scholarly journals Conditioned local limit theorems for random walks defined on finite Markov chains

2019 ◽  
Vol 176 (1-2) ◽  
pp. 669-735 ◽  
Author(s):  
Ion Grama ◽  
Ronan Lauvergnat ◽  
Émile Le Page
Keyword(s):  
Markov Chains ◽  
Random Walks ◽  
Limit Theorems ◽  
Local Limit ◽  
Finite Markov Chains ◽  
Local Limit Theorems

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.

Random walks and periodic continued fractions

1985 ◽  
Vol 17 (1) ◽  
pp. 67-84 ◽  
Author(s):  
Wolfgang Woess
Keyword(s):  
Random Walks ◽  
Harmonic Functions ◽  
Limit Theorems ◽  
Continued Fraction ◽  
Transition Probabilities ◽  
Local Limit ◽  
Periodic Sequence ◽  
Nearest Neighbour ◽  
Continued Fraction Expansions ◽  
Local Limit Theorems

Nearest-neighbour random walks on the non-negative integers with transition probabilities p0,1 = 1, pk,k–1 = gk, pk,k+1 = 1– gk (0 < gk < 1, k = 1, 2, …) are studied by use of generating functions and continued fraction expansions. In particular, when (gk) is a periodic sequence, local limit theorems are proved and the harmonic functions are determined. These results are applied to simple random walks on certain trees.

scholarly journals On the local limit theorems for psi-mixing Markov chains

2021 ◽  
Vol 18 (1) ◽  
pp. 1221
Author(s):  
Florence Merlevède ◽  
Magda Peligrad ◽  
Costel Peligrad
Keyword(s):  
Markov Chains ◽  
Limit Theorems ◽  
Local Limit ◽  
Local Limit Theorems

Random walks and periodic continued fractions

1985 ◽  
Vol 17 (01) ◽  
pp. 67-84 ◽  
Author(s):  
Wolfgang Woess
Keyword(s):  
Random Walks ◽  
Harmonic Functions ◽  
Limit Theorems ◽  
Continued Fraction ◽  
Transition Probabilities ◽  
Local Limit ◽  
Periodic Sequence ◽  
Nearest Neighbour ◽  
Continued Fraction Expansions ◽  
Local Limit Theorems

Nearest-neighbour random walks on the non-negative integers with transition probabilitiesp0,1= 1,pk,k–1=gk,pk,k+1= 1–gk(0 &lt;gk&lt; 1,k= 1, 2, …) are studied by use of generating functions and continued fraction expansions. In particular, when (gk) is a periodic sequence, local limit theorems are proved and the harmonic functions are determined. These results are applied to simple random walks on certain trees.

scholarly journals Local Limit Theorems for Sequences of Simple Random Walks on Graphs

2008 ◽  
Vol 29 (4) ◽  
pp. 351-389 ◽  
Author(s):  
D. A. Croydon ◽  
B. M. Hambly
Keyword(s):  
Random Walks ◽  
Limit Theorems ◽  
Local Limit ◽  
Random Walks On Graphs ◽  
Local Limit Theorems

Local Limit Theorems for random walks in a 1D random environment

2013 ◽  
Vol 101 (2) ◽  
pp. 191-200 ◽  
Author(s):  
D. Dolgopyat ◽  
I. Goldsheid
Keyword(s):  
Random Walks ◽  
Random Environment ◽  
Limit Theorems ◽  
Local Limit ◽  
Local Limit Theorems

scholarly journals Local limit theorems and asymptotic expansions for non-stationary Markov chains

1961 ◽  
Vol 1 (1-2) ◽  
pp. 231-313
Author(s):  
V. Statulevičius
Keyword(s):  
Markov Chains ◽  
Asymptotic Expansions ◽  
Limit Theorems ◽  
Local Limit ◽  
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: B. Статулявичюс. Локальные предельные теоремы и асимптотические разложения для неоднородных цепей Маркова V. Statulevičius. Lokalinės ribinės teoremos ir asimptotiniai išdėstymai nehomogeninėms Markovo grandinėms

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