EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 15, No. 2, 2022, 511-527 ISSN 1307-5543 – ejpam.com Published by New York Business Global Extensions of two classical Poisson limit laws to non-stationary independent sequences Aladji Babacar Niang1, Harouna Sangaré2, Tchilabalo Abozou Kpanzou3,∗, Gane Samb Lo4, Nafy Ngom1 1LERSTAD, Gaston Berger University, Saint-Louis, Sénégal. Imhotep Mathematical Center (IMC), 2DER MI, FST, Université des Sciences, des Techniques et des Technologies de Bamako (USTT-B), Mali. Affiliated to LERSTAD, Gaston Berger University, Saint-Louis, Sénégal 3 University of Kara, Kara, Togo. Affiliated to LERSTAD, Gaston Berger University, Saint- Louis, Sénégal 4 LERSTAD, Gaston Berger University, Saint-Louis, Sénégal (main affiliation) Department of Pure and Applied Mathematics, African university of Science and Technology, Abuja, Nigeria LSTA, Pierre and Marie Curie University, Paris VI, France (associated researcher) Abstract. In earlier stages in the introduction to asymptotic methods in probability theory, the weak convergence of sequences (Xn)n≥1 of binomial random variables (rv’s) to a Poisson law is classical and easy to prove. A version of such a result concerning sequences (Yn)n≥1 of negative binomial rv’s also exists. In both cases, Xn and Yn−n are by-row sums Sn[X] and Sn[Y ] of arrays of Bernoulli rv’s and corrected geometric rv’s respectively. When considered in the general frame of asymptotic theorems of by-row sums of rv’s of arrays, these two simple results in the independent and identically distributed scheme can be generalized to non-stationary data and beyond to non- stationary and dependent data. Further generalizations give interesting results that would not be found by direct methods. In this paper, we focus on generalizations to the non-stationary independent data in the frame of the central limit theorem for independent random variables. 2020 Mathematics Subject Classifications: 60B10, 60F05, 60G70, 62G30 Key Words and Phrases: Summands of independent and square integrable random variables; weak convergence of arrays; Poisson limits, binomial and negative binomial laws; Bernoulli and corrected Geometric laws; non-stationary. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v15i2.4323 Email addresses: niang.aladji-babacar@ugb.edu.sn (AB Niang), harounasangare@fst-usttb-edu.ml (H Sangaré), t.kpanzou@univkara.net (TA Kpanzou), gane-samb.lo@ugb.edu.sn (GS Lo), fany.ngom@ugb.edu.sn (N Ngom) https://www.ejpam.com 511 © 2022 EJPAM All rights reserved. AB Niang et al. / Eur. J. Pure Appl. Math, 15 (2) (2022), 511-527 512 1. Introduction 1.1. Preliminaries The approximation of a sequence of binomial probability laws (B(n, pn))n≥1 associated to a sequence of r.v’s (Zn)n≥1 [such that the sequence of probabilities (pn)n≥1 converges to zero and npn → λ > 0 as n→ +∞] to a Poisson law P(λ) is a classical and easy-to-prove result in probability theory. This approximation has very important applications in real- life problems, especially in lack of powerful computers. In this simple case, each Zn is a sum of n independent and identically distributed (iid) Bernoulli B(pn)-random variables {(Xj,n)1≤j≤n, n ≥ 1}, i.e., Zn = X1,n + X2,n + · · · + Xn,n. When we depart from the identical distributivity assumption, the problem may get more and rapidly complex, even if the independence assumption is still required. The situation becomes more interesting if the random variables Xj,n are non-stationary and independent. For the definition and more details on the Binomial and Poisson laws, see [4]. In this paper, we aim at giving non trivial generalizations of such results in the frame of the central limit theorem for independent random variables. Also, there is a negative version of the described result. Indeed, if we call a binomial law as a positive binomial law PB(n, p), n ≥ 1, 0 < p < 1 in opposition to a negative binomial law NB(n, p), we have the following two results concerning positive binomial and negative binomial laws respectively. Let us make this precision for once: throughout this paper, all limits are meant as n→ +∞ unless the contrary is specified. Proposition 1. Let (Xn)n≥1 be a sequence of random variables in some probability space (Ω,A,P) such that: 1) ∀n ≥ 1, Xn ∼ B(n, pn), n ≥ 1; 2) pn → 0 and npn → λ ∈ R+ \ {0} as n→ +∞. Then Xn P(λ). Next, we have: Proposition 2. Let (Xn)n≥1 be a sequence of random variables in some probability space (Ω,A,P) such that: AB Niang et al. / Eur. J. Pure Appl. Math, 15 (2) (2022), 511-527 513 1) ∀n ≥ 1, Xn ∼ NB(n, pn), n ≥ 1; 2) (1− pn) → 0 and n(1− pn) → λ ∈ R+ \ {0} as n→ +∞. Then Xn − n P(λ). Remark. These two results are proved in [5]. Although the proofs are direct, we do not encounter the second in some classical books as [1, 2], [3], [6]. For that reason, we give it below. Proof of Proposition 2. Let us use the convergence of characteristic functions. Let Xn be a sequence of NB(n, pn)- random variables and X be a P(λ) random variable. We have Sn = Xn − n = Z1 + · · ·+ Zn, where Z, Z1, · · · , Zn are independent random variables such that each Zi + 1 follows a geometric law of parameter pn, that is ΦZ(t) = pn 1− qneit , t ∈ R. So, for t ∈ R fixed, ΦSn(t) = exp ( n ( log pn − log ( 1− qne it ))) . We have, as n→ +∞, n log pn = n log(1− qn) = −nqn + o(nqn) and −n log ( 1− qne it ) = nqne it + o(nqn). Hence we get for any t ∈ R, ΦSn(t) = exp ( nqn(e it − 1) + o(nqn) ) → eλ(e it−1) = ΦP(λ)(t). � AB Niang et al. / Eur. J. Pure Appl. Math, 15 (2) (2022), 511-527 514 Our aim here is to provide non-trivial generalizations of such simple results to non- stationary and independent data. The used methods will later allow further generalizations even with dependent data. Let us prepare generalizations by transforming both results as sums of random variables. 1.2. The Central limit theorem frame It is known that a binomial random variable Xn ∼ B(n, pn) has the same law as a sum of n iid Bernoulli distributed random variables: Xn =d X1,n + · · ·+Xn,n, where X1,n, · · · , Xn,n are independent and follow all the B(pn)-law. Also Xn ∼ NB(n, pn) has the same law as a sum of n iid r.v’s: Xn =d X∗ 1,n + · · ·+X∗ n,n, where X∗ 1,n, · · · , X∗ n,n are independent and each X∗ j,n follows the geometric law G(pn). In the second, we rather use Xn − n = n∑ i=1 ( X∗ i,n − 1 ) =: n∑ i=1 Xi,n, where the