Estimate of remainder term in local limit theorems for number of renewals in the multidimensional case

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.

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Large deviations of generalized renewal process

Discrete Mathematics and Applications ◽

10.1515/dma-2020-0020 ◽

2020 ◽

Vol 30 (4) ◽

pp. 215-241

Author(s):

Gavriil A. Bakay ◽

Aleksandr V. Shklyaev

Keyword(s):

Large Deviations ◽

Renewal Process ◽

Limit Theorems ◽

Local Limit ◽

Asymptotic Formulas ◽

Additive Functionals ◽

Wide Range ◽

Finite Markov Chains ◽

Independent Identically Distributed ◽

Local Limit Theorems

AbstractLet (ξ(i), η(i)) ∈ ℝd+1, 1 ≤ i < ∞, be independent identically distributed random vectors, η(i) be nonnegative random variables, the vector (ξ(1), η(1)) satisfy the Cramer condition. On the base of renewal process, NT = max{k : η(1) + … + η(k) ≤ T} we define the generalized renewal process ZT = $\begin{array}{} \sum_{i=1}^{N_T} \end{array}$ξ(i). Put IΔT(x) = {y ∈ ℝd : xj ≤ yj < xj + ΔT, j = 1, …, d}. We find asymptotic formulas for the probabilities P(ZT ∈ IΔT(x)) as ΔT → 0 and P(ZT = x) in non-lattice and arithmetic cases, respectively, in a wide range of x values, including normal, moderate, and large deviations. The analogous results were obtained for a process with delay in which the distribution of (ξ(1), η(1)) differs from the distribution on the other steps. Using these results, we prove local limit theorems for processes with regeneration and for additive functionals of finite Markov chains, including normal, moderate, and large deviations.

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Local limit theorems for generalized scheme of allocation of particles into ordered cells

Discrete Mathematics and Applications ◽

10.1515/dma-2021-0026 ◽

2021 ◽

Vol 31 (4) ◽

pp. 293-307

Author(s):

Aleksandr N. Timashev

Keyword(s):

Weak Convergence ◽

Negative Binomial Distribution ◽

Limit Theorems ◽

Negative Binomial ◽

Sufficient Conditions ◽

Local Limit ◽

Number Of Components ◽

Normal Law ◽

Local Limit Theorems ◽

Generalized Scheme

Abstract A generalized scheme of allocation of n particles into ordered cells (components). Some statements containing sufficient conditions for the weak convergence of the number of components with given cardinality and of the total number of components to the negative binomial distribution as n → ∞ are presented as hypotheses. Examples supporting the validity of these statements in particular cases are considered. For some examples we prove local limit theorems for the total number of components which partially generalize known results on the convergence of this distribution to the normal law.

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