scholarly journals Heavy-tailed branching random walks on multidimensional lattices. A moment approach

2020 ◽  
pp. 1-22
Author(s):  
Anastasiya Rytova ◽  
Elena Yarovaya
Keyword(s):  
Random Walk ◽  
Asymptotic Behaviour ◽  
Continuous Time ◽  
Lattice Point ◽  
Positive Eigenvalue ◽  
Infinite Variance ◽  
Finite Variance ◽  
Branching Random Walks ◽  
Heavy Tailed ◽  
Number Of Particles

We study a continuous-time branching random walk (BRW) on the lattice ℤ d , d ∈ ℕ, with a single source of branching, that is the lattice point where the birth and death of particles can occur. The random walk is assumed to be spatially homogeneous, symmetric and irreducible but, in contrast to the majority of previous investigations, the random walk transition intensities a(x, y) decrease as |y − x|−(d+α) for |y − x| → ∞, where α ∈ (0, 2), that leads to an infinite variance of the random walk jumps. The mechanism of the birth and death of particles at the source is governed by a continuous-time Markov branching process. The source intensity is characterized by a certain parameter β. We calculate the long-time asymptotic behaviour for all integer moments for the number of particles at each lattice point and for the total population size. With respect to the parameter β, a non-trivial critical point β c  > 0 is found for every d ≥ 1. In particular, if β > β c the evolutionary operator generated a behaviour of the first moment for the number of particles has a positive eigenvalue. The existence of a positive eigenvalue yields an exponential growth in t of the particle numbers in the case β > β c called supercritical. Classification of the BRW treated as subcritical (β < β c ) or critical (β = β c ) for the heavy-tailed random walk jumps is more complicated than for a random walk with a finite variance of jumps. We study the asymptotic behaviour of all integer moments of a number of particles at any point y ∈ ℤ d and of the particle population on ℤ d according to the ratio d/α.

scholarly journals Strong Local Survival of Branching Random Walks is Not Monotone

2014 ◽  
Vol 46 (02) ◽  
pp. 400-421 ◽  
Author(s):  
Daniela Bertacchi ◽  
Fabio Zucca
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Generating Function ◽  
Continuous Time ◽  
Local Property ◽  
Branching Random Walk ◽  
Infinite Dimensional ◽  
Branching Random Walks ◽  
Local Survival ◽  
Branching Law

In this paper we study the strong local survival property for discrete-time and continuous-time branching random walks. We study this property by means of an infinite-dimensional generating functionGand a maximum principle which, we prove, is satisfied by every fixed point ofG. We give results for the existence of a strong local survival regime and we prove that, unlike local and global survival, in continuous time, strong local survival is not a monotone property in the general case (though it is monotone if the branching random walk is quasitransitive). We provide an example of an irreducible branching random walk where the strong local property depends on the starting site of the process. By means of other counterexamples, we show that the existence of a pure global phase is not equivalent to nonamenability of the process, and that even an irreducible branching random walk with the same branching law at each site may exhibit nonstrong local survival. Finally, we show that the generating function of an irreducible branching random walk can have more than two fixed points; this disproves a previously known result.

scholarly journals Strong Local Survival of Branching Random Walks is Not Monotone

2014 ◽  
Vol 46 (2) ◽  
pp. 400-421 ◽  
Author(s):  
Daniela Bertacchi ◽  
Fabio Zucca
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Generating Function ◽  
Continuous Time ◽  
Local Property ◽  
Branching Random Walk ◽  
Infinite Dimensional ◽  
Branching Random Walks ◽  
Local Survival ◽  
Branching Law

In this paper we study the strong local survival property for discrete-time and continuous-time branching random walks. We study this property by means of an infinite-dimensional generating function G and a maximum principle which, we prove, is satisfied by every fixed point of G. We give results for the existence of a strong local survival regime and we prove that, unlike local and global survival, in continuous time, strong local survival is not a monotone property in the general case (though it is monotone if the branching random walk is quasitransitive). We provide an example of an irreducible branching random walk where the strong local property depends on the starting site of the process. By means of other counterexamples, we show that the existence of a pure global phase is not equivalent to nonamenability of the process, and that even an irreducible branching random walk with the same branching law at each site may exhibit nonstrong local survival. Finally, we show that the generating function of an irreducible branching random walk can have more than two fixed points; this disproves a previously known result.

scholarly journals A Continuous-Time Random Walk Extension of the Gillis Model

Entropy ◽  
2020 ◽  
Vol 22 (12) ◽  
pp. 1431
Author(s):  
Gaia Pozzoli ◽  
Mattia Radice ◽  
Manuele Onofri ◽  
Roberto Artuso
Keyword(s):  
Random Walk ◽  
Continuous Time ◽  
Continuous Time Random Walk ◽  
Hitting Times ◽  
Occupation Times ◽  
Theoretical Predictions ◽  
Normal Diffusion ◽  
Heavy Tailed ◽  
The One ◽  
Waiting Periods

We consider a continuous-time random walk which is the generalization, by means of the introduction of waiting periods on sites, of the one-dimensional non-homogeneous random walk with a position-dependent drift known in the mathematical literature as Gillis random walk. This modified stochastic process allows to significantly change local, non-local and transport properties in the presence of heavy-tailed waiting-time distributions lacking the first moment: we provide here exact results concerning hitting times, first-time events, survival probabilities, occupation times, the moments spectrum and the statistics of records. Specifically, normal diffusion gives way to subdiffusion and we are witnessing the breaking of ergodicity. Furthermore we also test our theoretical predictions with numerical simulations.

scholarly journals Sample correlations of infinite variance time series models: an empirical and theoretical study

1998 ◽  
Vol 11 (3) ◽  
pp. 255-282 ◽  
Author(s):  
Jason Cohen ◽  
Sidney Resnick ◽  
Gennady Samorodnitsky
Keyword(s):  
Time Series ◽  
Correlation Function ◽  
Random Permutation ◽  
Infinite Variance ◽  
Finite Variance ◽  
More Care ◽  
Sample Correlation ◽  
Heavy Tailed ◽  
Random Limit ◽  
Permutation Scheme

When the elements of a stationary ergodic time series have finite variance the sample correlation function converges (with probability 1) to the theoretical correlation function. What happens in the case where the variance is infinite? In certain cases, the sample correlation function converges in probability to a constant, but not always. If within a class of heavy tailed time series the sample correlation functions do not converge to a constant, then more care must be taken in making inferences and in model selection on the basis of sample autocorrelations. We experimented with simulating various heavy tailed stationary sequences in an attempt to understand what causes the sample correlation function to converge or not to converge to a constant. In two new cases, namely the sum of two independent moving averages and a random permutation scheme, we are able to provide theoretical explanations for a random limit of the sample autocorrelation function as the sample grows.

Amplitudes of Composition Fluctuations in the Simplest Chemical Reaction Equilibrium

2004 ◽  
Vol 218 (9) ◽  
pp. 1033-1040 ◽  
Author(s):  
M. Šolc ◽  
J. Hostomský
Keyword(s):  
Random Walk ◽  
Discrete Time ◽  
Continuous Time ◽  
Time Scale ◽  
Numerical Study ◽  
Mean Amplitude ◽  
Time Interval ◽  
The Mean ◽  
Composition Fluctuations ◽  
Number Of Particles

AbstractWe present a numerical study of equilibrium composition fluctuations in a system where the reaction X1 ⇔ X2 having the equilibrium constant equal to 1 takes place. The total number of reacting particles is N. On a discrete time scale, the amplitude of a fluctuation having the lifetime 2r reaction events is defined as the difference between the number of particles X1 in the microstate most distant from the microstate N/2 visited at least once during the fluctuation lifetime, and the equilibrium number of particles X1, N/2. On the discrete time scale, the mean value of this amplitude, m̅(r̅), is calculated in the random walk approximation. On a continuous time scale, the average amplitude of fluctuations chosen randomly and regardless of their lifetime from an ensemble of fluctuations occurring within the time interval (0,z), z → ∞, tends with increasing N to ~1.243 N0.25. Introducing a fraction of fluctuation lifetime during which the composition of the system spends below the mean amplitude m̅(r̅), we obtain a value of the mean amplitude of equilibrium fluctuations on the continuous time scale equal to ~1.19√N. The results suggest that using the random walk value m̅(r̅) and taking into account a) the exponential density of fluctuations lifetimes and b) the fact that the time sequence of reaction events represents the Poisson process, we obtain values of fluctuations amplitudes which differ only slightly from those derived for the Ehrenfest model.

scholarly journals On a random-coefficient AR(1) process with heavy-tailed renewal switching coefficient and heavy-tailed noise

2006 ◽  
Vol 43 (02) ◽  
pp. 421-440
Author(s):  
Remigijus Leipus ◽  
Vygantas Paulauskas ◽  
Donatas Surgailis
Keyword(s):  
Unit Root ◽  
Domain Of Attraction ◽  
Stationary Solutions ◽  
Partial Sums ◽  
Infinite Variance ◽  
Finite Variance ◽  
Stable Law ◽  
Stable Lévy Process ◽  
Heavy Tailed ◽  
Independent Identically Distributed

We discuss the limit behavior of the partial sums process of stationary solutions to the (autoregressive) AR(1) equation X t = a t X t−1 + ε t with random (renewal-reward) coefficient, a t , taking independent, identically distributed values A j ∈ [0,1] on consecutive intervals of a stationary renewal process with heavy-tailed interrenewal distribution, and independent, identically distributed innovations, ε t , belonging to the domain of attraction of an α-stable law (0 &lt; α ≤ 2, α ≠ 1). Under suitable conditions on the tail parameter of the interrenewal distribution and the singularity parameter of the distribution of A j near the unit root a = 1, we show that the partial sums process of X t converges to a λ-stable Lévy process with index λ &lt; α. The paper extends the result of Leipus and Surgailis (2003) from the case of finite-variance X t to that of infinite-variance X t .

Nonuniversality of weighted random graphs with infinite variance degree

2017 ◽  
Vol 54 (1) ◽  
pp. 146-164 ◽  
Author(s):  
Enrico Baroni ◽  
Remco van der Hofstad ◽  
Júlia Komjáthy
Keyword(s):  
Continuous Time ◽  
Degree Sequence ◽  
Optimal Path ◽  
Configuration Model ◽  
Infinite Variance ◽  
Finite Variance ◽  
First Passage Percolation ◽  
First Passage ◽  
Continuous Time Processes ◽  
Graph Distances

AbstractWe prove nonuniversality results for first-passage percolation on the configuration model with independent and identically distributed (i.i.d.) degrees having infinite variance. We focus on the weight of the optimal path between two uniform vertices. Depending on the properties of the weight distribution, we use an example-based approach and show that rather different behaviours are possible. When the weights are almost surely larger than a constant, the weight and number of edges in the graph grow proportionally to log log n, as for the graph distances. On the other hand, when the continuous-time branching process describing the first-passage percolation exploration through the graph reaches infinitely many vertices in finite time, the weight converges to the sum of two i.i.d. random variables representing the explosion times of the continuous-time processes started from the two sources. This nonuniversality is in sharp contrast to the setting where the degree sequence has a finite variance, Bhamidi et al. (2012).

A NOTE ON THE LSE OF THREE-REGIME TAR MODEL WITH AN INFINITE VARIANCE

2018 ◽  
Vol 13 (02) ◽  
pp. 1850007
Author(s):  
YAXING YANG ◽  
SHIQING LING
Keyword(s):  
Asymptotic Behavior ◽  
Model Fitting ◽  
Simulation Method ◽  
Least Square ◽  
Theoretical Research ◽  
Infinite Variance ◽  
Finite Variance ◽  
Least Square Estimator ◽  
Heavy Tailed ◽  
Fitting In

The least square estimator (LSE) of three-regime TAR models has been well developed when the noise has a finite variance. However, little is known about the asymptotic behavior of the LSE of three-regime threshold autoregressive (TAR) models in the heavy-tailed setting and neither theory nor methodology is available for model fitting in this case. In this paper, we use simulation method to study the asymptotic behavior of the LSE of infinite variance three-regime TAR models. A real example is given for the NASDAQ daily volume from the year 2012 to 2014. The results in this paper may provide an empirical insight for further theoretical research of the LSE of three-regime TAR models with heavy-tailed innovations.

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