Asymptotic behaviour for random walks in random environments

1999 ◽  
Vol 36 (2) ◽  
pp. 334-349 ◽  
Author(s):  
S. Alili
Keyword(s):  
Random Walk ◽  
Limit Theorems ◽  
Transition Probabilities ◽  
Invariant Probability Measure ◽  
Random Environments ◽  
Regularity Conditions ◽  
Large Numbers ◽  
Random Transition ◽  
Independent Case ◽  
Uniquely Ergodic

In this paper we consider limit theorems for a random walk in a random environment, (Xn). Known results (recurrence-transience criteria, law of large numbers) in the case of independent environments are naturally extended to the case where the environments are only supposed to be stationary and ergodic. Furthermore, if ‘the fluctuations of the random transition probabilities around are small’, we show that there exists an invariant probability measure for ‘the environments seen from the position of (Xn)’. In the case of uniquely ergodic (therefore non-independent) environments, this measure exists as soon as (Xn) is transient so that the ‘slow diffusion phenomenon’ does not appear as it does in the independent case. Thus, under regularity conditions, we prove that, in this case, the random walk satisfies a central limit theorem for any fixed environment.

Asymptotic behaviour for random walks in random environments

1999 ◽  
Vol 36 (02) ◽  
pp. 334-349 ◽  
Author(s):  
S. Alili
Keyword(s):  
Random Walk ◽  
Limit Theorems ◽  
Transition Probabilities ◽  
Invariant Probability Measure ◽  
Random Environments ◽  
Regularity Conditions ◽  
Large Numbers ◽  
Random Transition ◽  
Independent Case ◽  
Uniquely Ergodic

In this paper we consider limit theorems for a random walk in a random environment, (X n ). Known results (recurrence-transience criteria, law of large numbers) in the case of independent environments are naturally extended to the case where the environments are only supposed to be stationary and ergodic. Furthermore, if ‘the fluctuations of the random transition probabilities around are small’, we show that there exists an invariant probability measure for ‘the environments seen from the position of (X n )’. In the case of uniquely ergodic (therefore non-independent) environments, this measure exists as soon as (X n ) is transient so that the ‘slow diffusion phenomenon’ does not appear as it does in the independent case. Thus, under regularity conditions, we prove that, in this case, the random walk satisfies a central limit theorem for any fixed environment.

Random walk in a random environment with correlated sites

2001 ◽  
Vol 38 (4) ◽  
pp. 1018-1032 ◽  
Author(s):  
T. Komorowski ◽  
G. Krupa
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Random Field ◽  
Random Environment ◽  
Law Of Large Numbers ◽  
Transition Probabilities ◽  
Direct Application ◽  
Random Environments ◽  
Large Numbers ◽  
Stationary Random Field

We prove the law of large numbers for random walks in random environments on the d-dimensional integer lattice Zd. The environment is described in terms of a stationary random field of transition probabilities on the lattice, possessing a certain drift property, modeled on the Kalikov condition. In contrast to the previously considered models, we admit possible correlation of transition probabilities at different sites, assuming however that they become independent at finite distances. The possible dependence of sites makes impossible a direct application of the renewal times technique of Sznitman and Zerner.

Random walk in a random environment with correlated sites

2001 ◽  
Vol 38 (04) ◽  
pp. 1018-1032 ◽  
Author(s):  
T. Komorowski ◽  
G. Krupa
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Random Field ◽  
Random Environment ◽  
Law Of Large Numbers ◽  
Transition Probabilities ◽  
Direct Application ◽  
Random Environments ◽  
Large Numbers ◽  
Stationary Random Field

We prove the law of large numbers for random walks in random environments on the d-dimensional integer lattice Z d . The environment is described in terms of a stationary random field of transition probabilities on the lattice, possessing a certain drift property, modeled on the Kalikov condition. In contrast to the previously considered models, we admit possible correlation of transition probabilities at different sites, assuming however that they become independent at finite distances. The possible dependence of sites makes impossible a direct application of the renewal times technique of Sznitman and Zerner.

scholarly journals DYNAMIC QUANTUM BERNOULLI RANDOM WALKS

2008 ◽  
Vol 11 (02) ◽  
pp. 213-229
Author(s):  
NADINE GUILLOTIN-PLANTARD ◽  
RENÉ SCHOTT
Keyword(s):  
Random Walk ◽  
Limit Theorem ◽  
Random Walks ◽  
Limit Theorems ◽  
Law Of Large Numbers ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Local Limit Theorem ◽  
Local Limit ◽  
Large Numbers

Quantum Bernoulli random walks can be realized as random walks on the dual of SU(2). We use this realization in order to study a model of dynamic quantum Bernoulli random walk with time-dependent transitions. For the corresponding dynamic random walk on the dual of SU(2), we prove several limit theorems (local limit theorem, central limit theorem, law of large numbers, large deviation principle). In addition, we characterize a large class of transient dynamic random walks.

The multitype branching random walk: temporal and spatial limit theorems

1989 ◽  
Vol 21 (03) ◽  
pp. 491-512
Author(s):  
B. Gail Ivanoff
Keyword(s):  
Random Walk ◽  
Random Field ◽  
Random Fields ◽  
Limit Theorems ◽  
Branching Random Walk ◽  
Initial Field ◽  
Large Numbers ◽  
Poisson Random Fields ◽  
The Mean ◽  
Temporal And Spatial

We consider a multitype branching random walk with independent Poisson random fields of each type of particle initially. The existence of limiting random fields as the generation number, is studied, when the intensity of the initial field is renormalized in such a way that the mean measures converge. Spatial laws of large numbers and central limit theorems are given for the limiting random field, when it is non-trivial.

The multitype branching random walk: temporal and spatial limit theorems

1989 ◽  
Vol 21 (3) ◽  
pp. 491-512 ◽  
Author(s):  
B. Gail Ivanoff
Keyword(s):  
Random Walk ◽  
Random Field ◽  
Random Fields ◽  
Limit Theorems ◽  
Branching Random Walk ◽  
Initial Field ◽  
Large Numbers ◽  
Poisson Random Fields ◽  
The Mean ◽  
Temporal And Spatial

We consider a multitype branching random walk with independent Poisson random fields of each type of particle initially. The existence of limiting random fields as the generation number, is studied, when the intensity of the initial field is renormalized in such a way that the mean measures converge. Spatial laws of large numbers and central limit theorems are given for the limiting random field, when it is non-trivial.

scholarly journals An Introduction to Bootstrap Theory in Time Series Econometrics

2021 ◽  
Author(s):  
Giuseppe Cavaliere ◽  
Heino Bohn Nielsen ◽  
Anders Rahbek
Keyword(s):  
Time Series ◽  
Limit Theorems ◽  
Financial Time Series ◽  
Convergence In Probability ◽  
Regularity Conditions ◽  
Time Series Econometrics ◽  
Large Numbers ◽  
Empirical Size ◽  
Asymptotic Validity ◽  
Bootstrap Theory

While often simple to implement in practice, application of the bootstrap in econometric modeling of economic and financial time series requires establishing validity of the bootstrap. Establishing bootstrap asymptotic validity relies on verifying often nonstandard regularity conditions. In particular, bootstrap versions of classic convergence in probability and distribution, and hence of laws of large numbers and central limit theorems, are critical ingredients. Crucially, these depend on the type of bootstrap applied (e.g., wild or independently and identically distributed (i.i.d.) bootstrap) and on the underlying econometric model and data. Regularity conditions and their implications for possible improvements in terms of (empirical) size and power for bootstrap-based testing differ from standard asymptotic testing, which can be illustrated by simulations.

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