scholarly journals On mixing and the local central limit theorem for hyperbolic flows

2018 ◽  
Vol 40 (1) ◽  
pp. 142-174 ◽  
Author(s):  
DMITRY DOLGOPYAT ◽  
PÉTER NÁNDORI
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Finite Horizon ◽  
Suspension Flow ◽  
Renewal Processes ◽  
Axiom A ◽  
Hyperbolic Flows ◽  
Local Central Limit Theorem ◽  
Lorenz Attractors

We formulate abstract conditions under which a suspension flow satisfies the local central limit theorem. We check the validity of these conditions for several systems including reward renewal processes, Axiom A flows, as well as the systems admitting Young’s tower, such as Sinai’s billiard with finite horizon, suspensions over Pomeau–Manneville maps, and geometric Lorenz attractors.

scholarly journals Quenched local limit theorem for random walks among time-dependent ergodic degenerate weights

2021 ◽  
Vol 179 (3-4) ◽  
pp. 1145-1181 ◽  
Author(s):  
Sebastian Andres ◽  
Alberto Chiarini ◽  
Martin Slowik
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Local Limit Theorem ◽  
Time Dependent ◽  
Moment Condition ◽  
Difference Operators ◽  
Random Conductance Model ◽  
Local Central Limit Theorem ◽  
Random Conductance

AbstractWe establish a quenched local central limit theorem for the dynamic random conductance model on $${\mathbb {Z}}^d$$ Z d only assuming ergodicity with respect to space-time shifts and a moment condition. As a key analytic ingredient we show Hölder continuity estimates for solutions to the heat equation for discrete finite difference operators in divergence form with time-dependent degenerate weights. The proof is based on De Giorgi’s iteration technique. In addition, we also derive a quenched local central limit theorem for the static random conductance model on a class of random graphs with degenerate ergodic weights.

A central limit theorem by remainder term for renewal processes

1992 ◽  
Vol 24 (2) ◽  
pp. 267-287 ◽  
Author(s):  
Allen L. Roginsky
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Remainder Term ◽  
Limit Theorems ◽  
Random Variables ◽  
Central Limit Theorems ◽  
Renewal Processes

Three different definitions of the renewal processes are considered. For each of them, a central limit theorem with a remainder term is proved. The random variables that form the renewal processes are independent but not necessarily identically distributed and do not have to be positive. The results obtained in this paper improve and extend the central limit theorems obtained by Ahmad (1981) and Niculescu and Omey (1985).

scholarly journals Occupation times of alternating renewal processes with Lévy applications

2018 ◽  
Vol 55 (4) ◽  
pp. 1287-1308 ◽  
Author(s):  
Nicos Starreveld ◽  
Réne Bekker ◽  
Michel Mandjes
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Random Variable ◽  
Levy Processes ◽  
Normal Random Variable ◽  
Occupation Time ◽  
Renewal Processes ◽  
Tail Asymptotics ◽  
Occupation Times

AbstractIn this paper we present a set of results relating to the occupation time α(t) of a processX(·). The first set of results concerns exact characterizations of α(t), e.g. in terms of its transform up to an exponentially distributed epoch. In addition, we establish a central limit theorem (entailing that a centered and normalized version of α(t)∕tconverges to a zero-mean normal random variable ast→∞) and the tail asymptotics of ℙ(α(t)∕t≥q). We apply our findings to spectrally positive Lévy processes reflected at the infimum and establish various new occupation time results for the corresponding model.

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