Xi,n’s are independent and each Xi,n follows the law G∗(pn) = G(pn) − 1, and such a law is called a corrected geometric law, for convenience. In both cases, we have to study an array X ≡ { {Xk,n, 1 ≤ k ≤ k(n)}, n ≥ 1 } , of random variables defined in the same probability space (Ω,A,P) such that here: 1) ∀n ≥ 1, k(n) = n; 2) ∀n ≥ 1, the variables X1,n, · · · , Xk(n),n are independent; 3) The sequence X1,n, · · · , Xk(n),n is stationary for n ≥ 1; 4) ∀k ∈ [1, k(n)], Xk,n ∼ B(pn) or Xk,n ∼ G(pn)− 1. AB Niang et al. / Eur. J. Pure Appl. Math, 15 (2) (2022), 511-527 515 We see that we are in the CLT frame and each of points (2) and (3) can be changed to lead to generalizations. Since, we want to generalize the limiting binomial laws and negative binomial laws, we keep the same hypotheses on the marginal laws ∀k ∈ [1, k(n)], Xk,n ∼ B(pk,n) or Xk,n ∼ G(pk,n)− 1 and try to answer to the questions (Q1) and (Q2) below: (Q1) Given an array X of random variables with k(n) → +∞ such that the elements of each row are independent and B(pk,n)-r.v’s, do we still have Sn[X] = k(n)∑ k=1 Xk,n P(λ), (1) when some of the assumptions (1), (2) and (3) [but mainly (2) and (3)] are violated, and under what sufficient conditions this should hold? (Q2) Given an array X of random variables with k(n) → +∞ such that the elements of each row are independent and G∗(pk,n)-r.v’s, do we still have (1) when some of the as- sumptions (1), (2) and (3) [but mainly (2) and (3)] are violated, and under what sufficient conditions this should hold? Although direct handlings of these questions might be possible, we think that a general and extensible solution resides in the CLT frame, since it will prepare further generaliza- tions for dependent data. Therefore, we organize the paper as follows. In Section 2, we recall the frame of the CLT problem as stated in [6]. In Section 3, we state and prove the results. We conclude the paper by conclusive remarks in Section 4. 2. Notation and G-CLT for summands of independent random variables Let us consider the array X ≡ { {Xk,n, 1 ≤ k ≤ kn = k(n)}, n ≥ 1 } , of square integrable random variables defined on the same probability space (Ω,A,P). We denote Fk,n as the cumulative distribution function (cdf ) of Xk,n. We also denote by ak,n = E(Xk,n) and σ2k,n = Var(Xk,n), 1 ≤ k ≤ k(n), if these expectations or variances exist. We also suppose that AB Niang et al. / Eur. J. Pure Appl. Math, 15 (2) (2022), 511-527 516 k(n) → +∞ as n→ +∞. The central limit theorem problem consists in finding, whenever possible, the weak limit law (in type) of the by-row sums of the array X, i.e. the summands: Sn[X] = k(n)∑ k=1 Xk,n, n ≥ 1. Historically, the CLT was discovered with the convergence of a binomial law (which has the same law as a sum of iid Bernoulli random variables) to the standard Gaussian law (due to Laplace, De Moivre, etc., around 1731, see [6] for a review). For a long period, the Gaussian limit was automatically meant in the CLT problem. Many authors, among them Lévy, Gnedenko, Kolmogorov, etc., characterized the class of possible limit laws under the Uniform Asymptotic Negligibility (UAN) condition, exactly as the class of in- finitely