class of natural exponential familiesw.r.t. the growing exponent parameter were investigated. It was shown that for the r.v. X θ with density of the form (1), constants a θ > and b θ can chosen so that a limit ( X θ − b θ ) /a θ −→ Y, θ → ∞ exists and that only possible limit distributions Y are thenormal and gamma distributions. Specifically, the following results were proved. • If P θ , θ ∈ Θ , is the cdf of the exponential family X θ whose pdf is given by (1) and ifthere exist constants a θ > and b θ ∈ R such that P θ (cid:16) x − b θ a θ (cid:17) −−−→ θ →∞ G weakly to somenon-degenerate cdf G then G belongs to the so called extended gamma family, whichincludes normal and gamma densities. • In the case of convergence to a non-degenerate limit G , a θ and b θ should be takensuch that P θ is centered and scaled by expectation and standard deviation. moment generating function (mgf) associated with p ( x ) ,i.e. M ( λ ) = R e λx p ( x ) dx . Specifically, as proved in [1], the following holds. If the function s ( λ ) = 1 / √ m ′′ ( λ ) , where m ( λ ) := ln M ( λ ) , is self-neglecting, then theexponential family X λ , λ ∈ Θ , is asymptotically normal. Otherwise, the only possible limitdistribution is gamma distribution. It is easy to check when the normal distribution appears as a limit in our case. Recalling(5), for the log–mgf (or cumulant generating function) of the introduced family we have m ( λ ) = A [1 − (1 − λc ) γ ] , so that m ′′ ( λ ) = Ac γ ( γ − (cid:2) (1 − λc ) γ − (cid:3) . Clearly, the above condition on s = 1 / √ m ′′ is only satisfied when γ ≥ . That means thatin all the cases w.r.t. to γ that are relevant for our study, the only possible limit distribution G is gamma distribution, i.e. according to the usual terminology, our distribution lies inthe domain of attraction ( DA ) of gamma distribution. Like briefly noted above, the proposed distributional family includes some well-known mod-els, which can be considered as special cases of exponential family with stability property.At the same time, it itself is subclass of another class – the Tweedie distributions [ ? ], which,in its turn, is a part of even a wider class of exponential dispersion models . For general definitions and characterizations of exponential dispersion models we refer to[ ? ]. The pdf of an exponential dispersion model (EDM) is represented in a way similar tothat of our class f ED ( y ) = p β ( y ) exp (cid:26) β ( θy − κ ( θ )) (cid:27) , where β > . A convenient representation of the Tweedie class is through the momentgenerating function M ( t ) = R exp( ty ) f ( y ) dy . The cumulant generating function of theEDM is m ED ( t ) = log M ED ( t ) = [ κ ( θ + tφ ) − κ ( θ )] /β . As the derivatives of the function κ ( · ) w.r.t θ provide the values of the cumulants of thedistribution, κ ( · ) is called the cumulant function. So that we have µ = κ ′ ( θ ) for the meanand βκ ′′ ( θ ) for the variance. Since the mapping from θ to µ is invertible, there exists afunction V ( · ) , called the variance function, such that κ ′′ ( θ ) = V ( µ ) .The Tweedie class is then defined as EDM with the variance function V ( · ) of a specificform, namely V ( µ ) = µ p for p ∈ ( −∞ , ∪ [1 , ∞ ) and V ( µ ) = exp( υµ ) for p = ∞ (here, υ = 0 ). The correspondent cumulant function κ for the Tweedie class can be representedas ( µ − p − − p , when p = 2log µ , when p = 2 . (18)7efining a new parameter α via ( p − − α ) = 1 , a representation for the moment generating function of the Tweedie class is obtained: M T w ( s ) = exp n (1 − p ) α θ α β (2 − p ) [(1 + s/θ ) α − o , when p = 1 , , (1 + sθ ) − /β , when p = 2 , exp n β e θ ( e s − o , when p = 1 . (19)From above, comparing with the chf (8) of our class, it is clear that our class is actuallya subclass of Tweedie distributions with some parameter p = 1 , and a certain combinationof other parameters.Correspondingly, the basic properties that we discussed above do hold for the Tweedieclass as well. For instance, it is known that the Tweedie distributions can be obtainedthrough exponential damping of stable distributions.The distributions of the Tweedie class are known to have a scale invariance property ,that can be expressed through the pdf f T w as f T w ( x ; µ, φ ) = cf T w ( cx ; cµ, c − p β ) , for c > , which also resembles stability property (in terms of scale invariance), however is not directlyrelated to the generalized stability property introduced by considering the relation (3).Note also that our subclass does not explicitly contain the parameter β in the aboveED notation (parameter related to the variance of the distribution).Besides, note that gamma distribution belonging to the Tweedie class, in our caseappears as a limit case θ −→ ∞ , like mentioned in the previous section. Two important special cases are visible immediately from the form of the characteristicfunctions f γ and f γ,θ .Recall the characteristic function representation of the well-known inverse Gaussiandistribution g IG ( t ) = e λµ · " − (cid:18) − µ itλ (cid:19) / (20)and note (as already briefly remarked in Case b of Paragraph 2.1) that it can be viewed as aspecial case of the introduced exponential family with stability — just plug the parameters’values ( A = λµ , c = µ λ , γ = ) into the general representation (8).Recall an important property of the inverse Gaussian distribution: • If X i has an IG ( µw i , λw i ) distribution for i = 1 , , . . . , n and X i are indepen-dent, then S = P ni =1 X i ∼ IG ( µw, λw ) , where w = n P i =1 w i . In other words, X ∼ IG ( µ, λ ) ⇒ tX ∼ IG ( tµ, tλ ) b and Case d ), so that the scale invariance appears to be naturally embedded in ourmodel.Another well known distribution appears as a limiting case of the Inverse Gaussiandistribution: the chf f Levy of the
Levy distribution is clearly a limit of g IG w.r.t. µ −→ ∞ (and respectively λ −→ ) which leads from (20) to f Levy ( t ) = e −√− i e ct . (21)Inverse Gaussian and Levy distributions are the important cases as they have simple ex-plicit forms for their pdf’s. The combination of the properties of our subclass relates in its purpose to other modifiedmodels such as the so-called Tempered Levy process.The idea behind Tempered Levy processes (also known as Truncated Levy flights [14])is to use a distribution which coincides with the stable one around zero (i.e. for smallfluctuations) and has heavy tails yet decreasing to zero fast enough to assure a finitevariance.The latter is achieved by truncating stable distributions, and one of the ways the trun-cation can be done is the so called smooth exponential truncation. The result correspondsto an infinitely divisible distribution enabling an analytical chf representation: φ α ; µ ( t ) = exp (cid:26) − a α cos( πα/ (cid:20) ( µ + t ) α/ cos (cid:18) α arctan (cid:18) | t | µ − µ α (cid:19)(cid:19)(cid:21)(cid:27) , where µ is the cut-off parameter. The asymptotic behavior of φ is given by φ α ; µ ( t ) ∼ t →∞ φ α ( t ) where φ α ( t ) is the chf of the (non-truncated) stable distribution, i.e. for small values of x ,the pdf p α ; µ ( x ) corresponding to the chf φ α ; µ behaves like a