VARIANCE-MEAN MIXTURES AS ASYMPTOTIC APPROXIMATIONS

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1 VARIANCE-MEAN MIXTURES AS ASYMPTOTIC APPROXIMATIONS Victor Korolev Vladimir Bening Lomonosov Moscow State University Leninskie Gory, Moscow, Russia Institute of Informatics Problems, Maria Grigoryeva Parexel International Osenny Bulvar, 23, Moscow, Russia Andrey Gorshenin Institute of Informatics Problems, Vavilova str., 44/2, Moscow, Russia MIREA, Faculty of Information Technology Alexer Zeifman Vologda State University S.Orlova, 6, Vologda, Russia Institute of Informatics Problems, Vavilova str., 44/2, Moscow, Russia Institute of Socio-Economic Development of Territories, a zeifman@mail.ru KEYWORDS Rom sequence, Rom index, Transfer theorem, Rom sum, Rom Lindeberg condition, Statistic constructed from sample with rom size, Normal variance-mean mixture. ABSTRACT We present a general transfer theorem for rom sequences with independent rom indexes in the double array limit setting. We also prove its partial inverse providing necessary sufficient conditions for the convergence of romly indexed rom sequences. Special attention is paid to the cases of rom sums of independent not necessarily identically distributed rom variables statistics constructed from samples with rom sizes. Using simple moment-type conditions we prove the theorem on convergence of the distributions of such sums to normal variance-mean mixtures. INTRODUCTION Rom sequences with independent rom indexes play an important role in modeling real processes in many fields. Most popular examples of the application of these models usually deal with insurance reliability theory (Kalashnikov, 1997; Bening Korolev, 2002), financial mathematics queuing theory (Bening Korolev, 2002; Gnedenko Korolev, 1996), chaotic processes in plasma physics (Korolev Skvortsova, 2006) where rom sums are principal mathematical models. More general romly indexed rom sequences arrive in the statistics of samples with rom sizes. Indeed, very often the data to be analyzed is collected or registered during a certain period of time the flow of informative events each of which brings a next observation forms a rom point process, so that the number of available observations is unknown till the end of the process of their registration also must be treated as a (rom) observation. The presence of rom indexes usually leads to that the limit distributions for the corresponding romly indexed rom sequences are heavytailed even in the situations where the distributions of non-romly indexed rom sequences are asymptotically normal see, e. g., (Gnedenko Korolev, 1996; Bening Korolev, 2005). The literature on rom sequences with rom indexes is extensive, see, e. g., the references above the references therein. The mathematical theory of rom sequences with rom indexes, in particular, rom sums, is well-developed. However, there still remain some unsolved problems. For example, necessary sufficient conditions for the convergence of the distributions of rom sums to normal variance-mean mixtures were found only recently for the case of identically distributed summs (see (Korolev, 2013; Grigoryeva Korolev, 2013)). The case of rom sums of non-identically distributed rom summs, moreover, more general statistics constructed from samples with rom sizes has not been considered yet. At the same time, normal variance-mean mixtures are widely used as mathematical models of statistical regularities in many fields. In particular, in O. Barndorff-Nielsen (Barndorff-Nielsen, Proceedings 28th European Conference on Modelling Simulation ECMS Flaminio Squazzoni, Fabio Baronio, Claudia Archetti, Marco Castellani (Editors) ISBN: / ISBN: (CD)

