Integro-Local Limit Theorems for Compound Renewal Processes under Cramér’S Condition. I

2018 ◽  
Vol 59 (3) ◽  
pp. 383-402 ◽  
Author(s):  
A. A. Borovkov ◽  
A. A. Mogulskii
Keyword(s):  
Limit Theorems ◽  
Local Limit ◽  
Renewal Processes ◽  
Cramér’S Condition ◽  
Compound Renewal Processes ◽  
Local Limit Theorems

scholarly journals Local limit theorems taking into account large deviations in the case when Cramer’s condition does not hold

1968 ◽  
Vol 8 (3) ◽  
pp. 553-579
Author(s):  
A. Nagaev
Keyword(s):  
Large Deviations ◽  
Limit Theorems ◽  
Local Limit ◽  
Cramér’S Condition ◽  
Local Limit Theorems

The abstracts (in two languages) can be found in the pdf file of the article. Original author name(s) and title in Russian and Lithuanian: А. В. Нагаев. Локальные предельные теоремы с учетом больших уклонений, когда не выполнено условие Крамера A. Nagajevas. Lokalinės ribinės teoremos dideliems atsilenkimams, kai nepatenkinta Kramerio sąlyga

scholarly journals Central and Local Limit Theorems for Numbers of the Tribonacci Triangle

Mathematics ◽  
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.

Large deviations of generalized renewal process

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.

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

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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