stable law of index α .This concept is then interpreted in terms of Levy processes and associated Levy mea-sures. Specifically, if ν ( x ) is a measure of a Levy process (generally, defined as the expectednumber, per unit time, of jumps whose size is less than x ), then the measure e ν ( dx ) = e − θx ν ( dx ) , corresponds to another Levy process, whose large sizes are "tempered" with exponentialdamping.The ideas are therefore similar to the one that lead to the introduction of the Tweedieclass, as well as to the characterization of our subclass, yet with certain methodologicaldifference: The idea of exponential tempering utilizes stable distribution as the initial dis-tribution whose tail is then exponentially smoothened, while in our approach the stabilityproperty appears naturally from the properties of the introduced class. That allows toview the exponential family with stability property as a natural extension of both classes:stable distributions and exponential families which is promising regarding their practi-cal applications, including the use of sufficient statistic naturally inherent in exponentialfamilies. 9 Modifications related to geometric stable laws
Preliminaries and definitions
First, recall the origin of geometric infinitely divisible (GID) distributions.The chf of an infinitely divisible (ID) r.v. X implies the representation φ ( t ) = ( φ n ( t )) n ,where the chf φ n ( t ) corresponds to some r.v. X ( n ) , which in turn implies that the sum X ( n )1 + · · · + X ( n ) n of iid copies of X ( n ) has the same distribution as X . The so-called transfer theorems (see e.g. [3]) state that the random sums X ( n )1 + · · · + X ( n ) ν n , (22)where ( ν n ) is a sequence of integer-valued r.v.’s such that ν n p −→ ∞ (in probability) while ν n /n d −→ ν (in distribution), converge (in distribution) to a r.v. Y whose chf has the form ω ( t ) = L ( − ln φ ( t )) , (23)where L is the Laplace transform of the r.v. ν .If ν n d = ν p has a geometric distribution with mean /p (where p is close to zero, and ν p is independent of X ′ i s ) which converges in distribution to the standard exponentialdistribution with Laplace transform L ( u ) = (1 + u ) − then (23) turns into ω ( t ) = (1 − ln φ ( t )) − , (24)and random variables whose chf has the form (24) were introduced in [6] as geometricinfinitely divisible (GID) random variables. Since sums such as (22) frequently appear inmany applied problems, GID have a variety of applications. The geometric stable distribu-tions (GS), also originated from [6] and developed in later works e.g. [3] and [5], appear asa natural subclass of GID and are widely used for modeling of stochastic processes. Thechf of GS laws has the form (24) with φ ( t ) being the chf of a stable r.v.It is natural to extend the exponential family with stability property introduced inprevious sections w.r.t geometric stability. Specifically, for the chf f γ,θ of the form (16),the corresponding geometric extension will have the chf ω γ,θ ( t ) = 11 − ln f γ,θ ( t ) , ∀ t ∈ R , θ ∈ [ a, b ] . (25)Since f γ,θ ( t ) = exp n e A · [1 − (1 − itc θ ) γ ] o , note that when e A = A · B γ < (which corre-sponds to certain ranges of γ ) then (25) can viewed as a sum of geometric progression withthe initial value (1 − e A ) − and common ratio e A − e A (1 − itc θ ) γ .Revisiting the cases considered in Paragraph 2.1 again, this time in relation to ω γ,θ ,the argument above leads to a convenient interpretation of Case a , which we sketch below,along with other cases w.r.t the ranges of γ . • Case a γ < . In