scholarly journals Large deviation estimates for branching random walks

2019 ◽  
Vol 23 ◽  
pp. 823-840
Author(s):  
Dariusz Buraczewski ◽  
Mariusz Maślanka
Keyword(s):  
Random Walk ◽  
Law Of Large Numbers ◽  
Large Deviation ◽  
First Passage Time ◽  
Passage Time ◽  
Branching Random Walk ◽  
First Passage ◽  
Branching Random Walks ◽  
The Central Limit Theorem ◽  
Large Numbers

For the branching random walk drifting to −∞ we study large deviations-type estimates for the first passage time. We prove the corresponding law of large numbers and the central limit theorem.

scholarly journals Analysis of random walks on a hexagonal lattice

2019 ◽  
Author(s):  
Antonio Di Crescenzo ◽  
Claudio Macci ◽  
Barbara Martinucci ◽  
Serena Spina
Keyword(s):  
Random Walk ◽  
Large Deviation ◽  
Transition Probabilities ◽  
First Passage Time ◽  
Hexagonal Lattice ◽  
Passage Time ◽  
First Passage ◽  
Time Problem ◽  
First Passage Time Problem ◽  
Applied Fields

Abstract We consider a discrete-time random walk on the nodes of an unbounded hexagonal lattice. We determine the probability generating functions, the transition probabilities and the relevant moments. The convergence of the stochastic process to a two-dimensional Brownian motion is also discussed. Furthermore, we obtain some results on its asymptotic behaviour making use of large deviation theory. Finally, we investigate the first-passage-time problem of the random walk through a vertical straight line. Under suitable symmetry assumptions, we are able to determine the first-passage-time probabilities in a closed form, which deserve interest in applied fields.

scholarly journals Tracking a random walk first-passage time through noisy observations

2012 ◽  
Vol 22 (5) ◽  
pp. 1860-1879 ◽  
Author(s):  
Marat V. Burnashev ◽  
Aslan Tchamkerten
Keyword(s):  
Random Walk ◽  
First Passage Time ◽  
Passage Time ◽  
First Passage

scholarly journals Fractional compound Poisson processes with multiple internal states

2018 ◽  
Vol 13 (1) ◽  
pp. 10 ◽  
Author(s):  
Pengbo Xu ◽  
Weihua Deng
Keyword(s):  
Stochastic Process ◽  
Random Walk ◽  
Waiting Time ◽  
Anomalous Diffusion ◽  
First Passage Time ◽  
Passage Time ◽  
Poisson Processes ◽  
First Passage ◽  
Internal States ◽  
Functional Distribution

For the particles undergoing the anomalous diffusion with different waiting time distributions for different internal states, we derive the Fokker-Planck and Feymann-Kac equations, respectively, describing positions of the particles and functional distributions of the trajectories of particles; in particular, the equations governing the functional distribution of internal states are also obtained. The dynamics of the stochastic processes are analyzed and the applications, calculating the distribution of the first passage time and the distribution of the fraction of the occupation time, of the equations are given. For the further application of the newly built models, we make very detailed discussions on the none-immediately-repeated stochastic process, e.g., the random walk of smart animals.

Random walk and first passage time on a weighted hierarchical network

2014 ◽  
Vol 25 (09) ◽  
pp. 1450037 ◽  
Author(s):  
Feng Zhu ◽  
Meifeng Dai ◽  
Yujuan Dong ◽  
Jie Liu
Keyword(s):  
Random Walk ◽  
First Passage Time ◽  
Passage Time ◽  
Scale Factor ◽  
Hierarchical Network ◽  
Self Similarity ◽  
First Passage ◽  
Hierarchical Networks ◽  
Average Value ◽  
Peripheral Node

This paper reports a weighted hierarchical network generated on the basis of self-similarity, in which each edge is assigned a different weight in the same scale. We studied two substantial properties of random walk: the first-passage time (FPT) between a hub node and a peripheral node and the FPT from a peripheral node to a local hub node over the network. Meanwhile, an analytical expression of the average sending time (AST) is deduced, which reflects the average value of FPT from a hub node to any other node. Our result shows that the AST from a hub node to any other node is related to the scale factor and the number of modules. We found that the AST grows sublinearly, linearly and superlinearly respectively with the network order, depending on the range of the scale factor. Our work may shed some light on revealing the diffusion process in hierarchical networks.

scholarly journals Tools to Estimate the First Passage Time to a Convex Barrier

2005 ◽  
Vol 42 (1) ◽  
pp. 61-81
Author(s):  
Ola Hammarlid
Keyword(s):  
Limit Distribution ◽  
Sequential Analysis ◽  
Large Deviation ◽  
First Passage Time ◽  
Stopping Time ◽  
Random Variable ◽  
Passage Time ◽  
First Passage ◽  
Linear Barrier ◽  
Asymptotic Normal

The first passage time of a random walk to a barrier (constant or concave) is of great importance in many areas, such as insurance, finance, and sequential analysis. Here, we consider a sum of independent, identically distributed random variables and the convex barrier cb(n/c), where c is a scale parameter and n is time. It is shown, using large-deviation techniques, that the limit distribution of the first passage time decays exponentially in c. Under a tilt of measure, which changes the drift, four properties are proved: the limit distribution of the overshoot is distributed as an overshoot over a linear barrier; the stopping time is asymptotically normally distributed when it is properly normalized; the overshoot and the asymptotic normal part are asymptotically independent; and the overshoot over a linear barrier is bounded by an exponentially distributed random variable. The determination of the function that multiplies the exponential part is guided by consideration of these properties.

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