Central and Local Limit Theorems for Numbers of the Tribonacci Triangle

: In this research, we continue studying limit theorems for combinatorial numbers satisfying a class of triangular arrays. Using the general results of Hwang and Bender, we obtain a constructive proof of the central limit theorem, specifying the rate of convergence to the limiting (normal) distribution, as well as a new proof of the local limit theorem for the numbers of the tribonacci triangle.


Introduction
The tribonacci triangle (also known as the Delannoy triangle [1], cf.OEIS Number sequence A008288 [2]) was introduced to derive the expansion of the tribonacci numbers (OEIS Number sequence A000073 [3]).The sum of the elements along a rising diagonal of this triangle (see Table 1) equals the tribonacci number: 1, 1, 2, 4, 7, 13, 24, . ...  (1) Barry [4] showed that the closed form for numbers satisfying the tribonacci triangle is Mathematics 2021, 9, 880.https://doi.org/10.3390/math9080880https://www.mdpi.com/journal/mathematics The tribonacci (Delannoy)-like triangles and their generalizations are intensively examined nowadays.Amrouche, Belbachir and Ramírez [1,5] have studied the unimodality of sequences located in the triangle's infinite transversals and derived the explicit formulation of the linear recurrence sequence satisfied by the sum of the elements lying over any finite ray of the generalized tribonacci matrix.Kuhapatanakul [6] has examined the connection between a generalized tribonacci triangle and a generalized Fibonacci sequence.The total positivity of Delannoy-like triangles has been considered by Mu and Zheng [7].Yang, Zheng and Yuan [8] have studied the inverses of the generalized Delannoy matrices.
The present research extends the investigations of the asymptotics for Delannoy numbers undertaken by Noble [9,10] and Wang, Zheng and Chen [11] (as well as our research into central and local limit theorems for combinatorial numbers satisfying a class of triangular arrays [12][13][14][15]).Noble has obtained asymptotic expansions for the central weighted Delannoy numbers (u r,r ) and the numbers along the the diagonal with slope 2 (u r,2r ).Wang, Zheng and Chen showed the asymptotic normality of Delannoy numbers, using the properties of the zeroes of Delannoy polynomials.In this work we provide a new constructive proof of the central limit theorem for the numbers of the tribonacci triangle, specifying the rate of convergence to the limiting distribution (in the process we receive the closed exact expression for the variance of the random variable, associated to the numbers of the tribonacci triangle, missing in the work of Wang et al.), together with a new proof of the local limit theorem.
The paper is organized in the following way.The first part is the introduction.In the second part, we specify the moment-generating function of the numbers of the tribonacci triangle and calculate exact expressions for the expectation and the variance of the random variable, associated to the numbers of the tribonacci triangle.The third and fourth sections are devoted to the central and local limit theorems.
Throughout this paper, we denote by Φ µ,σ (x) the cumulative distribution function of the normal distribution with the mean µ and the standard deviation σ; by ϕ µ,σ (x) we denote the corresponding density function.
All limits in the paper, unless specified, are taken as n → ∞.

Moment-Generating Function
Let us consider the generating function of the numbers given by Equation (1), Lemma 1. (Alladi and Hoggatt [16]).The bivariate generating function of the numbers of the tribonacci triangle is Let A n be an integral random variable with the probability mass function The moment-generating function of the random variable A n equals where S n stands for the sum of the n-th row of the triangle, Combining the definition of the generating function from Equations ( 2) and ( 5), we obtain Thus, the partial differentiation of the generating function G(x, y) at x = 0, yields us the moment-generating function Note that, since M n (0) = 1, we have a formula for the sum of the n-th row, Lemma 2. The moment-generating function of the random variable A n , associated to the numbers of the tribonacci triangle given by Equation (4), is Proof.Let us consider the denominator of G(x, e s ) from Equation (3), where Next, Now, using Equation (11), we calculate the n-th derivative of G(x, e s ), Thus, by Equation ( 7), yielding us the statement of the lemma (cf.Equation ( 6)).
Lemma 3. The expectation and the variance of the random variable A n , associated to the numbers of the tribonacci triangle from Equation (4), are

respectively.
Proof.By Equation ( 8), the first and second derivatives of the moment-generating function are and Note that (cf.Equation (9)) hence, Next, calculating the expectation, by Equations ( 12) and ( 14), we obtain yielding us the first statement of the lemma.
Calculating the variance (cf.Equations ( 13) and ( 15)), we obtain thus concluding the proof of the lemma.

Central Limit Theorem
Let, by Hwang [17], {Ω n } be a sequence of integral random variables, and Here O-term is uniform for |s| τ, s ∈ C, τ > 0, and , with u(s) and v(s) analytic for |s| τ and independent of n, u (0 κ n → +∞.We apply the following Hwang's result [17] to prove the central limit theorem for the numbers of Equation ( 1) and specify the rate of convergence to the limiting distribution.
Next we prove an auxiliary lemma.
The following theorem shows that the numbers of the tribonacci triangle from Equation ( 1) are asymptotically normal, and identifies the rate of convergence to the limiting distribution.
Theorem 2. Suppose that F n (x) is the cumulative distribution function of the random variable A n , then uniformly with respect to x, x ∈ R.
Proof.The logarithm of the moment-generating function equals Here . Note that the functions u(s), v(s), φ(n) and κ n satisfy the conditions (i)-(iii).Indeed, Hence, yielding us, by Equation ( 16), the statement of the theorem.

Local Limit Theorem
We apply a general local limit theorem, based on the nature of the bivariate generating function of Equations ( 2) and (3).Theorem 3. (Bender [18]) Let f (z, w) have a power series expansion f (z, w) = ∑ n,k 0 u n,k z n w k with non-negative coefficients and let a < b be real numbers.Define Suppose there exists ε > 0, δ > 0, a non-negative integer m, and functions A(s), r(s) such that (i) an A(s) is continuous and non-zero for s ∈ R(ε), (ii) an r(s) is non-zero and has a bounded third derivative for s ∈ R(ε), (iii) for s ∈ R(ε) and |z| |r(s)|(1 + δ) function is analytic and bounded, (iv) (r (α)/r(α) Now we can proceed with the local limit theorem for the numbers of the tribonacci triangle.
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