Chapter 2 Asymptotics

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1 Chapter Asymptotics.1 Asymptotic Behavior of Student s Pdf Proposition.1 For each x R d,asν, f ν,,a x g a, x..1 Proof Let a = 0. Using the well-known formula that Ɣz = π z e z z z 1 + O 1, as z,. z we find that, as ν, Ɣ π νπ d Ɣ ν νπ d e π ν e ν ν ν 1 π d.3 and, obviously, 1 + x 1 ν e 1 x 1,x.. Here and below is the equivalence sign. The statement.1 with a = 0 follows from 1.1,.,.3 and.. Let now a = 0 and y ν = [ a 1, a ν + x 1 1 ]. ν + d B. Grigelionis, Student s t-distribution and Related Stochastic Processes, 9 SpringerBriefs in Statistics, DOI: / _, The Authors 013

2 10 Asymptotics Because, as ν, uniformly in y see Appendix K ν νy π exp { ν 1 + y } y ν 1 + y y ν and 1 + y ν y ν, we shall have that K [ a 1, a ν + x 1 ] 1 π ν + d exp π e exp ν + d = K ν + d y ν { ν + d } y ν y ν + 1 y ν { 1 } ν + d a 1, a ν + x 1 y ν + 1 y ν From 1. and.5 we elementarily find that f ν,,a x ν ν { exp x 1, a } a 1, a π Ɣ ν π d ν + x 1 e, x ν + d { exp 1 } ν + d a 1, a ν + x 1 y ν + 1 y ν exp { x 1, a } π d e a 1,a e d ν + x y ν exp { x 1, a } e a 1,a e d exp π d { 1 } x 1 d y ν But y ν = [ ] ν + d a 1, a ν + x 1

3 .1 Asymptotic Behavior of Student s Pdf 11 { } 1 exp a 1, a..7 Thus,.6 and.7 imply that, for each x R d,asν, f ν,,a x exp { x 1, a } π d { exp 1 a 1, a + x 1 } = g a, x. Proposition. For each fixed xɛ R d and ν>0, as a 0, f ν,,a x f ν, x. Proof Indeed, as a 0, K [ a 1, a ν + x 1 ] 1 ν + d Ɣ [ ] 1 a 1, a ν + x 1 see Appendix and, having in mind formulas 1.1, 1., f ν,,a x ν ν Ɣ Ɣ ν π d ν + x 1 = f ν, x. Proposition.3 i As x, f ν, x c ν, x 1, where c ν, = Ɣ d+ν. π d Ɣ ν ii As x,a = 0, f ν,,a x c ν,,a x 1 +1

4 1 Asymptotics { [ exp a 1, a x 1 1 } ] + x 1, a, where c ν,,a = ν ν a 1, a +1 Ɣ ν. πd 1 Proof i Obviously follows from 1.1. ii Because, as x, K [ ] a 1, a ν + x 1 1 π [ ] a 1, a ν + x 1 1 exp { [ ] a 1, a ν + x 1 1 }, from 1. we find that, as x, f ν,,a x ν ν exp a 1, a 1 Ɣ ν πd 1 { exp { x 1, a } ν + x 1 +1 [ a 1, a ] ν + x 1 1 } c ν,,a x 1 +1 { [ exp a 1, a x 1 1 } ] + x 1, a. Corollary. Let d=1. i If a > 0, x, then f ν,σ,ax 1 νa ν σɣ ν x ν 1..8 σ ii If a > 0, x, then

5 .1 Asymptotic Behavior of Student s Pdf 13 iii If a < 0, x, then f ν,σ,ax 1 νa ν { σɣ ν x ν 1 exp a x } σ σ..9 f ν,σ,ax 1 σɣ ν iv If a < 0, x, then ν a σ f ν,σ,ax 1 σɣ ν ν { x ν 1 exp a x } σ..10 ν a σ ν x ν Asymptotic Distributions for Extremal and Record Values Let now d = 1 and {X n, n 1} a sequence of i.i.d. random variables with common Student s t-distribution function and let M n = max 1 j n X j. Proposition.5 i If pdf of L X 1 is f ν,σ, then, as n, L K 1 n ν 1 Mn ν, where means weak convergence of probability laws, ν is the Fréchet distribution { { exp x ν }, if x > 0 ν x = 0, if x 0, and K 1 = Ɣν+1 σ ν ν πɣ ν. ii If pdf of L X 1 is f ν,σ,a,a> 0, then, as n, L K n ν Mn ν, where