2 1977), (Barndorff-Nielsen, 1978) introduced the class of generalized hyperbolic distributions as a class of special variance-mean mixtures of normal laws in which the mixing is carried out in one parameter since location scale parameters of the mixed normal distribution are directly linked. The range of applications of generalized hyperbolic distributions varies from the theory of turbulence or particle size description to financial mathematics, see (Barndorff- Nielsen et al., 1982). The paper is organized as follows. Basic notation is introduced in section Notation. Auxiliary results. Here an auxiliary result on the asymptotic rapprochement of the distributions of romly indexed rom sequences with special scale-location mixtures is presented. In section General transfer theorem its inversion. The structure of limit laws we present discuss an improved version of a general transfer theorem for rom sequences with independent rom indexes in the double array limit setting. We also present its partial inverse providing necessary sufficient conditions for the convergence of romly indexed rom indexes. Following the lines of (Korolev, 1993), we first formulate a general result improving some results of (Korolev, 1993; Bening Korolev, 2002) by removing some superfluous assumptions relaxing some conditions. Special attention is paid to the case where the elements of the basic double array are formed as cumulative sums of independent not necessarily identically distributed rom variables. This case is considered in section Limit theorems for rom sums of independent rom variables. To prove our results, we use simply tractable moment-type conditions which can be easily interpreted unlike general conditions providing the weak convergence of rom sums of non-identically distributed summs in (Szasz, 1972; Kruglov Korolev, 1990) (Kruglov ZhangBo, 2002 ). In section A version of the central limit theorem for rom sums with a normal variance-mean mixture as the limiting law we present the theorem on convergence of the distributions of such sums to normal variance-mean mixtures. As a simple corollary of this result we can obtain some results of the recent paper by A.A. Toda Weak limit of the geometric sum of independent but not identically distributed rom variables (arxiv: v ). That paper demonstrates that there is still a strong interest to geometric sums of non-identically distributed summs to the application of the skew Laplace distribution which is a normal variance-mean mixture under exponential mixing distribution (Korolev Sokolov, 2012). Special attention is paid to the case where the elements of the basic double array are formed as statistics constructed from samples with rom sizes. This case is considered in section Limit theorems for statistics constructed from samples with rom sizes. Unfortunately, to fit the requirements to the size of the paper we had to omit the details of the proofs which will be published elsewhere. NOTATION. AUXILIARY RESULTS Assume that all the rom variables considered in this paper are defined on one the same probability space (Ω, F, P). In what follows the symbols d = = will denote coincidence of distributions weak convergence (convergence in distribution). A family {X j } j N of rom variables is said to be weakly relatively compact, if each sequence of its elements contains a weakly convergent subsequence. In the finite-dimensional case the weak relative compactness of a family {X j } j N is equivalent to its tightness: lim R sup n N P( X n > R) = 0 (see, e. g., (Loeve, 1977)). Let {S n,k }, n, k N, be a double array of rom variables. For n, k N let a n,k b n,k be real numbers such that b n,k > 0. The purpose of the constants a n,k b n,k is to provide weak relative compactness { of the family} of the rom variables Yn,k (S n,k a n,k )/b n,k in the cases where n,k N it is required. Consider a family {N n } n N of nonnegative integer rom variables such that for each n, k N the rom variables N n S n,k are independent. Especially note that we do not assume the row-wise independence of {S n,k } k 1. Let c n d n be real numbers, n N, such that d n > 0. Our aim is to study the asymptotic behavior of the rom variables Z n (S n,nn c n )/d n as n find rather simple conditions under which the limit laws for Z n have the form of normal variance-mean mixtures. In order to do so we first formulate a somewhat more general result following the lines of (Korolev, 1993), removing superfluous assumptions, relaxing the conditions generalizing some of the results of that paper. The characteristic functions of the rom variables Y n,k Z n will be denoted h n,k (t) f n (t), respectively, t R. Let Y be a rom variable whose distribution function characteristic function will be denoted H(x) h(t), respectively, x, t R. Introduce the rom variables U n = b n,nn /d n, V n = (a n,nn c n )/d n. Introduce the function g n (t) Eh(tU n ) exp{itv n } = { = exp it a n,k c } ( n tbn,k ) h, t R. d n d n k=1 It can be easily seen that g n (t) is the characteristic function of the rom variable Y U n + V n where the rom variable Y is independent of the pair (U n, V n ). Therefore, the distribution function

