scholarly journals On randomly spaced observations and continuous-time random walks

2016 ◽  
Vol 53 (3) ◽  
pp. 888-898
Author(s):  
Bojan Basrak ◽  
Drago Špoljarić
Keyword(s):  
Random Walks ◽  
Asymptotic Distribution ◽  
Renewal Process ◽  
Continuous Time ◽  
Random Variables ◽  
Limiting Behavior ◽  
Arrival Times ◽  
Sojourn Times ◽  
Continuous Time Random Walks ◽  
Heavy Tailed

AbstractWe consider random variables observed at arrival times of a renewal process, which possibly depends on those observations and has regularly varying steps with infinite mean. Due to the dependence and heavy-tailed steps, the limiting behavior of extreme observations until a given time t tends to be rather involved. We describe the asymptotics and extend several partial results which appeared in this setting. The theory is applied to determine the asymptotic distribution of maximal excursions and sojourn times for continuous-time random walks.

scholarly journals Chover-Type Laws of the Iterated Logarithm for Continuous Time Random Walks

2012 ◽  
Vol 2012 ◽  
pp. 1-13
Author(s):  
Kyo-Shin Hwang ◽  
Wensheng Wang
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Anomalous Diffusion ◽  
Renewal Process ◽  
Continuous Time ◽  
Waiting Times ◽  
Continuous Time Random Walk ◽  
Domains Of Attraction ◽  
Iterated Logarithm ◽  
Continuous Time Random Walks

A continuous time random walk is a random walk subordinated to a renewal process used in physics to model anomalous diffusion. In this paper, we establish Chover-type laws of the iterated logarithm for continuous time random walks with jumps and waiting times in the domains of attraction of stable laws.

Heterogeneous Memorized Continuous Time Random Walks in an External Force Fields

2014 ◽  
Vol 156 (6) ◽  
pp. 1111-1124 ◽  
Author(s):  
Jun Wang ◽  
Ji Zhou ◽  
Long-Jin Lv ◽  
Wei-Yuan Qiu ◽  
Fu-Yao Ren
Keyword(s):  
Random Walks ◽  
External Force ◽  
Continuous Time ◽  
Force Fields ◽  
Continuous Time Random Walks

scholarly journals Clustered continuous-time random walks: diffusion and relaxation consequences

2012 ◽  
Vol 468 (2142) ◽  
pp. 1615-1628 ◽  
Author(s):  
Karina Weron ◽  
Aleksander Stanislavsky ◽  
Agnieszka Jurlewicz ◽  
Mark M. Meerschaert ◽  
Hans-Peter Scheffler
Keyword(s):  
Random Walks ◽  
Power Law ◽  
Continuous Time ◽  
Waiting Times ◽  
Scaling Limit ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
The Novel ◽  
Fractional Diffusion Equations ◽  
Continuous Time Random Walks

We present a class of continuous-time random walks (CTRWs), in which random jumps are separated by random waiting times. The novel feature of these CTRWs is that the jumps are clustered. This introduces a coupled effect, with longer waiting times separating larger jump clusters. We show that the CTRW scaling limits are time-changed processes. Their densities solve two different fractional diffusion equations, depending on whether the waiting time is coupled to the preceding jump, or the following one. These fractional diffusion equations can be used to model all types of experimentally observed two power-law relaxation patterns. The parameters of the scaling limit process determine the power-law exponents and loss peak frequencies.

scholarly journals Limit theorems for some continuous-time random walks

2011 ◽  
Vol 43 (3) ◽  
pp. 782-813 ◽  
Author(s):  
M. Jara ◽  
T. Komorowski
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Random Walks ◽  
Quantum Transport ◽  
Continuous Time ◽  
Limit Theorems ◽  
Transport Theory ◽  
Sufficient Conditions ◽  
Continuous Time Random Walks ◽  
Quantum Transport Theory

In this paper we consider the scaled limit of a continuous-time random walk (CTRW) based on a Markov chain {Xn,n≥ 0} and two observables, τ(∙) andV(∙), corresponding to the renewal times and jump sizes. Assuming that these observables belong to the domains of attraction of some stable laws, we give sufficient conditions on the chain that guarantee the existence of the scaled limits for CTRWs. An application of the results to a process that arises in quantum transport theory is provided. The results obtained in this paper generalize earlier results contained in Becker-Kern, Meerschaert and Scheffler (2004) and Meerschaert and Scheffler (2008), and the recent results of Henry and Straka (2011) and Jurlewicz, Kern, Meerschaert and Scheffler (2010), where {Xn,n≥ 0} is a sequence of independent and identically distributed random variables.

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