Asymptotically exact constants in natural convergence rate estimates in the Lindeberg theorem

Following (Shevtsova, 2010) we introduce detailed classification of the asymptotically exact constants in natural estimates of the rate of convergence in the Lindeberg central limit theorem, namely in Esseen's, Rozovskii's, and Wang-Ahmad's inequalities and their structural improvements obtained in our previous works. The above inequalities involve algebraic truncated third-order moments and the classical Lindeberg fraction and assume finiteness only the second-order moments of random summands. We present lower bounds for the introduced asymptotically exact constants as well as for the universal and for the most optimistic constants which turn to be not far from the upper ones.

Let us also note that Esseen-type inequality (1) not only links the criteria of convergence with the rate of convergence, as Osipov's inequality does, but also provides a numerical demonstration of the Ibragimov's criteria [13] of the rate of convergence in the CLT to be of order order O(n −1/2 ). According to [13], in the i.i.d. case we have ∆ n = O(n −1/2 ) as n → ∞ if and only if max{|µ 1 (z)|, zσ 2 1 (z)} = O(1) as z → ∞. Inequality (1) trivially yields the sufficiency of the Ibragimov condition to ∆ n = O(n −1/2 ) as n → ∞.
Property (8) means that every function from G is asymptotically (as its argument goes to infinity) between a constant and a linear function. For example, besides g 0 , g 1 , the class G also includes the following functions: for all c > 0, δ ∈ [0, 1] and g ∈ G.
Extreme properties of the functions for every fixed set of distributions of X 1 , . . . , X n , so that the extreme values of the constants C E and C R in (11) and (12) with fixed n and F 1 , . . . , F n are attained at g = g 0 . Moreover, with the extreme functions g the fraction L • , n ∈ {L E,n , L R,n } satisfy the following relations for ε 1: Inequality (11) also generalizes and improves up to the values of the appearing absolute constant the classical Katz-Petrov inequality [15,20] (which is equivalent to the Osipov inequality [19]) because of involving the algebraic truncated third order moments instead of the absolutes ones and also a recent result of Wang and Ahmad [26] at the expense of moving the modulus and the least upper bound signs outside of the sum sign. In particular, inequality (11) established an upper bound of the constant in the Wang-Ahmad inequality C E (∞, 1) A E (1, 1) 2.73.
The main goal of the present work is construction of the lower bounds of the absolute constants C E (ε, γ), C R (ε, γ) in inequalities (11), (12), and also of the constants A E (ε, γ), A R (ε, γ) in inequalities (1), (2), in particular, we show that even in the i.i.d. case We consider various statements of the problem of construction of the lower bounds, namely, we introduce a detailed classification of the asymptotically exact constants and construct their lower bounds. As a corollary, we obtain two-sided bounds for the asymptotically exact constants A AE E (ε, γ) and A AE R (ε, γ) defined in (3) and (4) A AE R (1, 1) 1.80. The paper is organized as follows. In Section 2 we introduce exact, asymptotically exact and asymptotically best constants defining the corresponding statements for the construction of the lower bounds. Sections 4, 5, and 6 are devoted namely to the construction of the lower bounds for the introduced constants. Section 3 contains some auxiliary results which might represent an independent interest, in particular, the values of the fractions L E,n (g, ε, γ) and L E,n (g, ε, γ) are found for all n, ε, γ > 0 and some g ∈ G in the case where X 1 , . . . , X n have identical two-point distribution.

