Fine asymptotics for models with Gamma type moments

The aim of this paper is to give fine asymptotics for random variables with moments of Gamma type. Among the examples, we consider random determinants of Laguerre and Jacobi beta ensembles with varying dimensions (the number of observed variables and the number of measurements vary and may be different). In addition to the Dyson threefold way of classical random matrix models (GOE, GUE, GSE), we study random determinants of random matrices of the so-called tenfold way, including the Bogoliubov–de Gennes and chiral ensembles from mesoscopic physics. We show that fixed-trace matrix ensembles can be analyzed as well. Finally, we add fine asymptotics for the [Formula: see text]-dimensional volume of the simplex with [Formula: see text] points in [Formula: see text] distributed according to special distributions, which is strongly correlated to Gram matrix ensembles. We use the framework of mod-[Formula: see text] convergence to obtain extended limit theorems, Berry–Esseen bounds, precise moderate deviations, large and moderate deviation principles as well as local limit theorems. The work is especially based on the recent work of Dal Borgo et al. [Mod-Gaussian convergence for random determinants, Ann. Henri Poincaré (2018)].

Download Full-text

Certain Two-Dimensional Local Limit Theorems with Applications to the Statistics of Convex Polygons

SSRN Electronic Journal ◽

10.2139/ssrn.3536633 ◽

2020 ◽

Author(s):

Sakhavet Zarbaliev

Keyword(s):

Limit Theorems ◽

Local Limit ◽

Two Dimensional ◽

Convex Polygons ◽

Local Limit Theorems

Download Full-text

Central and Local Limit Theorems for Numbers of the Tribonacci Triangle

Mathematics ◽

10.3390/math9080880 ◽

2021 ◽

Vol 9 (8) ◽

pp. 880

Author(s):

Igoris Belovas

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Normal Distribution ◽

Rate Of Convergence ◽

Limit Theorems ◽

Local Limit Theorem ◽

Local Limit ◽

Constructive Proof ◽

The Central Limit Theorem ◽

Local Limit Theorems

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.

Download Full-text

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.

Download Full-text

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.

Download Full-text