6 1 Asymptotics K = νa σ ν νσɣ ν. iii If pdf of L X 1 is f ν,a,σ,a< 0, then, as n, L a ν σ M n ln n + 1 ln ln n + ln K 3, where is the Gumbel distribution and x = e e x, x R 1, K 3 = ν ν σ ν +3 ν+ Ɣ ν. Proof i From Proposition.3 i with d = 1 and the l Hospital s rule we have, as x, f ν,σ udu c ν,σ x ν = K1 x ν,.1 νσ σ x where c ν,σ = Ɣ ν+1 πɣ ν. σ ii The statement i is standard for Pareto-like distributions see, e.g., [1, ]. From Corollary. i and the l Hospital s rule we have that, as x, x f ν,σ,audu K x ν.13 iii and the conclusion is analogs to i. From Corollary. iii and the l Hospital s rule we find that, as x, x f ν,σ,audu σ a Ɣ ν ν a σ ν { x ν 1 exp a x } σ..1 The statement iii is standard for gamma-like distributions see, e.g., [1, ].

7 . Asymptotic Distributions for Extremal and Record Values 15 Now let us recall main results on limit theorems for record values in the sequences of i.i.d. random variables {X n, n 1} with a common continuous distribution function F which will be applied to the case of Student s t-distributions. The record times are L 1 = 1, L n+1 = min { k : k > n, X k > X Ln } for n = 1,,..., and the record values are R n = X Ln, n = 1,,...LetW x = log1 Fx be the integrated hazard function and the associate distribution function Ax = 1 e W x, x R 1.Letl a,b x = ax + b, a > 0, b R 1,bea group of affine homeomorphisms of R 1 with the composition law l a1,b 1 l a,b = l a1 a,a 1 b +b 1, the unit element l 1,0 and the inverse l 1 a,b = l a 1,a 1 b. The domain of attraction problem for record values using linear normalization was solved by Resnick see [3] also[]. It was proved that the class of all possible non-degenerated weak limit laws Q such that for suitable constants a n > 0, b n R 1, as n, L la 1 n,b n R n Q coincide with the class of laws log log G, where is a standard normal distribution and G is a l-max stable law, i. e. a non-degenerated distribution on R 1 such that for any n there exist constants a n > 0, b n R 1 satisfying G n x = G l an,b n x, x R 1. As in the classical extreme value theory this class can be factorized into three linear types, saying that probability distributions F 1 and F are of the same linear type it there exist constants a > 0, b R 1 such that F 1 x = F la,b x, x R 1. In the classical case these types are generated by the Fréchet distribution γ,the Gumbel distribution and the Weibull distribution { 1, if x 0, γ x = exp { x γ }, if x < 0, γ > 0, which correspond to generators of three types of the limiting laws for L l an,b n R n : γ x = { 0, if x 0, log x γ, if x > 0, γ > 0,

8 16 Asymptotics γ x = { log x γ, if x < 0, 1, if x 0, γ > 0, and the standard normal distribution x, x R 1. We say that F belongs to the record domain of attraction under linear normalization of the non-degenerated distribution Q F RDA l Q for short if there exist constants a n > 0 and b n R 1 such that L la 1 n,b n R n Q,asn. Duality theorem of Resnick says that F RDA l γ A MDA l γ, F RDA l γ A MDA l γ and F RDA l A MDA l, where MDA l Q denotes the maximum domain of attraction under linear normalization of the non-degenerated distribution Q see, e.g., [3]. As a corollary we find that in the case of heavy-tailed distributions F the record values cannot have non-degenerate limiting distributions if we use linear normalization. Indeed, for the Pareto-like distributions F, satisfying, as x, 1 Fx Kx δ, δ > 0, the associate distributions A satisfy, as x, 1 Ax e δ log x. In this case A MDA l γ MDA l γ MDA l. This fact is an argument to consider limit theorems for the record values using power normalization. Let p α,β x = α x β signx, α > 0, β > 0, x R 1. Observe that this class of functions form a group of homeomorphisms of R 1 with the composition law the unit element p 1,1 and the inverse p α1,β 1 p α,β = p α1 α β 1,β 1β, p 1 α,β = p α β 1,β 1. We say that F belongs to the record domain of attraction under power normalization of the non-degenerate distribution Q F RDA p Q for short if there exist constants α n > 0, β n > 0 such that, as n, L pα 1 n,β n R n Q. A non-degenerate distribution function G on R 1 is called p-max stable if for any n there exist constants α n > 0, β n > 0 such that G n x = Gp αn, β n x, x R 1.

