First-passage percolation processes with finite height

1985 ◽  
Vol 22 (4) ◽  
pp. 766-775
Author(s):  
Norbert Herrndorf
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
First Passage Percolation ◽  
Horizontal Strip ◽  
First Passage ◽  
Time Constants ◽  
Finite Height ◽  
Special Cases ◽  
Percolation Processes ◽  
Passage Times

We consider first-passage percolation in an infinite horizontal strip of finite height. Using methods from the theory of Markov chains, we prove a central limit theorem for first-passage times, and compute the time constants for some special cases.

First-passage percolation processes with finite height

1985 ◽  
Vol 22 (04) ◽  
pp. 766-775
Author(s):  
Norbert Herrndorf
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
First Passage Percolation ◽  
Horizontal Strip ◽  
First Passage ◽  
Time Constants ◽  
Finite Height ◽  
Special Cases ◽  
Percolation Processes ◽  
Passage Times

We consider first-passage percolation in an infinite horizontal strip of finite height. Using methods from the theory of Markov chains, we prove a central limit theorem for first-passage times, and compute the time constants for some special cases.

scholarly journals On the Markov Transition Kernels for First Passage Percolation on the Ladder

2011 ◽  
Vol 48 (02) ◽  
pp. 366-388 ◽  
Author(s):  
Eckhard Schlemm
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Asymptotic Variance ◽  
Almost Sure Convergence ◽  
First Passage Percolation ◽  
First Passage ◽  
Percolation Problem ◽  
Vertex Set ◽  
Step Transition

We consider the first passage percolation problem on the random graph with vertex set N x {0, 1}, edges joining vertices at a Euclidean distance equal to unity, and independent exponential edge weights. We provide a central limit theorem for the first passage times l n between the vertices (0, 0) and (n, 0), thus extending earlier results about the almost-sure convergence of l n / n as n → ∞. We use generating function techniques to compute the n-step transition kernels of a closely related Markov chain which can be used to explicitly calculate the asymptotic variance in the central limit theorem.

The exact order of normal approximation in bivariate renewal theory

1981 ◽  
Vol 13 (01) ◽  
pp. 113-128 ◽  
Author(s):  
Ibrahim A. Ahmad
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Normal Approximation ◽  
Renewal Theory ◽  
Rates Of Convergence ◽  
Partial Sums ◽  
Exact Order ◽  
First Passage ◽  
The Central Limit Theorem ◽  
Passage Times

Equivalence of rates of convergence in the central limit theorem between the vector of maximum sums and the corresponding first-passage variables is established. The bivariate case is studied. Analogous results about the equivalence between the vector of partial sums and corresponding renewal variables are also given and as a consequence we obtain a generalization of a theorem of Hunter (1974). Extension of the main result to more general first-passage times is also developed.

scholarly journals On the Markov Transition Kernels for First Passage Percolation on the Ladder

2011 ◽  
Vol 48 (2) ◽  
pp. 366-388 ◽  
Author(s):  
Eckhard Schlemm
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Asymptotic Variance ◽  
Almost Sure Convergence ◽  
First Passage Percolation ◽  
First Passage ◽  
Percolation Problem ◽  
Vertex Set ◽  
Step Transition

We consider the first passage percolation problem on the random graph with vertex set N x {0, 1}, edges joining vertices at a Euclidean distance equal to unity, and independent exponential edge weights. We provide a central limit theorem for the first passage times ln between the vertices (0, 0) and (n, 0), thus extending earlier results about the almost-sure convergence of ln / n as n → ∞. We use generating function techniques to compute the n-step transition kernels of a closely related Markov chain which can be used to explicitly calculate the asymptotic variance in the central limit theorem.

The exact order of normal approximation in bivariate renewal theory

1981 ◽  
Vol 13 (1) ◽  
pp. 113-128 ◽  
Author(s):  
Ibrahim A. Ahmad
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Normal Approximation ◽  
Renewal Theory ◽  
Rates Of Convergence ◽  
Partial Sums ◽  
Exact Order ◽  
First Passage ◽  
The Central Limit Theorem ◽  
Passage Times

Equivalence of rates of convergence in the central limit theorem between the vector of maximum sums and the corresponding first-passage variables is established. The bivariate case is studied. Analogous results about the equivalence between the vector of partial sums and corresponding renewal variables are also given and as a consequence we obtain a generalization of a theorem of Hunter (1974). Extension of the main result to more general first-passage times is also developed.

On the exact order of normal approximation in multivariate renewal theory

1985 ◽  
Vol 22 (02) ◽  
pp. 280-287 ◽  
Author(s):  
Ştefan P. Niculescu ◽  
Edward Omey
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Normal Approximation ◽  
Renewal Theory ◽  
Rates Of Convergence ◽  
Partial Sums ◽  
Exact Order ◽  
First Passage ◽  
The Central Limit Theorem

Equivalence of rates of convergence in the central limit theorem for the vector of maximum sums and the corresponding first-passage variables is established. A similar result for the vector of partial sums and the corresponding renewal variables is also given. The results extend to several dimensions the bivariate results of Ahmad (1981).

scholarly journals First Passage Times for Markov Additive Processes with Positive Jumps of Phase Type

2008 ◽  
Vol 45 (03) ◽  
pp. 779-799 ◽  
Author(s):  
Lothar Breuer
Keyword(s):  
Inverse Function ◽  
First Passage Times ◽  
Phase Type Distribution ◽  
First Passage ◽  
Additive Processes ◽  
Special Cases ◽  
Phase Type ◽  
Spectrally Negative Lévy Processes ◽  
Passage Times ◽  
Markov Additive Processes

The present paper generalises some results for spectrally negative Lévy processes to the setting of Markov additive processes (MAPs). A prominent role is assumed by the first passage times, which will be determined in terms of their Laplace transforms. These have the form of a phase-type distribution, with a rate matrix that can be regarded as an inverse function of the cumulant matrix. A numerically stable iteration to compute this matrix is given. The theory is first developed for MAPs without positive jumps and then extended to include positive jumps having phase-type distributions. Numerical and analytical examples show agreement with existing results in special cases.

An invariance principle in k-dimensional extended renewal theory

1979 ◽  
Vol 16 (3) ◽  
pp. 567-574 ◽  
Author(s):  
Attila Csenki
Keyword(s):  
Limit Theorem ◽  
Weak Convergence ◽  
Covariance Matrix ◽  
Invariance Principle ◽  
Exit Time ◽  
Renewal Theory ◽  
Random Vectors ◽  
First Exit Time ◽  
First Passage ◽  
Passage Times

Let ·be a sequence of k -dimensional i.i.d. random vectors and define the first-passage times for where (cvτ)v, τ= 1,· ··,k is the covariance matrix of In this paper the weak convergence of Zn in (D[0, ∞))k is proved under the assumption (0,∞) for all v = 1, ···, k. We deduce the result from the Donsker invariance principle by means of Theorem 5.5 of Billingsley (1968). This method is also used to derive a limit theorem for the first-exit time Mn = min{Nnt for fixed t1,···, tk > 0. The second result is an extension of a theorem of Hunter (1974) whose method of proof applies only if Ρ (ξ1 [0,∞)k) = 1 and μ ν = tv for all v = 1, ···, k.

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