3 G n (x) corresponding to the characteristic function g n (t) is the scale-location mixture of the distribution function H(x): G n (x) = EH ( (x V n )/U n ), x R. (1) In the double-array limit setting considered in this paper, to obtain non-trivial limit laws for Z n we require the following additional coherency condition: for any T (0, ) lim E sup hn,nn (t) h(t) = 0. (2) t T To clarify the sense of the coherency condition, note that if we had usual row-wise convergence of Y n,k to Y, then for any n N T [0, ) lim sup h n,k (t) h(t) = 0. (3) k t T So we can say that coherency condition (2) means that pure row-wise convergence (3) takes place on the average so that that the row-wise convergence as k is somehow coherent with the principal convergence as n. Remark 1. It can be easily verified that, since the values under the expectation sign in (2) are nonnegative bounded (by two), then coherency condition (2) is equivalent to that sup t T hn,nn (t) h(t) 0 in probability as n. Lemma 1. Let the family of rom variables {U n } n N be weakly relatively compact. Assume that coherency condition (2) holds. Then for any t R we have lim f n(t) g n (t) = 0. Lemma 1 makes it possible to use the distribution function G n (x) (see (1)) as an accompanying asymptotic approximation to F n (x) P(Z n < x). In order to obtain a limit approximation, in the next section we formulate prove the transfer theorem. GENERAL TRANSFER THEOREM AND ITS INVERSION. THE STRUCTURE OF LIMIT LAWS Theorem 1. Assume that coherency condition (2) holds. If there exist rom variables U V such that the joint distributions of the pairs (U n, V n ) converge to that of the pair (U, V ) : then (U n, V n ) = (U, V ) (n ), (4) Z n = Z d = Y U + V (n ). (5) where the rom variable Y is independent of the pair (U, V ). It is easy to see that relation (5) is equivalent to the following relation between the distribution functions F (x) H(x) of the rom variables Z Y : F (x) = EH ( (x V )/U ), x R, (6) that is, the limit law for normalized romly indexed rom variables Z n is a scale-location mixture of the distributions which are limiting for normalized non-romly indexed rom variables Y n,k. Among all scale-location mixtures, variancemean mixtures attract a special interest (to be more precise, we should speak of normal variance-mean mixtures). Let us see how these mixture can appear in the double-array setting under consideration. Assume that the centering constants a n,k c n are in some sense proportional to the scaling constants b n,k d n. Namely, assume that there exist ρ > 0, α n R β n R such that for all n, k N we have n,k α n d ρ n a n,k = bρ+1 there exist finite limits α = lim α n,, c n = d n β n, β = lim β n. Then under condition (4) n ( bn,nn (U n, V n ) =, a n,n n c ) n = d n d n = ( U n, α n Un ρ+1 ) ( + β n = U, αu ρ+1 + β ), so that in accordance with theorem 2 the limit law for Z n takes the form ( x β αu ρ+1 ) P(Z < x) = EH, x R. U If ρ = 1, then we obtain the pure variance-mean mixture ( x β αu 2 ) P(Z < x) = EH, x R. U We will return to the discussion of convergence of romly indexed sequences, more precisely, of rom sums, to normal scale-location mixtures in Sect. 5. In order to present the result that is a partial inversion of theorem 1, for fixed rom variables Z Y with the characteristic functions f(t) h(t) introduce the set W(Z Y ) containing all pairs of rom variables (U, V ) such that the characteristic function f(t) can be represented as f(t) = Eh(tU)e itv, t R, (7) P(U 0) = 1. Whatever rom variables Z Y are, the set W(Z Y ) is always nonempty since it trivially contains the pair (0, Z). It is easy to see

4 that representation (7) is equivalent to relation (6) between the distribution functions F (x) H(x) of the rom variables Z Y. The set W(Z Y ) may contain more that one element. For example, if Y is the stard normal rom variable Z = d W 1 W 2 where W 1 W 2 are independent rom variables with the same stard exponential distribution, then along with the pair (0, W 1 W 2 ) the set W(Z Y ) contains the pair ( W 1, 0 ). In this case F (x) is the symmetric Laplace distribution. Let L 1 (X 1, X 2 ) be the Lévy distance between the distributions of rom variables X 1 X 2 : if F 1 (x) F 2 (x) are the distribution functions of X 1 X 2, respectively, then L 1 (X 1, X 2 ) = inf{y 0 : F 2 (x y) y F 1 (x) F 2 (x + y) + y, x R}. As is well known, the Lévy distance metrizes weak convergence. Let L 2 ( (X1, X 2 ), (Y 1, Y 2 ) ) be any probability metric which metrizes weak convergence in the space of two-dimensional rom vectors. An example of such a metric is the Lévy Prokhorov metric (see,e. g., (Zolotarev, 1997)). Theorem 2. Let the family of rom variables {U n } n N be weakly relatively compact. Assume that coherency condition (2) holds. Then a rom variable Z such that Z n = Z (n ) (8) with some c n R exists if only if there exists a weakly relatively compact sequence of pairs (U n, V n) W(Z Y ), n N, such that lim L ( 2 (Un, V n ), (U n, V n) ) = 0. (9) Remark 2. It should be noted that in (Korolev, 1993) some subsequent papers a stronger less convenient version of the coherency condition was used. Furthermore, in (Korolev, 1993) the subsequent papers the statements analogous to lemma 1 theorems 1 2 were proved under the additional assumption of the weak relative compactness of the family {Y n,k } n,k N. LIMIT THEOREMS FOR RANDOM SUMS OF INDEPENDENT RANDOM VARI- ABLES Let {X n,j } j 1, n N, be a double array of row-wise independent not necessarily identically distributed rom variables. For n, k N denote S n,k = X n, X n,k. (10) If S n,k is a sum of independent rom variables, then the condition of weak relative compactness of the sequence {U n } n N used in the preceding section can be replaced by the condition of weak relative compactness of the family {Y n,k } n,k N which is in fact considerably less restrictive. Indeed, let, for example, the rom variables S n,k possess moments of some order δ > 0. Then, if we choose b n,k = (E S n,k a n,k δ ) 1/δ, then by the Markov inequality lim sup P ( Y n,k > R ) 1 lim R n,k N R R δ = 0, that is, the family {Y n,k } n,k N is weakly relatively compact. Theorem 3. Assume that the rom variables S n,k have the form (10). Let the family of rom variables {Y n,k } n,k N be weakly relatively compact. Assume that coherency condition (2) holds. Then convergence (8) of normalized rom sums Z n to some rom variable Z takes place with some c n R if only if there exists a weakly relatively compact sequence of pairs (U n, V n) W(Z Y ), n N, such that condition (9) holds. A VERSION OF THE CENTRAL LIMIT THEOREM FOR RANDOM SUMS WITH A NORMAL VARIANCE-MEAN MIXTURE AS THE LIMITING LAW Let {X n,j } j 1, n N, be a double array of row-wise independent not necessarily identically distributed rom variables. As in the preceding section, let S n,k = X n, X n,k, n, k N. The distribution function of the rom variable X n,j will be denoted F n,j (x). Denote µ n,j = EX n,j, σn,j 2 = DX n,j assume that 0 < σn,j 2 <, n, j N. Denote ( ) A n,k = µ n, µ n,k = ESn,k, Bn,k 2 = σn, σn,k 2 ( ) = DSn,k It is easy to make sure that ES n,nn = EA n,nn, DS n,nn = EBn,N 2 n + DA n,nn, n N. In order to formulate a version of the central limit theorem for rom sums with the limiting distribution being a normal variance-mean mixture, assume that nonrom sums, as usual, are centered by their expectations normalized by by their mean square deviations put a n,k = A n,k, b n,k = Bn,k 2, n, k N. Although it would have been quite natural to normalize rom sums by their mean square deviations as well, for simplicity we will use slightly different normalizing constants put d n = EB 2 n,n n. Let Φ(x) be the stard normal distribution function, Φ(x) = 1 2π x e z2 /2 dz, x R.

