Multidimensional Random Walks Conditioned to Stay Ordered via Generalized Ladder Height Functions

2020 ◽  
pp. 39-65
Author(s):  
Osvaldo Angtuncio-Hernández
Keyword(s):  
Random Walks ◽  
Height Functions ◽  
Ladder Height

Stationarity properties of neural networks

1998 ◽  
Vol 35 (04) ