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 Identification and isotropy characterization of deformed random fields through excursion sets

2018 ◽  
Vol 50 (3) ◽  
pp. 706-725
Author(s):  
Julie Fournier
Keyword(s):  
Random Field ◽  
Euler Characteristic ◽  
Random Fields ◽  
Weak Form ◽  
Invariance Property ◽  
Basic Domain ◽  
Excursion Sets ◽  
The Mean ◽  
Isotropic Random

Abstract A deterministic application θ:ℝ2→ℝ2 deforms bijectively and regularly the plane and allows the construction of a deformed random field X∘θ:ℝ2→ℝ from a regular, stationary, and isotropic random field X:ℝ2→ℝ. The deformed field X∘θ is, in general, not isotropic (and not even stationary), however, we provide an explicit characterization of the deformations θ that preserve the isotropy. Further assuming that X is Gaussian, we introduce a weak form of isotropy of the field X∘θ, defined by an invariance property of the mean Euler characteristic of some of its excursion sets. We prove that deformed fields satisfying this property are strictly isotropic. In addition, we are able to identify θ, assuming that the mean Euler characteristic of excursion sets of X∘θ over some basic domain is known.

Limit theorems for the minimal position in a branching random walk with independent logconcave displacements

2000 ◽  
Vol 32 (01) ◽  
pp. 159-176 ◽  
Author(s):  
Markus Bachmann
Keyword(s):  
Random Walk ◽  
Limit Theorems ◽  
Random Number ◽  
Time Lag ◽  
Extreme Value Distribution ◽  
Extreme Value ◽  
Limiting Distribution ◽  
Branching Random Walk ◽  
Minimal Position ◽  
Conditional Limit

Consider a branching random walk in which each particle has a random number (one or more) of offspring particles that are displaced independently of each other according to a logconcave density. Under mild additional assumptions, we obtain the following results: the minimal position in the nth generation, adjusted by its α-quantile, converges weakly to a non-degenerate limiting distribution. There also exists a ‘conditional limit’ of the adjusted minimal position, which has a (Gumbel) extreme value distribution delayed by a random time-lag. Consequently, the unconditional limiting distribution is a mixture of extreme value distributions.

scholarly journals Fractional random fields associated with stochastic fractional heat equations

2005 ◽  
Vol 37 (01) ◽  
pp. 108-133 ◽  
Author(s):  
M. Ya. Kelbert ◽  
N. N. Leonenko ◽  
M. D. Ruiz-Medina
Keyword(s):  
Random Field ◽  
Random Fields ◽  
Evolution Equations ◽  
Heat Equations ◽  
Mean Square ◽  
Fractional Evolution Equations ◽  
The Mean

This paper introduces a convenient class of spatiotemporal random field models that can be interpreted as the mean-square solutions of stochastic fractional evolution equations.

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.

Limit theorems for local particle numbers in branching random walk

2012 ◽  
Vol 85 (3) ◽  
pp. 403-405 ◽  
Author(s):  
E. Vl. Bulinskaya
Keyword(s):  
Random Walk ◽  
Limit Theorems ◽  
Branching Random Walk

scholarly journals Exact moderate and large deviations for linear random fields

2018 ◽  
Vol 55 (2) ◽  
pp. 431-449 ◽  
Author(s):  
Hailin Sang ◽  
Yimin Xiao
Keyword(s):  
Large Deviations ◽  
Random Field ◽  
Nonparametric Regression ◽  
Random Fields ◽  
Limit Theorems ◽  
Large Deviation ◽  
Strong Limit ◽  
Field Errors

Abstract By extending the methods of Peligrad et al. (2014), we establish exact moderate and large deviation asymptotics for linear random fields with independent innovations. These results are useful for studying nonparametric regression with random field errors and strong limit theorems.

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.

scholarly journals Deviation inequalities for Banach space valued martingales differences sequences and random fields

2019 ◽  
Vol 23 ◽  
pp. 922-946 ◽  
Author(s):  
Davide Giraudo
Keyword(s):  
Banach Space ◽  
Linear Regression ◽  
Random Field ◽  
Random Fields ◽  
Law Of Large Numbers ◽  
Partial Sums ◽  
Martingale Differences ◽  
Large Numbers ◽  
Deviation Inequalities ◽  
The Law

We establish deviation inequalities for the maxima of partial sums of a martingale differences sequence, and of an orthomartingale differences random field. These inequalities can be used to give rates for linear regression and the law of large numbers.

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