EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 14, No. 1, 2021, 19-42 ISSN 1307-5543 – ejpam.com Published by New York Business Global Asymptotic laws for upper and strong record values in the extreme domain of attraction and beyond Gane Samb Lo1,2,3,∗, Mohammad Ahsanullah4, Moumouni Diallo5, Modou Ngom1 1 LERSTAD, Gaston Berger University, Saint-Louis, Sénégal 2 Department of Pure and Applied Mathematics, African University of Science and Technology, Abuja, Nigeria 3 LSTA, Pierre and Marie Curie University, Paris VI, France 4 Department of Management Sciences. Rider University. Lawrenceville, New Jersey, USA 5 Université des Sciences Sociale et de Gestion de Bamako (USSGB), Faculté des Sciences Économiques et de Gestion (FSEG) Abstract. Asymptotic laws of record values have usually been investigated as limits in type. In this paper, we use functional representations of the tail of cumulative distribution functions in the extreme value domain of attraction to directly establish asymptotic laws of record values, not necessarily as limits in type and their rates of convergences. Results beyond the extreme value domain are provided. Explicit asymptotic laws concerning very usual laws and related rates of convergence are listed as well. Some of these laws are expected to be used in fitting distribution. 2020 Mathematics Subject Classifications: 62G30, 60G70, 60Fxx Key Words and Phrases: Record values and record times, normal Asymptotic theory, extreme value theory, generalized extreme values distributions 1. Introduction Let X, X1, X2, · · · be a sequence of independent real-valued randoms, defined on the same probability space (Ω,A,P), with common cumulative distribution function F , which has the lower and upper endpoints, the first asymptotic moment function and the generalized inverse function defined by lep(F ) = inf{x ∈ R, F (x) > 0}, uep(F ) = sup{x ∈ R, F (x) < 1}, R(x, F ) = 1 1− F (y) ∫ uep(F ) x (1− F (y)) dy, x ∈]lep(F ), uep(F )[ ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v14i1.3872 Email addresses: gane-samb.lo@ugb.edu.sn (G. S. Lo), ahsan@rider.edu (M. Ahsanullah), moudiallo1@gmail.com (M. Diallo), ngom.modou1@ugb.edu.sn (M. Ngom) http://www.ejpam.com 19 c© 2021 EJPAM All rights reserved. G. S. Lo et al. / Eur. J. Pure Appl. Math, 14 (1) (2021), 19-42 20 and F−1(u) = inf{x ∈ R, F (x) ≥ u} for u ∈]0, 1[ and F−1(0) = F−1(0+). respectively. Finally, let us consider the sequence of strong record values X(1) = X1, X (n), · · · (see [7]) and the sequence of record times U(1) = 1, U(2), · · · . Before beginning an asymptotic theory, we should be sure that we have an infinite se- quence (X(n))n≥1. For a bounded random variable with finite upper bound uep(F ) such that P(X = uep(F )) > 0, we have (X(n) < uep(F )) finitely often. This happens for classical integer-valued and bounded random variables as Binomial laws. In such cases, the asymptotic theory is meaningless. But, an interesting question would be the charac- terization the infinite random sequence (nk)k≥1 such that Xnk = uep(F ) for all k ≥ 1. In all other cases, even if uep(F ) is bounded, the sequence (X(n))n≥1 is infinite. So, the results of this paper apply to cdf ’s F such that P(X = uep(F )) = 0. In that context, asymptotic laws have been proposed in the literature by many authors like [8], [10], [9], etc., in relation with Extreme Value Theory, as limits in type in the form (∃(An)n≥1 ⊂ R+ \ {0}), ∃ (Bn)n≥1 ⊂ R, X(n) −Bn An Z, (1) where stands for the convergence in distribution and Z is a non-degenerate random variable. The motive beneath this search is the following. If we denote by M(n) = max(X1, · · · , Xn) as the n-th maximum for n ≥ 1, it is clear that we have ∀ n ≥ 1, X(n) = M(U(n)). (2) Since