Addendum: Integral Limit Theorems Taking Large Deviations into Account when Cramer’s Condition Does Not Hold

1969 ◽  
Vol 14 (3) ◽  
pp. 534-534
Author(s):  
A. V. Nagaev
Keyword(s):  
Large Deviations ◽  
Limit Theorems ◽  
Integral Limit ◽  
Cramér’S Condition

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

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.

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