decomposable distributions. The longtime association of CLT ’s with Gaussian limits explains that some authors reserve the vocable CLT for Gaussian limits and for other possible limits, they use different vocables. Here we use the vocable of G-CLT to cover all possible limit laws G beyond the Gaussian law. Here we suppose that the Xk,n’s are integrable with finite variances. For an array X, we define some important hypotheses used in the formulation of the CLT problem. (1) The UAN condition: for any ε > 0, U(n, ε,X) = sup 1≤k≤kn P(|Xk,n − ak,n| ≥ ε) → 0. (2) (2) The Bounded Variance Hypothesis (BVH): there exists a constant c > 0, such that sup n≥1 MV (n,X) ≤ c, where MV (n,X) = Var(Sn[X]), n ≥ 1. (3) The Variance Convergence Hypothesis (VCH): MV (n,X) → c ∈]0,+∞[. AB Niang et al. / Eur. J. Pure Appl. Math, 15 (2) (2022), 511-527 517 According to the state of the art in CLT ’s theory for centered, square integrable and in- dependent by-row arrays of random variables, the summands weakly converge to a proba- bility law associated to the cdf G and to the characteristic function (cha.f ) ψG under the UAN condition and the BVH if and only if the sequence of distribution functions (df ) Kn(x) = k(n)∑ k=1 ∫ x −∞ y2dFk,n(y), x ∈ R, n ≥ 1, pre-weakly converges to a df K, denoted Kn pre K, that is for any continuity point x of K denoted as [x ∈ C(K)], we have Kn(x) → K(x), and the cha.f ψG(◦) of G is given by exp(ψ[K](◦)) with ∀u ∈ R, ψ[K](u) = ∫ eiux − 1− iux x2 dK(x). If we have the VCH, the convergence criterion is replaced by the weak convergence Kn K. Moreover, the limit law G is necessarily an infinitely decomposable law. In the non centered case, with the same hypotheses above on the random variables of the array, the summands weakly converge to a probability law associated to the cdf G∗ and to the cha.f ψG∗ under the UAN condition and the BVH if and only if k(n)∑ k=1 ak,n → a, a ∈ R and the sequence of distribution functions (df ) K∗ n(x) = k(n)∑ k=1 ∫ x −∞ y2dFk,n(y + ak,n), x ∈ R, n ≥ 1, pre-weakly converges to a df K∗ and the cha.f ψG∗(◦) of G∗ is given by exp(ψ[K∗](◦)) with ∀u ∈ R, ψ[K∗](u) = ∫ eiux − 1− iux x2 dK∗(x). AB Niang et al. / Eur. J. Pure Appl. Math, 15 (2) (2022), 511-527 518 If we have the VCH, the convergence criterion is replaced by the weak convergence K∗ n K∗. Moreover, the limit law G∗ is of the form G∗ = G + a, with G is neces- sarily a centered and infinitely decomposable law. By specializing the limit law as a Gaussian law or a Poisson law, which clearly are infinitely decomposable laws, we have the following characterizations. C1. Under the conditions ( ∀n ≥ 1, ∀1 ≤ k ≤ k(n), ak,n = 0 ) and k(n)∑ k=1 σ2k,n = 1, the summands Sn[X] of the arrayX converges to standard Gaussian law and max1≤k≤k(n) σ 2 k,n → 0 if and only if the following Lynderberg-Gaussian condition holds: ∀ε > 0, Ln,G(ε) = k(n)∑ k=1 ∫ (|x|≥ε) x2 dFk,n(x) → 0. (3) C2. Under the conditions max 1≤k≤k(n) σ2k,n → 0 and k(n)∑ k=1 σ2k,n → λ, λ > 0, the summands Sn[X] of the array X converges to a translated Poisson law P(a, λ) ≡ a+ P(λ), a ∈ R, if and only if k(n)∑ k=1 ak,n → a+ λ and the following Lynderberg Poisson-type condition holds: ∀ε > 0, Ln,P (ε) = k(n)∑ k=1 ∫ (|x−1|≥ε) x2 dFk,n(x+ ak,n) → 0. (4) 3. Statements of the results As announced, we focus here on the non-stationary independent scheme. AB Niang et al. / Eur. J. Pure Appl. Math, 15 (2) (2022), 511-527 519 First, we consider uniform conditions of the convergence of the probabilities pk,n (in the Bernoulli case) and qk,n (in the corrected geometric case) to zero to unveil refined versions of the extensions. Later, we will provide more general conditions. Theorem 1. Let X = { {Xk,n, 1 ≤ k ≤ kn = k(n)}, n ≥ 1 } , be an array of by-row independent Bernoulli random variables, that is: (1) ∀n ≥ 1, ∀1 ≤ k ≤ k(n), Xk,n ∼ B(pk,n), with 0 < pk,n < 1 and: (2) sup1≤k≤k(n) pk,n → 0; (3) ∑ 1≤k≤k(n) pk,n → λ ∈]0, +∞[. Then we have Sn[X] P(λ). Proof of Theorem 1. Throughout this proof, the notation `k,n = on(1), for k ranging over some set In means that the sequence `k,n goes to zero as n → +∞ uniformly in k ∈ In. So Assumption (2) means that pk,n = on(1) and qk,n = 1− pk,n = 1 + on(1). We have to check the UAN condition. By using Chebychev’s inequality, we have, for any ε > 0, U(n, ε,X) = sup 1≤k≤kn P(|Xk,n − ak,n| ≥ ε) ≤ ε−2 sup 1≤k≤kn Var(Xk,n) = ε−2 pk,n qk,n = ε−2 on(1)(1 + on(1)) → 0. The VCH also holds since MV (n,X) = ∑ 1≤k≤k(n) Var(Xk,n) AB Niang et al. / Eur. J. Pure Appl. Math, 15 (2) (2022), 511-527 520 = ∑ 1≤k≤k(n) pk,n qk,n = (1 + on(1)) ∑ 1≤k≤k(n) pk,n → λ. Besides ∑ 1≤k≤k(n) E(Xk,n) = ∑ 1≤k≤k(n) pk,n → λ. So, we are in the position of applying the conditions of weak convergence to a Poisson law by checking the Poisson Lynderbeg condition (4). We have for any ε > 0 Ln,P (ε) =: k(n)∑ k=1 Ln,k,P (ε), with Ln,k,P (ε) = ∫ (|x−1|≥ε) x2 dFk,n(x+ pk,n) = ∫ (|Xk,n−pk,n−1|≥ε) |Xk,n − pk,n|2 dPk,n = pk,n ( 1(|Xk,n−pk,n−1|≥ε)|Xk,n − pk,n|2 ) (Xk,n=1) + (1− pk,n) ( 1(|Xk,n−pk,n−1|≥ε)|Xk,n − pk,n|2 ) (Xk,n=0) = pk,n1(|pk,n|≥ε) (1− pk,n) 2 + (1− pk,n)1(|pk,n+1|≥ε) p 2 k,n = pk,n1(|on(1)|≥ε)(1 + on(1)) 2 + on(1)(1 + on(1))1(|on(1)+1|≥ε) pk,n. We only need to get (4) for 0 < ε < ε0, for a fixed ε0 > 0. Let us fix ε0 = 1/2. So, for n large enough, 1(|on(1)|≥ε) = 0 and 1(|on(1)+1|≥ε) = 1 and hence k(n)∑ k=1 Ln,k,P (ε) = on(1)(1 + on(1)) k(n)∑ k=1 pk,n AB Niang et al. / Eur. J. Pure Appl. Math, 15 (2) (2022), 511-527 521 = on(1)(1 + on(1))(λ+ o(1)) → 0. The proof is complete. � Theorem 2. Let X = { {Xk,n, 1 ≤ k ≤ kn = k(n)}, n ≥ 1 } , be an array of by-row-independent corrected geometric random variables, that is: (1) ∀n ≥ 1, ∀1 ≤ k ≤ k(n), Xk,n ∼ G∗(pk,n), with 0 < pk,n = 1− qk,n < 1 and: (2) sup1≤k≤k(n) qk,n → 0; (3) ∑ 1≤k≤k(n) qk,n → λ ∈]0, +∞[. Then we have Sn[X] P(λ). Proof of Theorem 2. Assumption (2) of the theorem means that qk,n = on(1), pk,n = 1 + on(1) and (1/pk,n) i = 1 + on(1), i = 1, 2, 3. We have to check the UAN condition. By using Chebychev’s inequality, we have, for any ε > 0, U(n, ε,X) = sup 1≤k≤kn P(|Xk,n − ak,n| ≥ ε) ≤ ε−2 sup 1≤k≤kn Var(Xk,n) = ε−2 sup 1≤k≤kn qk,n p2k,n = ε−2on(1)(1 + on(1)) → 0. The VCH also holds since MV (n,X) = ∑ 1≤k≤k(n) Var(Xk,n) AB Niang et al. / Eur. J. Pure Appl. Math, 15 (2) (2022), 511-527 522 = ∑ 1≤k≤k(n) qk,n p2k,n = (1 + on(1)) ∑ 1≤k≤k(n) qk,n → λ. Besides ∑ 1≤k≤k(n) E(Xk,n) = ∑ 