for any F in the extremal domain of attraction D, we have that for some γ ∈ R, (∃(an)n≥1 ⊂ R+ \ {0}), (∃(bn)n≥1 ⊂ R, M(n)− an bn Zγ , (3) where the cdf of Zγ is the Generalized Extreme Value distribution defined by Gγ(x) = exp(−(1 + γx)1/γ), with 1 + γx > 0, and G0(x) = exp(− exp(−x)) for x ∈ R. In Extreme value Theory, Formula (3) is rephrased as F is attracted by Gγ denoted by F ∈ D(Gγ). From Formulas (2) and (3) and from the fact that U(n) → +∞ as n → +∞, the inves- tigation of the validity of (1) was justified enough. The results of the cited authors and G. S. Lo et al. / Eur. J. Pure Appl. Math, 14 (1) (2021), 19-42 21 others were positive with the stunning result that the cdf of Z should be on the form Φ(g(x)), x ∈ R, where Φ is the cdf of the standard normal law and g satisfies one of three definitions (in which c is a positive constant) g(x) = x, x ∈ R. g(x) = −∞1(x<0) + (c log x) 1(x≥0), x ∈ R. g(x) = (−c log−x) 1(x<0) +∞ 1(x>0), x ∈ R. Instead of using this mathematically appealing approach based on functional equations, an other approach consisting in directly finding the asymptotic laws of X(n), not necessarily in the form of Formula (1) is possible and we proceed to it here. That approach is based on representations of F ∈ D due to Karamata and to de Haan for example. Our achievement is the finding of the asymptotic laws of the records for all F ∈ D and the related rates of convergence : first, for γ 6= 0, outside the frame of Formula (1), that is as limits in type, and without any further condition and secondly, for γ = 0, within the frame of Formula (1), under a general regularity condition. That regularity condition generally holds for usual cdf ’s. We also give general conditions to ensure the asymptotic normality of the record values for F not necessarily in the extremal domain. As well general rates of convergence are given. These rates can be explicitly stated for usual cdf ’s in D. Finally, we give detailed asymptotic laws of the records of a list of remarkable cdf ’s with specific coefficients. In this paper that we want short, we use many results from Extreme Value Theory and Record Values Theory. So, for more details, we refer the reader to the books of [7] and [9], for an easy introduction to records and to those of [2], [1], [10] and [6], concerning Extreme Value Theory. To end this introduction, we recall two important tools of extreme value theory that form the basis of our method. The first is the following proposition. Suppose that X ≥ 0, that is F (0) = 0. In that case, we define Y = logX with cdf G(x) = F (ex), x ∈ R and we have the proposition below, Proposition 1. (see [5]) We have the following equivalences. (1) If γ > 0, F ∈ D(Gγ) ⇔ (G ∈ D(G0) and R(x,G)→ γ as x→ uep(G)). (2) If γ = 0, G. S. Lo et al. / Eur. J. Pure Appl. Math, 14 (1) (2021), 19-42 22 F ∈ D(G0) ⇔ (G ∈ D(G0) and R(x,G)→ 0 as x→ uep(G)). (3) If γ < 0, F ∈ D(Gγ) ⇔ (G ∈ D(Gγ)). In the second place, we recall the following representations of cdf ’s in the extreme value domain that repeatedly will be used in the sequel. Proposition 2. ([3] and [1]) We have the following characterizations for the three ex- tremal domains. (a) F ∈ D(Hγ), γ > 0, if and only if there exist a constant c and functions a(u) and `(u) of u→ u ∈]0, 1] satisfying (a(u), `(u))→ (0, 0) as u→ 0, such that F−1 admits the following representation of Karamata F−1(1− u) = c(1 + a(u))u−γ exp (∫ 1 u `(t) t dt ) . (4) (b) F ∈ D(Hγ), γ < 0, if and only if uep(F ) < +∞ and there exist a constant c and functions a(u) and `(u) of u ∈]0, 1] satisfying (a(u), `(u))→ (0, 0) as u→ 0, such that F−1 admit the following representation of Karamata uep(F )− F−1(1− u) = c(1 + a(u))u−γ exp (∫ 1 u `(t) t dt ) . (5) (c) F ∈ D(H0) if