9 . Asymptotic Distributions for Extremal and Record Values 17 Probability distributions F 1 and F are of the same power type if there exist constants α>0, β>0such that F 1 x = F p α,β x, x R 1. The class of non-degenerated limiting distributions for L pα 1 n,β n R n, asn, is equal to the class of law ˆ log log Ĝ, where G is a p-max stable law K, and is factorized to the six power types, generated by the distribution functions see [5, 6]: { 0, if x 1, ˆ 1,γ x = γ log log x, if x > 1, γ > 0, 0, if x 0, ˆ,γ x = γ log log x, if 0 < x < 1, 1, if x 1, γ > 0, 0, if x 1, ˆ 3,γ x = γ log log x, if 1 < x < 0, 1, if x 0, γ > 0, { γ log log x, if x < 1, ˆ,γ x = 1, if x 1, γ > 0, { 0, if x 0, ˆ 5 x = log x, if x > 0, and { log x, if x < 0, ˆ 6 x = 1, if x 0. There are the valid analog of Resnick s duality theorem and the principle of equivalent tails, which says that if continuous distribution functions F 1 and F are such that rf 1 = rf and 1 F 1 x 1 F x, asx rf 1, then F 1 RDA p Q if and only if F RDA p Q with the same normalizing constants, where rf = sup{x : Fx <1} and Q is a non-degenerate limiting distribution for record values using power normalization. The following analog of classical R. von Mises theorem [7] holds true. Theorem.6 [8]. Assume that the integrated hazard function W x is differentiable in some neighborhood of rf. Then: i if rf = and then F RDA p ˆ 1,γ ; W xx log x lim = γ, γ > 0, x W x

10 18 Asymptotics ii if 0 < rf < and lim x rf then F RDA p ˆ,γ ; iii if rf = 0 and then F RDA p ˆ 3,γ ; iv if rf <0 and lim x rf W xx log rf x = γ, γ > 0, W x W xx log x lim = γ, γ > 0, x 0 W x W x x log x rf W x = γ, γ > 0, then F RDA p ˆ,γ ; v if W is twice differentiable in some neighborhood of rf and W lim W x x x rf W x + 1 xw = 0,.15 x then for 0 < rf F RDA p ˆ 5 and for rf 0 F RDA p ˆ 6. Proposition.7 i If pdf of F is f ν,σ, then F RDA p ˆ 5. ii If pdf of F is f ν,σ,a,a> 0, then F RDA p ˆ 5. iii If pdf of F is f ν,σ,a,a< 0, then F RDA l. Proof i From the principle of equivalent tails and.1 it is enough to check.15 with rf = and the integrated hazard function Indeed, W x = ν ln x ln K 1. W x W x + 1 xw x = ν x νx + 1 ν 0.

11 . Asymptotic Distributions for Extremal and Record Values 19 ii From the principle of equivalent tails and.13 it is enough to check.15 with rf = and the integrated hazard function Again we find that W x = ν ln x ln K. W x W x + 1 xw x = ν x ν + ν 0. x iii From.1 and the principle of equivalent tails it is enough to consider the integrated hazard function W x = ν + 1 ln x + a σ x ln K 3, where K 3 = σ a Ɣ ν The corresponding associated distribution ν ν a. σ { } ν 1 Ax = exp + 1 ln x + a σ x ln K 3 { } a exp σ x, as x. Using again the principle of equivalent tails, Resnick s duality theorem and criteria from the classical extreme value theory we easily find that A MDA l and thus F RDA l. References 1. Embrechts, P., Klüppelberg, C., Mikosch, T.: Modelling Extremal Events for Insurance and Finance. Springer, Berlin Leadbetter, M.R., Lindgren, G., Rootzen, H.: Extremes and Related Properties of Random Sequences and Processes. Springer, Berlin Resnik, S.I.: Limit laws for record values. Stoch. Processes Appl. 1, Tata, M.N.: On outstanding values in a sequence of random variables. Z. Wahrscheinlichkeitstheor. vewr. Geb. 11,

12 0 Asymptotics 5. Mohan, N.R., Ravi, S.: Max domains of attraction of univariate and multivariate p-max stable laws. Teor. Veroyatnost. i Primenen, 37, Pantcheva, E.: Limit theorems for extreme order statistics under nonlinear normalization. In: Lecture Notes in Math., vol. 1155, pp Springer, Berlin, von Mises, R.: La distribution de la plus grande de n valeurs. Revue Mathématique de l Union Interbalkanique Athens 1, Grigelionis, B.: Limit theorems for record values using power normalization. Lith. Math. J. 6,

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