1≤k≤k(n) qk,n pk,n = (1 + on(1)) ∑ 1≤k≤k(n) qk,n → λ. Here again, we are in the position of applying the conditions of weak convergence to a Poisson law by checking the Poisson Lynderbeg condition (4). We have for any 0 < ε < 1/2 Ln,P (ε) =: k(n)∑ k=1 Ln,k,P (ε), with Ln,k,P (ε) = ∫ (|x−1|≥ε) x2 dFk,n(x+ ak,n) = ∫(∣∣∣∣Xk,n− qk,n pk,n −1 ∣∣∣∣≥ε ) ∣∣∣∣Xk,n − qk,n pk,n ∣∣∣∣2 dPXk,n = +∞∑ j=0 pk,nq j k,n ( 1(∣∣∣∣Xk,n− qk,n pk,n −1 ∣∣∣∣≥ε ) ∣∣∣∣Xk,n − qk,n pk,n ∣∣∣∣2) (Xk,n=j) = pk,n1(∣∣∣∣ qk,npk,n +1 ∣∣∣∣≥ε )( qk,n pk,n )2 (for j = 0) + pk,nqk,n1(∣∣∣∣ qk,npk,n ∣∣∣∣≥ε )( 1− qk,n pk,n )2 (for j = 1) + +∞∑ j=2 pk,nq j k,n1 (∣∣∣∣j−1− qk,n pk,n ∣∣∣∣≥ε )( j − qk,n pk,n )2 (for j ≥ 2). By the same remarks used in the precedent proof, we have for n large enough, 1(∣∣∣∣ qk,npk,n +1 ∣∣∣∣≥ε ) = 1(|on(1)(1+on(1))+1|≥ε) = 1 AB Niang et al. / Eur. J. Pure Appl. Math, 15 (2) (2022), 511-527 523 and next pk,n1(∣∣∣∣ qk,npk,n +1 ∣∣∣∣≥ε ) ∣∣∣∣ qk,npk,n ∣∣∣∣2 = (1 + on(1))q 2 k,n = on(1)(1 + on(1)) qk,n. (L1) Also, we have for n large enough, 1(∣∣∣∣ qk,npk,n ∣∣∣∣≥ε ) = 1(|on(1)(1+on(1))|≥ε) = 0 and next pk,nqk,n1(∣∣∣∣ qk,npk,n ∣∣∣∣≥ε )( 1− qk,n pk,n )2 = 0. (L2) Now, for n large enough and for any j ≥ 2, 1(|j−1+on(1)(1+on(1))|≥ε) = 1 and thus, Aj,k,n := pk,nq j k,n1 (∣∣∣∣j−1− qk,n pk,n ∣∣∣∣≥ε )( j − qk,n pk,n )2 = ( (1 + on(1))qk,n ) qj−1 k,n 1(∣∣∣∣j−1+on(1)(1+on(1)) ∣∣∣∣≥ε )( j + on(1)(1 + on(1)) )2 = ( (1 + on(1))qk,n ) qj−1 k,n (j + on(1)(1 + on(1))) 2 ≤ ( (1 + on(1))qk,n ) qj−1 k,n (j + 1)2 ≤ 2 ( (1 + on(1))qk,n ) qj−1 k,n (j2 + 1), where we apply the C2-inequality in the last line. So we have ∑ j≥2 Aj,k,n ≤ 2(1 + on(1)) qk,n B(n, k), (L3) with AB Niang et al. / Eur. J. Pure Appl. Math, 15 (2) (2022), 511-527 524 B(n, k) = ∑ j≥2 qj−1 k,n + ∑ j≥2 j2qj−1 k,n =: B(n, k, 1) +B(n, k, 2). We have B(n, k, 1) = (∑ j≥0 qjk,n ) − 1 = qk,n pk,n = on(1)(1 + on(1)). (L4a) Next B(n, k, 2) = ∑ j≥2 jqj−1 k,n + ∑ j≥2 j(j − 1)qj−1 k,n = ( +∞∑ j=1 jqj−1 k,n − 1 ) + ( qk,n +∞∑ j=2 j(j − 1)qj−2 k,n ) = ({ +∞∑ j=0 qjk,n }′ − 1 ) + qk,n { +∞∑ j=0 qjk,n }′′ = ( 1 p2k,n − 1 ) + ( qk,n 2 p3k,n ) = 1− p2k,n p2k,n + 2qk,n p3k,n = pk,n(1− p2k,n) + 2qk,n p3k,n = pk,nqk,n(1 + pk,n) + 2qk,n p3k,n = qk,n (pk,n(1 + pk,n) + 2) p3k,n ≤ 4qk,n p3k,n = 4on(1)(1 + on(1)). (L4b) Hence B(n, k) ≤ Con(1)(1 + on(1)), (L4c) for some C > 0 by (L4a) and (L4b). AB Niang et al. / Eur. J. Pure Appl. Math, 15 (2) (2022), 511-527 525 Finally, by putting together (L1), (L2), (L3) and (L4c), we get Ln,P (ε) ≤ on(1)(1 + on(1))(λ+ o(1)) { 1 + 2C(1 + on(1)) } → 0. This completes the proof. � A simple application. Let us give a simple application to a classical example. We suppose that we observe occurrences of landing crashes at some airport (A) over a period T > 0. We know that those crashes are usually of very low probabilities. Over n landings, we denote Xn the number of crashes at times k(n). Usually, we suppose that the data (of landing crashes) are observations of iid Bernoulli B(pn) and then, the approximation Xn ≈ Z ∼ P(λ), with λ = npn can be used. That formula was systematically used with limited performance of computers. However, with powerful computers, we no-longer need that approximation to