and only if there exist a constant d and a slowly varying function s(u) such that F−1(1− u) = d+ s(u) + ∫ 1 u s(t) t dt, 0 < u < 1, (6) and there exist a constant c and functions a(u) and `(u) of u ∈]0, 1] satisfying (a(u), `(u))→ (0, 0) as u→ 0, G. S. Lo et al. / Eur. J. Pure Appl. Math, 14 (1) (2021), 19-42 23 such that the function s(u) of u ∈]0, 1[ admits the representation s(u) = c(1 + a(u)) exp (∫ 1 u `(t) t dt ) . (7) Moreover, if F−1(1− u) is differentiable for small values of u such that r(u) = −u(F−1(1− u))′ = u dF−1(1− u)/du is slowly varying at zero, then (6) may be replaced by F−1(1− u) = d+ ∫ u0 u r(t) t dt, 0 < u < u0 < 1, (8) which will be called a reduced de Haan representation of F−1. The rest of the paper is organized as follows. The results are stated in Section 2. Examples and Applications are given in Section 3. The proofs are stated in Section 4. The compu- tations related to examples in Section 3 are detailed in the Appendix Section 6 (Appendix I, page 37). The paper is closed by a conclusion in Section 5. 2. Results Before stating our results, we recall that any F ∈ D is associated to a pair of functions (a(u), b(u)) of u ∈ [0, 1] as defined in the representations of Proposition 2 for F ∈ D(Gγ), γ 6= 0. In the special case where γ = 0, the pair of functions (a(◦), b(◦)) is used in the representation of the function s(u) of u ∈ [0, 1] in equation (6). We will need the following condition. Let us define, for any n ≥ 1, a finite sum of n standard exponential random variables S(n) = E1,n + · · ·+ En,n, and denote Vn = exp(−S(n)) and vn = exp(−n), n ≥ 1 and finally set the hypothesis (Ha) : sup {∣∣∣u v − 1 ∣∣∣ , min(vn, Vn) ≤ u, v ≤ max(vn, Vn) } →P 0 as n→ +∞, (Hb) : (∃α > 0), √ n s(vn)→ α as n→ +∞, G. S. Lo et al. / Eur. J. Pure Appl. Math, 14 (1) (2021), 19-42 24 where →P stands for the convergence in probability. Here are our results that cover the whole extreme value domain of attraction. For γ 6= 0, we need any condition. Let us begin by asymptotic laws for F ∈ D. Theorem 1. Let F ∈ D(Gγ), γ ∈ R. We have : (a) If γ > 0, the asymptotic law of X(n) is lognormal, precisely( X(n) F−1 (1− e−n) )n−1/2 LN(0, γ2), where LN(m,σ2) is the lognormal law of parameters m and σ > 0. (b) If γ > 0 and X ≥ 0, Y = logX ∈ D(G0)) and R(x,G) → γ as x → uep(G) and we have Y (n) −G−1 (1− e−n)√ n N (0, γ2). (c) If γ < 0, the asymptotic law of X(n) is lognormal, precisely( uep(F )−X(n) uep(F )− F−1 (1− e−n) )n−1/2 exp(N (0, γ2)). (d) Suppose that γ = 0 and R(x,G)→ 0 as x→ uep(G). If (Ha) and (Hb) hold both, we have X(n) − F−1 ( 1− e−n ) N (0, α2). More precisely, we have : Given γ = 0, R(x,G)→ 0 as x→ uep(G) and (Ha), the above asymptotic normality is valid if and only if (Hb) holds. Beyond distributions in D, we may use the delta-method as follows. Drawing lessons from Theorem 1, we might be tempted to generalize point (a) by imposing that F−1 satisfies, for some coefficient γ, ∀λ > 0, F−1(1− λu)/F−1(1− u) = λγ(1 + o(1)), u ∈]0, 1[. But, by Extreme Value Theory, this would imply that F ∈ D(Gγ) and nothing new would happen. But trying a generalization from Point (c) would be successful. Let us define the G. S. Lo et al. / Eur. J. Pure Appl. Math, 14 (1) (2021), 19-42 25 following hypothesis : (Ga) F is differentiable in some left neighborhood of uep(F ). (Gb) The function s(x) = e−x [ F−1(1− t) ]′ t=e−x , ex < u0 < 1, for some u0 ∈]0, 1[ decreases to 0 as x→ +∞ and is such that : for any sequence (xn, yn)n≥1 such that lim sup n→+∞ |xn − yn|/ √ n < +∞, we have, for some α > 0, lim n→+∞ √ n s (exp(min(xn, yn))) = lim n→+∞ √ n s (exp(max(xn, yn))) = α. We have the following generalization. Theorem 2. If F satisfies Assumptions (Ga) and (Gb), then we have X(n) − F−1 ( 1− e−n ) N (0, α2). Comments I . A firm look at the results shows that for any F ∈ D, we found the direct asymptotic law of X(n) or that of a function of X(n), mainly logX(n). For example, Point (d) of Theorem 1 cannot be applied when X follows a lognormal law but can be applied to exp(X). This leads to the following rule for all F ∈ D : (e) If F ∈ D(Gγ), γ 6= 0, we apply Points (a) or (c) without any further condition. (f) If F ∈ D(G0) and exp(X) ∈ D(Gγ) for some γ > 0, we apply Point (b) without any further condition. (g) If F ∈ D(G0) and s(u)→ 0 as u→ 0. If (Ha) and (Hb) hold, we conclude by applying Point (d). If not (as it is for a lognormal law), we search whether X1 = exp(X) ∈ D(Gγ) for some γ > 0 or X1 = exp(X) fulfills (Ha) and (Hb). If yes, we conclude by Point (b) or by Point (d). If not, we consider X2 = exp(X1), and we continue until we reach Xp = exp(Xp−1) ∈ D(Gγ) for some γ > 0 or Xp = exp(Xp−1) for some p ≥ 1. Finally, we handle the rates of convergences in the theorems stated above. Let us introduce the following notations. G. S. Lo et al. / Eur. J. Pure Appl. Math, 14 (1) (2021), 19-42 26 Theorem 3. Let F ∈ D(Gγ), γ ∈ R. Then, there exists a probability space (Ω,A,P) holding a sequence of independent standard exponential random variables (En)n≥1 and a Brownian Process {W (t), t ≥ 0} such that the record values X(n), n ≥ 1, of the sequence Xj = F−1 ( 1− eEj ) , j ≥ 1, satisfy the following representations below under the appro- priate conditions. Here, Sn = E1 + ... + En, n ≥ 1, are the partial sums of the sequence (En)n≥1, S∗n = n−1/2(Sn − n), vn = e−n and Vn = e−Sn. Below, the function a(u), b(u) and s(u) of u ∈]0, 1[ are those in the representations in Proposition 2. By denoting W ∗n = n−1/2W (n) and cn = n−1/2 log n, we have W ∗n ∼ N (0, 1) and |S∗n −W ∗n | = OP(cn). Further, we have the following results. (a) Let γ > 0. Suppose that 1− 1 + a(Vn) 1 + a(vn) = O(an), sup{|b(t)|, 0 ≤ t ≤ vn ∨ Vn} = OP(bn) . (9) Then, we have ( X(n) F−1 (1− e−n) )n−1/2 = exp(γS∗n) +OP(an ∨ bn) = exp(γW ∗n) +OP(an ∨ bn ∨ cn). (b) Let γ > 0 and X ≥ 0, Y = logX ∈ D(G0)) and R(x,G)→ γ as x→ uep(G) and we have Y (n) −G−1 (1− e−n)√ n = γS∗n +OP(an ∨ bn) = γW ∗n +OP(an ∨ bn ∨ cn). (c) Let γ < 0. Then, by using the rates of convergence in Formula (9), we have ( uep(F )−X(n) uep(F )− F−1 (1− e−n) )n−1/2 = exp(γS∗n) +OP(an ∨ bn) = exp(γW ∗n) +OP(an ∨ bn ∨ cn). (d) Suppose that γ = 0 and R(x,G) → 0 as x → uep(G). Suppose that (Ha) and (Hb) hold both. If G. S. Lo et al. / Eur. J. Pure Appl. Math, 14 (1) (2021), 19-42 27 sup {∣∣∣∣s(u) s(v) − 1 ∣∣∣∣ , min(vn, Vn) ≤ u, v ≤ max(vn, Vn) } = OP(dn), and √ ns(vn)−α = O(en), we have X(n) − F−1 ( 1− e−n ) = αS∗n +OP(∨dn ∨ en) = αW ∗n +OP(cn ∨ dn ∨ en). Comments II . In the domain of extremal attraction, most of the cdf ’s which are used in applications are differentiable in a left-neighborhood of the upper endpoint. In such a case, we may take a ≡ 0 in Representation (4) and (5) in Proposition 2. By solving easy differential equations, we have the representation for b(u) = −u(G−1(1− u))′ − γ, u ∈ (0, 1) and a ≡ 0 (10) for γ > 0 and b(u) = u(uep(F )− F−1(1− u)) F ′(F−1(1− u)) , u ∈ (0, 1) (11) for γ < 0, whenever we have b(u)→ 0 as u→ 0. Consequently, the rate of convergence is reduced to OP(bn ∨ cn). For γ = 0, Representation (8) in Proposition 2 holds for s(u) = −u(F−1(1− u))′, 0 < u < 1, whenever it is slowly varying at zero and the rate of convergence dn becomes useless. In such cases, the rate of convergence is reduced to OP(dn ∨ cn). Furthermore, based on the limit Sn/n→ 1 as n→ +∞, we get that have for any η ∈]0, 1[, lim inf n→+∞ P ( e−n/η ≤ e−Sn ≤ e−ηn ) = 1. (12) So we may replace the rates of convergence dn and bn by dn(η) and bn(η) defined as follows, for η ∈]0, 1[, sup{|b(t)|, 0 ≤ t ≤ e−ηn} = O(bn(η)) (13) G. S. Lo et al. / Eur. J. Pure Appl. Math, 14 (1) (2021), 19-42 28 and sup {∣∣∣∣s(u) s(v) − 1 ∣∣∣∣ , e−n/η ≤ u, v ≤ e−ηn} = O(dn(η)).. (14) Specific rates of convergence will be given in the examples below as illustrations ♦ 3. Examples and applications Let us begin to explain how to apply the results for γ = 0. Generally, we may find the function s(u), with u ∈]0, 1[, by the π-variation formula ∀ λ > 0, F−1(1− λu)− F−1(1− u) s(u) → − log λ as u→ 0. Another method concerns the special case where F is differentiable on a left neighborhood of uep(F ). It is proved in [5] that if u ( F−1(1− u) )′ is slowly varying at zero, we have for some u0 ∈]0, 1[, s(u) = −u ( F−1(1− u) )′ for u ∈]0, u0[. Checking hypothesis, (Ha) and (Hb) can be done with the function s(u) with u ∈]0, 1[, as explained above. Here are some specific examples of asymptotic laws and related rates of convergence. The details for each case are given in the Appendix (Section 6). Let us recall that {W (t), t ≥ 1} is a Brownian motion defined on the same probability space as the sequence of records. We begin for light tails : I - F ∈ D(G0). (1) X follows an exponential law E(λ), λ > 0. By Point (b) of Theorem 1, X(n) − n√ n N (0, λ−2). But for a fixed n, we have for a random variable V following a gamma law of parameters n ≥ 1 and 1, X(n) − n√ n ∼ V − E(V ) Var(V )1/2 . G. S. Lo et al. / Eur. J. Pure Appl. Math, 14 (1) (2021), 19-42 29 The rate of convergence is X(n) − n√ n = S∗n = W ∗n +OP(n−1 log n). (2) X follows a standard normal law N (0, 1). By Point (d) of Theorem 1, X(n) − (2n)1/2 N (0, 1/2). The rate of convergence is given by X(n) − (2n)1/2 = S∗n +OP ( (log n)2 n ) = W ∗n +OP ( log n n ) . (3) X follows a Rayleigh law of parameter ρ > 0, with cdf 1− F (x) = exp(−ρx2), x ≥ 0. By Point (d) of Theorem 1, we have X(n) − ( n ρ )1/2 N (0, ρ−1/4). We also have X(n) − ( n ρ )1/2 = ρ−1/2S∗n +OP ( 1√ n ) . (4) X follows the logistic law, with cdf F (x) = 1 1 + e−x , x ∈ R. By Point (b) of Theorem 1, we have X(n) − n√ n N (0, 1). G. S. Lo et al. / Eur. J. Pure Appl. Math, 14 (1) (2021), 19-42 30 The rate of convergence is given, for any η ∈]0, 1[, X(n) − n√ n = S∗n +OP ( e−nη 1− e−n/η ) . (5) X > 0 follows a standard lognormal law, that is logX follows a standard normal law. We have logX(n) − (2n)1/2 N (0, 1/2). The rate of convergence is given by logX(n) − (2n)1/2 = S∗n +OP(n−1(log n)2). (6) X > 0 follows a Gumbel law with cdf F (x) = exp ( −e−x ) , x ∈ R. By Point (b) of Theorem 1, we have X(n) − n√ n N (0, 1). The rate of convergence is given, for any η ∈]0, 1[, by X(n) − n√ n = S∗n +O ( e−ηn ) . II - F ∈ D(Gγ), γ > 0. (7) X follows a log-logistic law of parameter p > 0, with cdf F (x) = xp 1 + xp , x ≥ 0. By Point (a) of Theorem 1, G. S. Lo et al. / Eur. J. Pure Appl. Math, 14 (1) (2021), 19-42 31 ( e−n/pX(n) )−1/2 LN(0, p2). For the rate of convergence, we take η ∈]0, 1[ and bn(η) = e−ηn p ( 1− e−n/η ) . We have ( e−n/pX(n) )−1/2 − exp (S∗n) = OP(bn). (8) X follows a sing-Maddala law of parameters a > 0, b > 0 and c > 0, with cdf 1− F (x) = ( 1 1 + axb )c , x ≥ 0. By Point (a) of theorem 1, we have( a1/b exp(−n/(bc))X(n) )1/√n LN(0, (bc)−2). The rate of convergence is given as follows. Let η ∈]0, 1[, and bn(η) = e−ηn/c b ( 1− e−n/(cη) ) . We have ( a1/b exp(−n/(bc))X(n) )1/√n = exp(S∗n) +OP(bn(η)). 