scholarly journals Local asymptotics for the area under the random walk excursion

2018 ◽  
Vol 50 (2) ◽  
pp. 600-620
Author(s):  
Elena Perfilev ◽  
Vitali Wachtel
Keyword(s):  
Random Walk ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Tail Behaviour ◽  
Local Asymptotics ◽  
Negative Drift ◽  
Local Central Limit Theorem

Abstract We study the tail behaviour of the distribution of the area under the positive excursion of a random walk which has negative drift and light-tailed increments. We determine the asymptotics for local probabilities for the area and prove a local central limit theorem for the duration of the excursion conditioned on the large values of its area.

scholarly journals Quantitative recurrence results for random walks

2018 ◽  
Vol 18 (03) ◽  
pp. 1850003
Author(s):  
Nuno Luzia
Keyword(s):  
Random Walk ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Walks ◽  
Central Limit ◽  
Lattice Random Walks ◽  
Asymptotic Number ◽  
Quantitative Recurrence ◽  
Recurrence Theorem ◽  
Local Central Limit Theorem

First, we prove an almost sure local central limit theorem for lattice random walks in the plane. The corresponding version for random walks in the line has been considered previously by the author. This gives us an extension of Pólya’s Recurrence Theorem, namely we consider an appropriate subsequence of the random walk and give the asymptotic number of returns to the origin and other states. Secondly, we prove an almost sure local central limit theorem for (not necessarily lattice) random walks in the line or in the plane, which will also give us quantitative recurrence results. Finally, we prove a version of the almost sure central limit theorem for multidimensional random walks. This is done by exploiting a technique developed by the author.

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.

Heavy traffic limit theorems in fluctuation theory

1978 ◽  
Vol 10 (04) ◽  
pp. 852-866
Author(s):  
A. J. Stam
Keyword(s):  
Random Walk ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Walks ◽  
Central Limit ◽  
Limit Theorems ◽  
Heavy Traffic ◽  
The Central Limit Theorem ◽  
Heavy Traffic Limit ◽  
Oscillating Random Walk

Let be a family of random walks with For ε↓0 under certain conditions the random walk U (∊) n converges to an oscillating random walk. The ladder point distributions and expectations converge correspondingly. Let M ∊ = max {U (∊) n , n ≧ 0}, v 0 = min {n : U (∊) n = M ∊}, v 1 = max {n : U (∊) n = M ∊}. The joint limiting distribution of ∊2σ∊ –2 v 0 and ∊σ∊ –2 M ∊ is determined. It is the same as for ∊2σ∊ –2 v 1 and ∊σ–2 ∊ M ∊. The marginal ∊σ–2 ∊ M ∊ gives Kingman's heavy traffic theorem. Also lim ∊–1 P(M ∊ = 0) and lim ∊–1 P(M ∊ < x) are determined. Proofs are by direct comparison of corresponding probabilities for U (∊) n and for a special family of random walks related to MI/M/1 queues, using the central limit theorem.

Sign in / Sign up

Export Citation Format

Share Document