5 Theorem 4. Assume that the following conditions hold: (i) for every n N there exist real numbers α n such that µ n,j = α nσn,j 2, n, j N, EB 2 n,nn lim α n = α, 0 < α < ; (ii) (the rom Lindeberg condition) for any ɛ > 0 E [ 1 N n Bn,N 2 n j=1 x µ n,j >ɛb n,nn (x µ n,j) 2 df n,j(x) ] 0 as n. Then the convergence of the normalized rom sums S n,nn EB 2 n,n n = Z (11) to some rom variable Z as n takes place if only if there exists a rom variable U such that ( x αu P(Z < x) = EΦ ), x R, (12) U B 2 n,n n EB 2 n,n n = U (n ). (13) The proof uses a result of V. V. Petrov (Petrov, 1979) improved in (Korolev Popov, 2012), the reasoning used to prove theorem 7 in Sect. 3, Chapt. V of (Petrov, 1987) relation (3.8) the fact recently proved in (Korolev, 2013) that normal variance-mean mixtures are identifiable. Remark 3. In accordance with what has been said in remark 1, the rom Lindeberg condition (ii) can be used in the following form: for any ɛ > 0 1 N n Bn,N 2 n j=1 x µ n,j >ɛb n,nn (x µ n,j) 2 df n,j(x) 0 in probability as n. LIMIT THEOREMS FOR STATISTICS CONSTRUCTED FROM SAMPLES WITH RANDOM SIZES Let {X n,j } j 1, n N, be a double array of rowwise independent not necessarily identically distributed rom variables. For n, k N let T n,k = T n,k (X n,1,..., X n,k ) be a statistic, i.e., a real-valued measurable function of X n,1,..., X n,k. For each n 1 we define a r.v. T n,nn by setting T n,nn (ω) T n,nn(ω)(x n,1 (ω),..., X n,nn(ω)(ω)), ω Ω. Let θ n be real numbers, n N. In this section we will assume that the rom variables S n,k have the form S n,k = T n,k θ n, n, k N. Concerning the normalizing constants we will assume that there exist finite real numbers α, α n, β, β n σ n > 0 such that α n α, β n β (n ) (14) for all n, k N so that b n,k = 1 σ n k, d n = 1 σ n n, a n,k = α n n σ n k, c n = β n (15) σ n n Y n,k = σ n k(tn,k θ n ) α n n/k Z n = σ n n(tn,nn θ n ) β n. As this is so, σ 2 n can be regarded as the asymptotic variance of T n,k as k whereas the bias of T n,k is α n n/(kσn ). The l 1 -distance between distribution functions P 1 P 2 will be denoted P 1 P 2 : P 1 P 2 = P 1 (x) P 2 (x) dx. Recall that the distribution function of the rom variable Y n,k is denoted H n,k (x). Let Φ(x) be the stard normal distribution function. In what follows we will assume that the statistic T n,k is asymptotically normal in the following sense: for any γ > 0 lim E H n,n n Φ = 0. (16) Theorem 5. Let the family of rom variables {n/n n } n N be weakly relatively compact, the normalizing constants have the form (15) satisfy condition (14). Assume that the statistic T n,k is asymptotically normal so that condition (16) holds. Then a rom variable Z such that σ n n(tn,nn θ n ) β n = Z (n ) exists if only if there exists a nonnegative rom variable W such that (x R) P(Z < x) = 0 ( x β αw Φ )dp(w < w), w P(N n < nx) = P(W 1 < x) (n ).

6 CONCLUDING REMARKS The class of normal variance-mean mixtures is very wide, in particular, contains the class of generalized hyperbolic distributions which, in turn, contains (a) symmetric skew Student distributions (including the Cauchy distribution) with inverse gamma mixing distributions; (b) variance gamma distributions (including symmetric non-symmetric Laplace distributions) with gamma mixing distributions; (c) normal\\inverse Gaussian distributions with inverse Gaussian mixing distributions including symmetric stable laws. By variance-mean mixing many other initially symmetric types represented as pure scale mixtures of normal laws can be skewed, e. g., as it was done to obtain non-symmetric exponential power distributions in (Grigoryeva Korolev, 2013). According to theorem 4, all these laws can be limiting for rom sums of independent non-identically distributed rom variables. For example, to obtain the skew Student distribution for Z it is necessary sufficient that in (12) (13) the rom variable U has the inverse gamma distribution (Korolev Sokolov, 2012). To obtain the variance gamma distribution for Z it is necessary sufficient that in (12) (13) the rom variable U has the gamma distribution (Korolev Sokolov, 2012). In particular, for Z to have the asymmetric Laplace distribution it is necessary sufficient that U has the exponential distribution. Remark 4. Note that the non-rom sums in the coherency condition are centered, whereas in (11) the rom sums are not centered, if α 0, then the limit distribution for rom sums becomes skew unlike usual non-rom summation, where the presence of the systematic bias of the summs results in that the limit distribution becomes just shifted. So, if non-centered rom sums are used as models of some