4. Proofs (I) - Proof of Theorem 1. We begin by describing the main tools which are based on following results of record theory. Suppose that {T, Tj > 0, 1 ≤ j ≤ k} are (k + 1) non-negative real-valued, iid random variables and define X0 = 0, Tj = Xj −Xj−1, 1 ≤ j ≤ k. G. S. Lo et al. / Eur. J. Pure Appl. Math, 14 (1) (2021), 19-42 32 It is clear that if T ∼ E(λ), λ > 0, then the absolutely continuous pdf of T = (T1, · · · , Tk)t is given by fT (t1, · · · , tk) = λke−λtk 1(0≤t1≤···≤tk). (15) Suppose if Tj ’s are independent and follow an exponential law E(λ), λ > 0, we have r(x) = dF (x)/dx 1− F (x) = λ and f(x) = λe−λx, x ≥ 0. As stated in page 3 of [7], the joint distribution of the k first record values (T (1), · · · , T (k)) of the sequence (Tn)n≥1 is the one given in Formula (15). As a consequence, we have Fact 1. If the Tj’s are independent and follow an exponential law E(λ), then the k-th record value, k ≥ 1, has the same law as the sum of k independent E(λ)-random variables E1,k, · · · , Ek,k, i.e. T (k) =d E1,k + · · ·+ Ek,k, where =d stands for the equality in distribution. By the Renyi’s representation, we can represente the random variable X of cdf F by a standard exponential random variable E X =d F −1 (1− e−E) . It comes that, by considering iid sequence (Xn)n≥1 and (En)n≥1 from X and E, and by denoting the two n-th record values X(n) and E(n) from the two sequences respectively, we have the following representations X(n) =d F −1 ( 1− e−E(n) ) , where E(n) = E1,n + · · ·+ En,n = S(n). In the sequel, we can and do use the equality : X(n) = F−1 ( 1− e−S(n) ) . Let us apply the representations by using the simple central limit theorem S(n) − n√ n N (0, 1) as n→ +∞. In the sequel, any unspecified limit is meant as n→ +∞. G. S. Lo et al. / Eur. J. Pure Appl. Math, 14 (1) (2021), 19-42 33 Let us suppose that X ∈ D(G1/γ). If X ≥ 0, we will consider Y = logX of cdf G defined by G(x) = F (ex), x ∈ R. Let us prove the theorem 1. (a) - Asymptotic law of X(n) for γ > 0. We recall that Vn = e−S(n) and vn = e−n, n ≥ 1. By Representation (4), we have F−1 ( 1− e−S(n) ) = (1 + a(Vn))V −γn exp (∫ 1 Vn b(t) t dt ) , n ≥ 1 and F−1 ( 1− e−n ) = (1 + a(vn))v−γn exp (∫ 1 vn b(t) t dt ) , n ≥ 1. We get that Vn →P 0, (1 + a(Vn))/(1 + a(vn)) ≡ 1 + pn →P 1. We get log ( X(n) F−1 (1− e−n) ) = pn(1 + oP(1))− γ(S(n) − n) + ∫ Vn vn b(t) t dt. (16) We have ∣∣∣∣∫ 1 vn b(t) t dt ∣∣∣∣ ≤( sup 0≤t≤(vn∨Vn) |b(t)| ) |S(n) − n|. (17) By combining the two last formulae, we have n−1/2 log ( X(n) F−1 (1− e−n) ) N (0, γ2). (b) - Asymptotic law of Y (n) for γ > 0. From the previous theorem, it is immediate for the following result. It is clear that G−1 = logF−1. So, the previous theorem implies n−1/2 ( Y (n) −G−1 ( 1− e−n )) N (0, γ2). Here, it is clear that Y ∈ D(G0) and R(x,G)→ γ as x→ uep(G). Hence this result says that n−1/2 ( X(n) − F−1 ( 1− e−n )) N (0, γ2), if F ∈ D(G0) and R(x, F )→ γ as x→ uep(F ). G. S. Lo et al. / Eur. J. Pure Appl. Math, 14 (1) (2021), 19-42 34 (c) - Asymptotic law of Y (n) for γ < 0. We have P(X = uep(F )) = 0. By using Representation (5), we may and do prove this point exactly as for Point (a). (d) - Asymptotic law of Y (n) for γ = 0. We did not have yet the general law. Let us learn for a no-trivial example. (A) - X ∼ N (0, 1). Let us recall the expansion of the tail of F as follows F−1(1− s) = (2 log(1/s))1/2 − log 4π + log log(1/s) 2(2 log(1/s))1/2 (18) + O((log log(1/s))2(log 1/s)−1/2). We get X(n) = (2S(n)) 1/2 − log 4π + logS(n) 2(2S(n)))1/2 +O(S −1/2 (n) logS(n)) = (2S(n)) 1/2 − log 4π + logS(n) 2(2S(n))1/2 +OP(n−1(log n)2) (19) = (2S(n)) 1/2 +OP(n−1/2 log n). (20) Furthermore F−1(1− e−n) = (2n)1/2 − log 4π + log n 2(2n)1/2 +OP(n−1(log n)2) (21) = (2n)1/2 +OP(n−1/2 log n). (22) Combining relations (20) and (22) leads to X(n) − F−1(1− e−n) = 1√ 2 S(n) − n√ n (n/ζn)1/2 +OP(n−1(log n)2), (23) with n∧S(n) < ζn < n∨S(n) and next, by the weak law of large numbers, (n/ζn)1/2 →P 1 and thus X(n) − (2n)1/2 N (0, 1/2). (24) (B) - General proof. It is known that s(u) ∼ R(F−1(1 − u), F ) and so, s(u) → 0 as u→ 0. By Representation (6) of Proposition 2 and