this case, f γ< ,θ is a chf of a probability distribution only when e A < , which coincides with the condition under which the representation (25) can beinterpreted as a geometric progression. Thus the cdf Ω γ< ,θ corresponding to the chf10 γ< ,θ has the representation as a series of convolutions of the gamma distributionwith itself: Ω γ< ,θ ( x ) = 11 − e A ∞ X n =0 e A − e A ! n F ∗ nγ,θ ( x ) , (26)where F γ,θ ( x ) = 1 − F γ,θ ( x ) , and F γ,θ is the cdf of gamma distribution with shapeparameter γ = − γ and scale parameter c θ . • Case b γ ∈ (0 , . In this case f γ ∈ (0 , ,θ is a chf only when e A > , so the representationof the form (26) is not valid. However, as the representation (25) implies that any ω γ,θ is a chf as soon as f γ,θ is a chf, it follows that ω γ ∈ (0 , ,θ is the chf (providedthat e A > ), and the corresponding r.v. is a geometric analogue of exponentiallytransformed one-sided stable r.v., corresponding to the chf f γ ∈ (0 , ,θ . • Case d γ ∈ (1 , . Recall the representation (27) meaning that f γ ∈ (1 , ,θ is a chflinked via the transformation R (cid:2) φ (2 , c θ ) ( t ) (cid:3) x p e γ,θ ( x ) dx with the pdf p e γ,θ of Case b ( e γ ∈ (0 , ) and with the chf φ (2 , c θ ) of the normal distribution (with mean andvariance (2 , c θ ) ). Then ω γ ∈ (1 , ,θ given by (25) is also a chf, the geometric analogueof f γ ∈ (1 , ,θ , provided that e A > and additionally c θ < . • Case e γ = 2 . As chf f γ =2 ,θ corresponds to the Normal r.v., ω γ =2 ,θ is the chf of thegeometric analogue of the Normal r.v.Check that this case can also be interpreted in the way similar to the Case b , so that ω γ =2 ,θ can be viewed as an exponentially transformed Laplace r.v. • Case f γ > . This is a new case in our framework, since f γ> ,θ ( t ) does not corre-spond to a proper probability distribution (in the representation R (cid:2) e P ( t ) (cid:3) x p θ ( x ) dx that we used in Case b and Case d , the polynomial P ( t ) can not have a degreegreater than , due to Marcinkiewicz’s theorem [12]).However, ω γ> ,θ ( t ) = (1 − ln f γ> ,θ ( t )) − is a proper chf. That could be seen if weassume γ ∈ (2 , , denote e γ := γ/ and a := 1 / (1 − e A ) and consider ω γ ∈ (2 , ,θ ( t ) = 11 − e A (1 − itc θ ) γ = 11 + e A (cid:2) (1 − itc θ ) e γ (cid:3) = a − (1 − a ) (cid:2) ( − izc θ ) e γ (cid:3) with z = t + i/c . Clearly, the r.h.s. of the above can be represented as L L ( − log f stab e γ ( z )) where f stab e γ ( z ) = exp { A [ − ( − izc θ ) e γ ] } is the chf of one-sided stable random vari-able, as L L ( u ) = 1 / (1 − b u ) is the Laplace transform of the Laplace distribu-tion. Clearly ω g ( z ) := 1 / (cid:16) − (1 − a ) (cid:2) ( − izc θ ) e γ (cid:3) (cid:17) is a chf of geometric analogue ofone-sided stable r.v. (provided that ( a − > ), and since ω g ( − i/c ) = 1 /a , then ω γ ∈ (2 , ,θ ( t ) = ω g ( z − i/c ) /ω g ( − i/c ) . Then ω γ ∈ (2 , ,θ is itself a chf. Note that thecondition (1 − a ) > means that it should be that e A > .When γ > , we can argue analogously. Note that the r.v. corresponding to the chf ω γ> ,θ is nevertheless not geometric stable, nor it is geometrically infinitely divisible.11 Summary, interpretation and further extensions of the model
