Local Limit Theorems for Large Deviations

Wolfgang Richter

doi:10.1137/1102014

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.

Integro-Local Limit Theorems Including Large Deviations for Sumsof Random Vectors.

Theory of Probability and Its Applications ◽

10.1137/s0040585x97976623 ◽

1999 ◽

Vol 43 (1) ◽

pp. 1-12 ◽

Cited By ~ 11

Author(s):

A. A. Borovkov ◽

A. A. Mogulskii

Keyword(s):

Large Deviations ◽

Limit Theorems ◽

Local Limit ◽

Random Vectors ◽

Local Limit Theorems

Multi-Dimensional Local Limit Theorems for Large Deviations

Theory of Probability and Its Applications ◽

10.1137/1103009 ◽

1958 ◽

Vol 3 (1) ◽

pp. 100-106 ◽

Cited By ~ 24

Author(s):

V. Richter

Keyword(s):

Large Deviations ◽

Limit Theorems ◽

Local Limit ◽

Local Limit Theorems

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

Lithuanian Mathematical Journal ◽

10.15388/lmj.1968.20590 ◽

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

Integro-Local Limit Theorems Including Large Deviations for Sums of Random Vectors. II

Theory of Probability and Its Applications ◽

10.1137/s0040585x97978026 ◽

2001 ◽

Vol 45 (1) ◽

pp. 3-22 ◽

Cited By ~ 11

Author(s):

A. A. Borovkov ◽

A. A. Mogulskii

Keyword(s):

Large Deviations ◽

Limit Theorems ◽

Local Limit ◽

Random Vectors ◽

Local Limit Theorems

The discounted local limit theorems for large deviations

Lietuvos matematikos rinkinys ◽

10.15388/lmr.2008.18152 ◽

2008 ◽

Vol 48 ◽

Author(s):

Leonas Saulis ◽

Dovilė Deltuvienė

Keyword(s):

Large Deviations ◽

Density Function ◽

Limit Theorems ◽

Distribution Density ◽

Normal Approximation ◽

Random Variables ◽

Local Limit ◽

Distribution Density Function ◽

Local Limit Theorems

Theorems of large deviations, both in the Cramer zone and the Linnik power zones, for the normal approximation of the distribution density function of normalized sum Sv = \sum∞ k=0 vkXk, 0 < v < 1, of i.i.d. random variables (r.v.) X0, X1, . . . satisfying the generalized Bernstein’s condition are obtained.