Hypothesis (Ha) together lead to G. S. Lo et al. / Eur. J. Pure Appl. Math, 14 (1) (2021), 19-42 35 X(n) − F−1(1− en) = s(Vn)− s(vn) + ∫ Vn vn s(u) u du (25) = s(vn) ( s(Vn) s(vn) − 1 ) + (1 + oP(dn))s(vn)(S(n) − n). = {s(vn) √ n} ( 1√ n ( s(Vn) s(vn) − 1 )) + {s(vn) √ n}(1 +OP(dn)) S∗n = (α+OP(en)) ( 1√ n ( s(Vn) s(vn) − 1 )) + (α+OP(en))(1 +OP(dn))S∗n = αS∗n +OP(en ∨ dn) = αW ∗n +OP(cn ∨ en ∨ dn). (26) From there, the conclusion is immediate by using Hypothesis (Hb) � (II) - Proof of Theorem 2. We have g(x) = F−1(1 − e−x) , g′(x) = S(x), x ∈ ]lep(F ), uep(F )[. The mean value theorem gives, for ∀n > 0, X(n) − F−1(1− e−n) = S(n) − n√ n (√ nS(exp(−ζn)) ) , (27) where ζn ∈] min(n, S(n)), max(n, S(n))[. From there, the conclusion is direct � Proof of Theorem 3. We will prove that theorem in a special space but it will be valid in any probability space. Following [4], we consider the probability space (Ω,A,P) holding a sequence (En)n≥1 and a Wiener process W such that |(Sn − n)−W (n)|√ n = O ( log n√ n ) . (28) We set Un = 1 − exp(−En), n ≥ 1 and finally Xn = F−1(1 − exp(−En)), n ≥ 1. From that point, all the notations above remain valid. The proofs of the different points of the theorems derive easily from the proof of the same points in Theorem 1. Here are the details. G. S. Lo et al. / Eur. J. Pure Appl. Math, 14 (1) (2021), 19-42 36 Proof of Points (a) and (b). By combining Formulas (16) and (17) with Formula (28) approximation, we easily isolate a multiple of S∗n or of W ∗n and find the rates of conver- gence as stated for Point (b). Point (a) is obtained by an exponential transformation. Proof of Point (c). This is proved exactly as Point (a). Proof of Point (d). From Formulas (25) and (28), we simply use the rates of convergence in Hypothesis (Ha) and (Hb) to conclude � 5. Conclusion After the statements of the asymptotic laws of the strong record values from iid random variables and their rates of convergences, and after some examples have been given, it should be interesting to have a review of such asymptotic laws for cdf ’s as much as possible, F ∈ D. G. S. Lo et al. / Eur. J. Pure Appl. Math, 14 (1) (2021), 19-42 37 6. Appendix Appendix. Let us give the details concerning the results listed in Section 3. (1) X follows an exponential law E(λ), λ > 0. We have exp(X) ∈ D(G−1) and F−1(1− e−n) = n. We apply Point (b) to conclude that X(n) − n√ n N (0, λ−2). The rate of convergence is the one in the approximation between S∗n and W ∗n which is cn. (2) X follows a standard normal law N (0, 1). The result of this point is justified by Formula 24, page 34. To find the rate of convergence, we proceed to a direct proof based on Formulas (19) and (21). We get X(n) − F−1(1− e−n) = (2S(n)) 1/2(2n)1/2 (L1) − ( log 4π + logS(n) 2(2S(n)))1/2 − log 4π + logS(n) 2(2S(n))1/2 ) (L2) + OP(n−(log n)2). (L3) By using the result of the application of the mean value theorem in Formula (23) (page 34) in Line (L1), by applying the mean value theorem in Line (L2) above and by using the fact (n/ζn)→ 1, we get X(n) − F−1(1− e−n) = 1√ 2 S(n) − n√ n + 1√ 2 S(n) − n√ n ( (n/ζn)1/2 − 1 ) − OP(n−1(log n)) + OP(n−1/2 log n). Finally, by using (n/ζn)−1 = OP(n−1(log n)) and (Sn−n)/ √ n = OP(1), we conclude that X(n) − F−1(1− e−n) = S∗n +OP(n−1/2 log n), which was the target. (3) X follows a Rayleigh law of parameter ρ > 0. We have G. S. Lo et al. / Eur. J. Pure Appl. Math, 14 (1) (2021), 19-42 38 F−1(1− u) = ( −1 ρ log u )1/2 , u ∈]0, 1[ and s(u) = −u ( F−1(1− u) )′ = 1 2ρ(−(1/ρ) log u)1/2 → 0 as u→ 0. Furthermore, s(u) is decreasing in u ∈]0, 1[ and s(Vn)/s(vn)→ 0 as n→ +∞. Finally, √ ns(vn)→ ρ−1/2/2. We conclude the case by applying Point (d) of Theorem 1. For finding the rate of convergence, we have en = 0 