Exact, asymptotically exact and asymptotically best constants
Following [16,7,27,3,4,5,6,22,23,24], let us define exact, asymptotically exact and asymptotically best constants in inequalities (11), (12). Let F be a set of all d.f.'s with zero means and finite second order moments. Denote Note that the fractions L E,n (g, ε, γ), L R,n (g, ε, γ) also depend on d.f.'s F 1 , . . . , F n , but we omit these arguments for the sake of brevity. The constants C E (g, ε, γ) and C R (g, ε, γ) are the minimal possible (exact) values of the constants C E (ε, γ) and C R (ε, γ) in inequalities (11), (12) for the fixed function g ∈ G, while their universal values sup g∈G C E (g, ε, γ), sup g∈G C R (g, ε, γ), that provide the validity of the inequalities under consideration for all g ∈ G are called exact constants and namely they are the minimal possible (exact) values of the constants C E (ε, γ) and C R (ε, γ) in (11) and (12), respectively. In order not to introduce excess notation and following the above convention, we use namely these exact values for the definitions of C E (ε, γ) and C R (ε, γ) in the present work: and call them the exact constants. Note that hence, every lower bound for the constants A E (ε, γ), A R (ε, γ) serves as a lower bound for the constants C E (ε, γ), C R (ε, γ) as well.
The least upper bounds in (18) are taken without any restrictions on the values of the fractions L E,n (g, ε, γ), L R,n (g, ε, γ), while inequalities (11), (12) represent the most interest with small values of these fractions, when the normal approximation is adequate and only the concrete estimates of its accuracy are needed. That is why it is interesting to study not only the absolute, but also the asymptotic constants in (11), (12), some of which were already introduced in (3), (4). Since we are interested in the lower bounds, we assume that X 1 , . . . , X n have identical distributions. Generalization of the definitions introduced below to the non-i.id. case is not difficult (see, for example, definitions (3) and (4) of the asymptotically exact constants for the general case).
For each of the fractions L n ∈ {L E,n , L R,n } appearing in inequalities (11), (12) we define the asymptotically best constants the upper asymptotically exact constants the asymptotically exact constants the lower asymptotically exact constants the conditional upper asymptotically exact constants In order not to introduce excess indexes, we use identical notation for the asymptotic constants in (11), (12), in what follows every time specifying the inequality in question. Recall that the fractions L n appearing in (20)-(24) depend also on the common d.f. F of the random summands X 1 , . . . , X n . All the introduced constants, except (20), assume double array scheme. The values of L n are infinitesimal in (20), (22), and (23). The distinction between (22) and (23) is in the upper bound and the limit with respect to n, so that C AE (g, ε, γ) C AE (g, ε, γ), where the strict inequality may also take place, as it happens indeed, for example, with the similar constants in the classical Berry-Esseen inequality [3,4,5,6,22,23]. The constants in the classical Berry-Esseen inequality similar to those defined in (20)- (24) were firstly considered in [7] for (20), [16] for (22), [14,27] for (21), [22] for (23), and [24] for (24). The upper asymptotically exact (21) and the conditional upper asymptotically exact (24) constants are linked by the following relation C * AE C AE by definition, and we shall construct lower bounds namely for C * AE . As for the constants C AE , we introduce them here to pay tribute to the classical works [14,27]. The function g in (20), (21), (24) may be arbitrary from the class G, while the constants C AB (g, ε, γ) and C AE (g, ε, γ) (see (22) and (23)) are defined not for all g ∈ G. For example, C AB (g 1 , ε, γ) and C AE (g 1 , ε, γ) are not defined in any of the inequalities (11), (12), since the corresponding fractions L E,n (g 1 , ε, γ), L R,n (g 1 , ε, γ) are bounded from below by one uniformly with respect to ε and γ (see (13) and (14)) and, hence, cannot be infinitesimal.

Two-point distributions
Most of the lower bounds will be obtained by the choice of a two-point distribution for the random summands. The present section contains the corresponding required results. In particular, we will find values of the fractions L E,n (g, ε, γ), L R,n (g, ε, γ) with g = g * , g c , g 0 , g 1 for all ε, γ > 0 (theorem 1). We will also investigate the uniform distance between the d.f. of (26) and the standard normal d.f. Φ and find the corresponding extreme values of the argument of d.f.'s (see theorem 2):