compute the related p-values P(Xn > t) of the statistical tests since we know the explicit form of Sn[X]. In the software R, the code 1− pbiniom(t, pn) gives the desired values. Now suppose we can use the independence hypothesis only and not the stationary dis- tribution. Hence the distribution of Xn is the convolution product of Bernoulli B(pk,n) distributions and its law is not simple. So, the simplest way to compute P(Xn > t) should be using the approximation P(P(λ) > t) with λ = p1,n+ · · ·+pk(n),n. So it is better to use the non-stationary scheme since the stationary hypothesis is usually a working hypothesis, not confirmed, and pn is computed as the average number of crashes. These two theorems actually are still particular cases of two more general results. Theorem 3. Let X = { {Xk,n, 1 ≤ k ≤ kn = k(n)}, n ≥ 1 } , be an array of by-row-independent Bernoulli random variables, that is: (GP1) ∀n ≥ 1, ∀1 ≤ k ≤ k(n), Xk,n ∼ B(pk,n), with 0 < pk,n < 1 and: (GP2) sup1≤k≤k(n) pk,n(1− pk,n) → 0; (GP3) ∑ 1≤k≤k(n) pk,n(1− pk,n) → λ ∈]0, +∞[ and ∑ 1≤k≤k(n) pk,n → λ; AB Niang et al. / Eur. J. Pure Appl. Math, 15 (2) (2022), 511-527 526 (GP4) for 0 < ε < 1, n ≥ 1, 1 ≤ k ≤ k(n), and for B(ε, k, n) = 1∑ j=0 1(∣∣∣∣j−qk,n−1 ∣∣∣∣≥ε )( j − pk,n )2 pjk,nq 1−j k,n , we have B(ε, n) = k(n)∑ k=1 B(ε, k, n) → 0. Then we have Sn[X] P(λ). Theorem 4. Let X = { {Xk,n, 1 ≤ k ≤ kn = k(n)}, n ≥ 1 } , be an array of by-row-independent corrected geometric random variables, that is: (GN1) ∀n ≥ 1, ∀1 ≤ k ≤ k(n), Xk,n ∼ G∗(pk,n), with 0 < pk,n = 1− qk,n < 1 and: (GN2) sup1≤k≤k(n)(qk,n/p 2 k,n) → 0; (GN3) for h ∈ {1, 2}, ∑ 1≤k≤k(n)(qk,n/p h k,n) → λ ∈]0, +∞[. (GN4) for 0 < ε < 1, n ≥ 1, 0 ≤ k ≤ k(n), and for B(ε, k, n) = +∞∑ j=0 1(∣∣∣∣j− qk,n pk,n −1 ∣∣∣∣≥ε )( j − qk,n pk,n )2 pk,nq j k,n, we have B(ε, n) = k(n)∑ k=1 B(ε, k, n) → 0. Then we have Sn[X] P(λ). REFERENCES 527 Proofs of Theorems 3 and 4. In the proofs of Theorems 1 and 2, the UAN, the CVH, the Poisson Lynderberg condition and the convergence of ESn[X] are the general conditions for Sn[X] P(λ). In Theorems 3 and 4, uniform convergence simplifies these conditions, making the proofs lighter. � 4. Concluding remarks The extensions we provide are the first general results. The central limit theorem frame seems to be the appropriate way to get more general extensions. Theorems 1 and 2 can be done by direct methods. However, general forms in Theorems 3 and 4 could hardly be obtained in direct methods. They are products of the CLT frame. References [1] W Feller. An introduction to Probability Theory and its Applications. Volume I. Third Editions. John Wiley & Sons Inc, New-York, 1968. [2] W Feller. An introduction to Probability Theory and its Applications. Volume II. Third Editions. John Wiley & Sons Inc, New-York, 1968. [3] A Gut. Probability : A Graduate Course. Springer Science+Business Media, Inc, Singapore, 2005. [4] GS Lo. Mathematical Foundations of Probability Theory. SPAS Books Series., Calgary, 2018. [5] GS Lo, TA Kpanzou, M Ngom, and AB Niang. Weak Convergence (IA): Sequences of random vectors. SPAS Books Series., Calgary, 2021. [6] M Loève. Probability Theory I. Springer-Verlag., New-York, 1977.