As the classification of different cases with respect to the ranges of γ spreads out to f γ , f γ,θ and ω γ,θ , it might be useful to summarize the relevant information in a summary table.Case a γ < Case b γ ∈ (0 , Case d γ ∈ (1 , Case f γ > f γ exp (cid:8) A [ b γ ( t ) − (cid:9) • A < with z := t + i/c : exp { − [ − izc ] γ A } • A > with z := t + i/c : exp (cid:8) − [( − izc ) ] γ A (cid:9) • A > — f γ,θ e − e A [ b γ ( t ) − • e A < with z := t + i/c : exp n − [ − izc θ ] γ e A o • e A > with z := t + i/c θ : exp n − [( − izc θ ) ] γ e A o • e A > — Geometr. ω γ,θ α ∞ P n =0 α n f nγ • e A < − ln f γ ∈ (0 , ,θ ( t )) − • e A > (1 − ln f γ ∈ (1 , ,θ ( t )) − • e A > − b − b [ ( − izc θ ) e γ ] • e A > While in Section 4 a new class was introduced whose chf φ is linked with the chf f γ,θ through φ ( t ) = L ( − ln f γ,θ ( t )) , the chf f γ,θ itself, for < γ < , can be represented in theabove form. Indeed, according to Case d of Paragraph 2.1, we have the following mixturerepresentation for f γ ∈ (1 , f γ ∈ (1 , ( t ) = Z (cid:2) φ (2 , c ) ( t ) (cid:3) x p e γ ∈ (0 , ( x ) dx = Z e x ln φ (2 , c ) ( t ) p e γ ∈ (0 , ( x ) dx = L e γ ( − ln φ (2 , c ) ( t )) , (27)and the same is valid for f γ ∈ (1 , ,θ as well.Recall that a r.v. S with chf of the form L N ( − ln f X ( t )) can be interpreted as a randomsum (with random but finite number of summands), if the Laplace transform L N corre-sponds to a discrete random variable (a realization of a counting process). One of the mostcelebrated examples is the Poisson process whose Laplace transform is exp (cid:8) λ (cid:0) e − t − (cid:1)(cid:9) ,so that the chf exp { λ ( φ ( t ) − } corresponds to the compound Poisson r.v. S = N ( t ) P i =1 X i .According to (13), in our case the LT of the r.v. associated with the counting process is L N ( u ) = exp (cid:8) A (cid:2) − (1 + cu ) e γ (cid:3)(cid:9) with < e γ < . Clearly, it corresponds to a continuous r.v.; moreover, as discussed in Case b , the chf f e γ ∈ (0 , of this r.v. is obtained from the one-sided stable chf by a complex shift, s.t. the corresponding r.v. relates to the exponentialtransformation of one-sided stable density. Hence the chf (27) cannot be interpreted as achf of a compound r.v. in its usual form N ( t ) P i =1 X i .However, the corresponding r.v. can be interpreted in terms of continuous-time ana-logues of random sums — the increments of a subordinated Levy process . For the chf of any12evy process X t on R we have (see e.g. [2]) E (cid:2) e izX t (cid:3) = e tψ X ( u ) , z ∈ R where ψ X , usually called the characteristic exponent of the Levy process, is defined via φ X ( u ) = e ψ X ( u ) , with φ X being the chf of the r.v. X .Furthermore, for a subordinated Levy process Y t = X S ( t ) with subordinator S ( t ) (whichcan be viewed is a random compression of time) we have E (cid:2) e izY t (cid:3) = e t · l S ( ψ X ( z )) , z ∈ R , (28)where l S ( z ) defined via E (cid:2) e uS ( t ) (cid:3) = e t · l S ( u ) is called the Laplace exponent of S . For the chfof the increment Y in unit time, (28) turns into e l S ( ψ X ( z )) .Recall that our chf f γ ∈ (1 , , according to (27), can be represented as f γ ∈ (1 , ( z ) = L e γ ( − ln φ (2 , c ) ( z )) = exp n A h − (1 − ln φ (2 , c ) ( z )) e γ io = e l e γ ( ln φ (2 , c ) ( z ) ) , where l e γ is the Laplace exponent of the Levy process whose increments have the chf f e γ ∈ (0 , ,i.e. f e γ ∈ (0 , ( iu ) = L e γ ( − u ) = e l e γ ( u ) . Therefore f γ ∈ (1 , can be interpreted as a chf of the r.v. Y , i.e. of the increments of subordinated Levy process Y t = X S ( t ) in unit time t = 1 . Theincrements of the underlying process X t have normal distribution with parameters (2 , c ) ,while the increments of the subordinator S ( t ) are distributed according to the exponentiallytransformed one-sided Levy distribution whose chf is given by f e γ ∈ (0 , .A possible way to extend the model is to use any relevant chf as the component φ inthe representation (27). That would also lead to an interpretation of the correspondingdistribution as a distribution of the increments of a subordinated Levy process Y t = X S ( t ) ,with the same subordinator S ( t ) but with a different underlying process X t . A Derivation of the chf in the explicit form