Computation of the uniform distance
In the present section we compute the uniform distance ∆ 1 (p) between the d.f. of (26) and the standard normal d.f. Let us denote Note that Ψ(x) is an even function, therefore, it suffices to investigate it only for x 0.
Lemma 1 (see [2]). The function Ψ(x) is positive for x 0, increases for 0 < x < x Φ , decreases for x > x Φ and attains its maximal value C Φ = 0.54093 . . . in the point x Φ = 0.213105 . . . , where x Φ is the unique root of the equation Remark. The statement about the interval of monotonicity is absent in the formulation of the corresponding lemma in [2], but these intervals were investigated in the proof.
The definition of Ψ and lemma 1 immediately imply that the function is positive for p ∈ (0, 1), increases for 0 < p < p Φ , decreases for p Φ < p < 1 and max p∈(0,1) Proof. The function φ(p) monotonically increases as a superposition of monotonically increasing functions. Note that the derivative φ ′ (p) takes only positive values, hence, we may define its logarithm changes its sign in the points p ∈ (0, 1) that are the roots of the equation which has a unique solution p = 1/2 on (0, 1). Since then u(p) is strictly decreasing on (0, 1), and hence, φ ′ (p) is also strictly decreasing on (0, 1).
Theorem 2. Let X 1 have distribution (26), then Let p q. Then f 1 (p) f 4 (p), and it suffices to show that Let us prove that f 5 (p) := f 3 (p) − f 4 (p) 0. Using lemma 2, we conclude that the derivative The inequality f 2 (p) f 3 (p) follows from the properties of the function Ψ with the account of Ψ 1 2 ∧ Ψ(1 − 0) = Ψ 1 2 . Thus, the theorem has been proved in the case p q. The validity of the statement of the theorem for p < q follows from that f 1 (p) = f 4 (q) and f 2 (p) = f 3 (q).
Lemma 3 (see [1,2]). For an arbitrary d.f. F with zero mean and unit variance we have

Remark. In [1, (2.32), (2.33)] it is proved that
In [2] it is proved that in the inequality the equality is attained at a two-point distribution.

Lower bounds for exact constants
In the present section we construct lower bounds for the quantities and for all ε, γ > 0. Recall that g 0 (z) = min{z, B n }, g 1 (z) = max{z, B n }, g * (z) = z, g c (z) = 1, z 0.
By virtue of invariance of the fractions L E,n (g, ε, γ), L R,n (g, ε, γ) with respect to scale transform of g and extreme property of g 1 (see (15)) we have on one hand, and inf g∈G C E (g, ε, γ) C E (g 1 , ε, γ) by definition of the lower bound, on the other hand. Therefore where the second equality is proved similar to the first one. Let us show that Indeed, for arbitrary n ∈ N, F 1 , . . . , F n ∈ F and g ∈ G due to the extremality of g 0 (see (15)) we have ∆ n C E (g 0 , ε, γ)L E,n (g 0 , ε, γ) C E (g 0 , ε, γ)L E,n (g, ε, γ), hence, C E (ε, γ) C E (g 0 , ε, γ), on one hand. On the other hand, this inequality may hold true only with the equality sign, since C E (ε, γ) = sup g∈G C E (g, ε, γ) by definition. The same reasoning holds also true for C R (ε, γ).

In particular,
and for the both constants C • ∈ {C E , C R } with ε = 1, γ = 1 the following lower bound holds: Proof. Let n = 1 and the r.v. X 1 have the two-point distribution (26) with p ∈ [ 1 2 , 1). Then, by theorem 2, we have ∆ 1 (F 1 ) = ∆ 1 (p), and all the constants C • ∈ {C E , C E }, A • ∈ {A E , A R } can be bounded from below as .
Values of C AB (γ) for some γ and the corresponding extreme values of p are given in table 4.  Table 4: Values of the lower bound C AB (γ) (see (35)), rounded down, for the asymptotically best constant C AB (g 0 , ε, γ) from inequality (11) for some γ. The second line contains rounded extreme values of p in (35).

Lower bounds for the asymptotically exact constants
First of all note that asymptotically exact constants C AE (g 1 , ε, γ) and the lower asymptotically exact constants C AE (g 1 , ε, γ) are defined for none of the inequalities (11), (12), since the corresponding fractions L E,n (g 1 , ε, γ), L R,n (g 1 , ε, γ) are bounded from below by one uniformly with respect to ε and γ (see (13) and (14)) and, hence, cannot be infinitesimal.