A limit theorem for random walks with drift

1967 ◽  
Vol 4 (01) ◽  
pp. 144-150 ◽  
Author(s):  
C. C. Heyde
Keyword(s):  
Random Walk ◽  
Limit Theorem ◽  
Random Walks ◽  
First Passage Time ◽  
Random Variables ◽  
Passage Time ◽  
Random Walk Process ◽  
First Passage ◽  
Time Out ◽  
Image Position

Let Xi , i = 1, 2, 3, … be a sequence of independent and identically distributed random variables. Write and for x ≧ 0 define M(x) + 1 is then the first passage time out of the interval (– ∞, x] for the random walk process Sn.

A limit theorem for random walks with drift

1967 ◽  
Vol 4 (1) ◽  
pp. 144-150 ◽  
Author(s):  
C. C. Heyde
Keyword(s):  
Random Walk ◽  
Limit Theorem ◽  
Random Walks ◽  
First Passage Time ◽  
Random Variables ◽  
Passage Time ◽  
Random Walk Process ◽  
First Passage ◽  
Time Out ◽  
Image Position

Let Xi, i = 1, 2, 3, … be a sequence of independent and identically distributed random variables. Write and for x ≧ 0 define M(x) + 1 is then the first passage time out of the interval (– ∞, x] for the random walk process Sn.

Invariance Theorems for First Passage Time Random Variables

1972 ◽  
Vol 15 (2) ◽  
pp. 171-176 ◽  
Author(s):  
A. K. Basu
Keyword(s):  
First Passage Time ◽  
Random Variables ◽  
Passage Time ◽  
Wiener Measure ◽  
First Passage ◽  
Image Position

Let X1X2,… be i.i.d. r.v. with EX=μ>0, and E(X-μ)2 = σ2<∞.Let Sk=X1+…+Xk and vx=max{k:Sk≤x}, x≥0 and vx=0 if X1>x. Billingsley [1] proved if X1≥0 thenconverges weakly to the Wiener measure W.Let τx(ω)=inf{k≥1|Sk>x}. In §2 we prove thatconverges weakly to the Wiener measure when the X's may not necessarily be nonnegative. Also we indicate that this result can be extended to the nonidentical case.

Calculations of first passage time of delayed tree-like networks

2015 ◽  
Vol 29 (28) ◽  
pp. 1550200
Author(s):  
Shuai Wang ◽  
Weigang Sun ◽  
Song Zheng
Keyword(s):  
Random Walks ◽  
First Passage Time ◽  
Passage Time ◽  
The Novel ◽  
Mean First Passage Time ◽  
First Passage ◽  
Initial Node ◽  
Network Parameters ◽  
Self Similar ◽  
The Mean

In this paper, we study random walks in a family of delayed tree-like networks controlled by two network parameters, where an immobile trap is located at the initial node. The novel feature of this family of networks is that the existing nodes have a time delay to give birth to new nodes. By the self-similar network structure, we obtain exact solutions of three types of first passage time (FPT) measuring the efficiency of random walks, which includes the mean receiving time (MRT), mean sending time (MST) and mean first passage time (MFPT). The obtained results show that the MRT, MST and MFPT increase with the network parameters. We further show that the values of MRT, MST and MFPT are much shorter than the nondelayed counterpart, implying that the efficiency of random walks in delayed trees is much higher.

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.

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