The function f to be found satisfies the balance equations f θ ( t ) = f ( t − iθ ) f ( − iθ ) and f θ ( t ) = f α ( θ ) ( β ( θ ) t ) ∀ t ∈ R , θ ∈ [ a, b ] , ∈ [ a, b ] . In terms of the log-function g ( t ) = log f ( t ) , it turns into g ( t − iθ ) = g ( − iθ ) + α ( θ ) g ( β ( θ ) t ) . In order to prove that g can be represented in the form (5), let us differentiate g ( t − iθ ) w.r.t. to t and to θ . First differentiate it w.r.t. to tg ′ ( t − iθ ) = α ( θ ) · β ( θ ) · g ′ ( β ( θ ) t ) . (29)Considering a limit w.r.t θ → and assuming that α (0) β (0) = 0 , we get g ′ ( t ) = Ag ′ ( bt ) , (30)where b := β (0) and A := α (0) β (0) . Two major cases are possible: b = 1 and b = 0 . Let usstart with the first one. 13 = 1 Without loss of generality, assume that | b | < . Let us differentiate (30) further k times g ( k +1) ( t ) = A · b k g ( k +1) ( bt ) , and consider it at t = 0 g ( k +1) (0) = A · b k g ( k +1) (0) . Considering a limit w.r.t k → ∞ , we see that < Ab k → , which means that g ( k +1) (0) = 0 when k > K ( for some large enough K ) . It follows from the latter that g ( t ) is some polynomial g ( t ) = P ( t ) , so that the chf f ( t ) isthe exponent of a polynomial f ( t ) = e P ( t ) . Due to Marcinkiewicz’s theorem [12], we canconclude that the degree of P should not be higher than , e.g. it is a quadratic polynomial.In other words, b = 1 in (30) corresponds to the case when f ( t ) is the chf of the Normaldistribution considered in Section 2.1 as Case e. b = 1 Specifically, b = 1 means that not just β (0) = 1 but also α (0) = 1 , as follows from (29).Denoting Λ( θ ) := α ( θ ) β ( θ ) and keeping in mind that Λ(0) = 1 , let us now differentiate(29) w.r.t to θ which gives − ig ′′ ( t − iθ ) = Λ ′ ( θ ) g ′ ( β ( θ ) t ) + Λ( θ ) β ′ ( θ ) g ′′ ( β ( θ ) t ) t and considering a limit w.r.t θ → , we get − ig ′′ ( t ) = Λ ′ (0) g ′ ( t ) + tβ ′ (0) g ′′ ( t ) , which leads to − ( tβ ′ (0) + i ) g ′′ ( t ) = Λ ′ (0) g ′ ( t ) Denoting u ( t ) := g ′ ( t ) , we get an equation u ′ ( t ) u ( t ) = − Λ ′ (0)( i + tβ ′ (0)) (31)The solution of this equation gives ln u ( t ) = − c ln( i + tβ ′ (0)) + c ⇐⇒ u ( t ) = c ( tβ ′ (0) + i ) c ( where c := e c ) , i.e. g ′ ( t ) = c ( tβ ′ (0) + i ) − c , which means that g ( t ) = c (1 − c ) β ′ (0) (cid:0) β ′ (0) t + i (cid:1) − c + c . Denote − c =: γ and, in order to express the constant c explicitly, consider the aboveat t = 0 c (1 − c ) β ′ (0) i γ = − c , so that we can denote A := c (1 − c ) β ′ (0) and rearrange to get g ( t ) = A (cid:2) ( β ′ (0) t + i ) γ − i γ (cid:3) . A := − A i γ and c := β ′ (0) , after rearrangement it finally turnsinto g ( t ) = A [1 − (1 − ict ) γ ] . Throughout the derivation, we assumed that β ′ (0) = 0 . Let us now consider the oppositecase. β ′ (0) = 0 It easy to check that this corresponds to a degenerate case, as follows from (31) u ′ ( t ) u ( t ) = − const = c ⇐⇒ ln u ( t ) = ct + c , which implies that u ( t ) = e ce ct . Eventually, the general form of the function g in question is g ( t ) = A [1 − (1 − itc ) γ ] . References [1] A. Balkema, C. Kluppelberg, and S. Resnick (1999). Limit laws for exponential families.
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