since √ n s(vn) = (2 √ ρ) = α. The rate of convergence corresponding to dn is obtained by remarking that s(◦) is decreas- ing to zero and so sup {∣∣∣u v − 1 ∣∣∣ , min(vn, Vn) ≤ u, v ≤ max(vn, Vn) } ≤ ∣∣∣∣s(e−n ∨ e−Sn) s(e−n ∧ e−Sn) − 1 ∣∣∣∣ ∣∣∣∣s(e−n ∨ e−Sn) s(e−n ∧ e−Sn) − 1 ∣∣∣∣ ≤ ∣∣∣∣∣ ( Sn n )1/2 − 1 ∣∣∣∣∣+ ∣∣∣∣∣ ( Sn n )−1/2 − 1 ∣∣∣∣∣ = OR ( 1√ n ) . This closes the discussions on the rate of convergence. (4) X follows the logistic law. It is immediate that exp(X) ∈ D(G−1) and we have F−1(1− u) = log(u/(1− u)), u ∈]0, 1[. We conclude with Point (b) of Theorem 1. The rate of convergence can be found from that Z = exp(X) of cdf H(t) = t(1 + t)−1, t > 0. G. S. Lo et al. / Eur. J. Pure Appl. Math, 14 (1) (2021), 19-42 39 We have that H ∈ G1 and, for γ = 1, −u(logH−1(1− u))′ − γ = u(1− u)−1. By applying Point (1) of Theorem 3, we take for any η ∈]0, 1[ and bn(η) = e−ηn p ( 1− e−n/η ) and get ( Z(n) n− log(1− e−n) )1/ √ n = exp(S∗n) +OP(bn). (5) X > 0 follows a standard lognormal law, that is logX follows a standard normal law. Since logX(n) has the same law as the n-th record Z(n) from iid N (0, 1) random variables. So we have logX(n) − (2n)1/2 → N (0, 1/2). The conclusion is done by taking the logarithm of both members. We still can use the rate of convergence from normal records to have: logX(n) − (2n)1/2 = S∗n +OP(n−1(log n)2). (6) X > 0 follows a Gumbel law. We have F−1(1− u) = − log log(1/(1− u)), u ∈]0, 1[ and for any λ > 0, F−1(1− λu)− F−1(1− u)→ − log λ as u→ 0. So, exp(X) ∈ D(G1). From there, an application of Point (b) of Theorem 1 closes the case. The rate of convergence can be found from that of Z = exp(X) of cdf G. S. Lo et al. / Eur. J. Pure Appl. Math, 14 (1) (2021), 19-42 40 H(t) = exp(−1/t), t > 0. We have that H ∈ G1 and, for γ = 1, −u(logH−1(1− u))′ = −u [ log 1 − log(1− u) ]′ = −u (1− u) log(1− u) = u (1− u)(u− u2/2 +O(u3)) = 1 (1− 3u/2 +O(u2)) . Hence −u(logH−1(1− u))′ − γ = 3u/2(1 + o(1)). By applying Point (1) of Theorem 3, we take for any η ∈]0, 1[ and bn(η) = 1.5e−ηn and get ( Z(n) n− log(1− e−n) )1/ √ n = exp(S∗n) +OP(bn). The conclusion is done by taking the logarithm of both members. (7) X follows a log-logistic law of parameter p > 0, with cdf F (x) = xp 1 + xp , x ≥ 0. We have F ∈ D(G1/p) since F−1(1− u) = u−1/p(1− u)1/p, u ∈]0, 1[. By Point (a) of Theorem 1, G. S. Lo et al. / Eur. J. Pure Appl. Math, 14 (1) (2021), 19-42 41 ( e−n/pX(n) )1/√n LN(0, p2). To find the rate of convergence, we apply the recommendations in Comments II (page 27). We have b(u) = −u(G−1(1− u))′ − (1/p) = u p(1− u) , u ∈]0, 1[. For any 0 < η < 1, we get the rate of convergence bn(η) = e−ηn p ( 1− e−n/η ) . (8) X follows a sing-Maddala law of parameters a > 0, b > 0 and c > 0. We have 1− F (x) = x−bc(x−b + a)−c ≡ x−bcL(x), x ≥ 0, and L is a slowly varying function at +∞. So F ∈ G1/(bc). Applying the Point (a) of Theorem 1, when combined with F−1(1− u) = a−1/bu−1/(bc)(1− u1/c)1/b, u ∈]0, 1[, and with, F−1(1− e−n) = a−1/ben/(bc)(1− e−n/c)1/b, for n ≥ 1. To find the rate of convergence, we check that we have γ = 1/(bc) and b(u) = −u(G−1(1− u))′ − (1/(bc)) = u1/c b(1− u1/c) , u ∈]0, 1[. For any 0 < η < 1, we get the rate of convergence bn(η) = e−ηn/c b ( 1− e−n/(cη) )� REFERENCES 42 References [1] L. de Haan. On regular variation and its application to the weak convergence of sample extremes. Mathematical Center Tracts,Amsterdam. (MR0286156), 1970. [2] J. Galambos. The Asymptotic theory of Extreme Order Statistics. Wiley, Nex-York. (MR0489334), 1985. [3] Karamata J. Some theorems concerning slowly varying. Mathematics Research Cen- ter, Tech. Rep.,no 369. University of Winconsin, Madison., 1962. [4] Komlós J., Major P., and Tusnády. 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