real phenomena the limit variance-mean mixture is skew, then it can be suspected that the summs are actually biased. Remark 5. In limit theorems of probability theory mathematical statistics, centering normalization of rom variables are used to obtain non-trivial asymptotic distributions. It should be especially noted that to obtain reasonable approximation to the distribution of the basic rom variables (in our case, S n,nn ), both centering normalizing values should be non-rom. Otherwise the approximate distribution becomes rom itself, say, the problem of evaluation of quantiles becomes senseless. ACKNOWLEDGEMENTS The research is supported by the Russian Foundation for Basic Research (projects a, a, a) the Grant of the President of the Russian Federation (project MK ). REFERENCES Barndorff-Nielsen, O.E Exponentially decreasing distributions for the logarithm of particle size. Proc. Roy. Soc. Lond., Ser. A 353, Barndorff-Nielsen, O.E Hyperbolic distributions distributions of hyperbolae. Sc. J. Statist. 5, Barndorff-Nielsen, O.E., J. Kent, M. Sørensen Normal variance-mean mixtures z- distributions. Int. Statist. Rev. 50, No.2, Bening, V.E.; V.Yu. Korolev Generalized Poisson Models their Applications in Insurance Finance. VSP, Utrecht. Bening, V.E. V.Yu. Korolev On an application of the Student distribution in the theory of probability mathematical statistics. Theory of Probability its Applications 49, No.3, Gnedenko, B.V.; V.Yu. Korolev Rom Summation: Limit Theorems Applications. CRC Press, Boca Raton. Grigoryeva, M.E. V.Yu. Korolev On convergence of the distributions of rom sums to asymmetric exponential power laws. Informatics its Applications 7, No.4, Loéve, M Probability Theory. 3rd ed. Springer, New York. Kalashnikov, V.V Geometric Sums: Bounds for Rare Events with Applications. Academic Publishers, Dordrecht, Kluwer. Korolev, V.Yu On limit distributions of romly indexed rom sequences. Theory of Probability its Applications 37, No.3, Korolev, V.Yu Generalized hyperbolic laws as limit distributions for rom sums. Theory of Probability its Applications 58, No.1, Korolev, V.Yu. S.V. Popov Improvement of convergence rate estimates in the central limit theorem under weakened moment conditions. Doklady Mathematics 86, No.1, Korolev, V.Yu.; N.N. Skvortsova (Eds) Stochastic Models of Structural Plasma Turbulence. VSP, Utrecht.

7 Korolev, V.Yu. I.A. Sokolov Skew Student distributions, variance gamma distributions their generalizations as asymptotic approximations. Informatics its Applications 6, No.1, Kruglov, V.M.; V.Yu. Korolev Limit Theorems for Rom Sums. Moscow University Publishing House, Moscow. Kruglov, V.M. Zhang Bo Weak convergence of rom sums. Theory of Probability its Applications 46, No.1, is: ALEXANDER ZEIFMAN is Doctor of Science in physics mathematics; professor, Dean of the Faculty of Applied Mathematics Computer Technologies, Vologda State University; senior scientist, Institute of Informatics Problems, Russian Academy of Sciences; leading scientist, Institute of Socio-Economic Development of Territories, Russian Academy of Sciences. His address is: a zeifman@mail.ru Petrov, V.V A limit theorem for sums of independent, nonidentically distributed rom variables. Journal of Soviet Mathematics 20, No.3, Petrov, V.V Limit Theorems for Sums of Independent Rom Variables. Nauka, Moscow. Szász, D Limit theorems for the distributions of the sums of a rom number of rom variables. Annals of Mathematical Statistics 43, No.6, Zolotarev, V.M.; N.N. Skvortsova Modern Theory of Summation of Rom Variables. VSP, Utrecht. AUTHOR BIOGRAPHIES VICTOR KOROLEV is Doctor of Science in physics mathematics, professor, Department of Mathematical Statistics, Faculty of Computational Mathematics Cybernetics, M.V. Lomonosov Moscow State University; leading scientist, Institute of Informatics Problems, Russian Academy of Sciences. His address is: victoryukorolev@yex.ru VLADIMIR BENING is Doctor of Science in physics mathematics, professor, Department of Mathematical Statistics, Faculty of Computational Mathematics Cybernetics, M.V. Lomonosov Moscow State University; senior scientist, Institute of Informatics Problems, Russian Academy of Sciences. His address is: bening@yex.ru ANDREY GORSHENIN is Cidate of Science (PhD) in physics mathematics, senior scientist, Institute of Informatics Problems, Russian Academy of Sciences; associate professor, Faculty of Information Technology, Moscow State Institute of Radio Engineering, Electronics Automation. His address is: agorshenin@ipiran.ru MARIA GRIGORYEVA is biostatistician II, Parexel International, Moscow. Her address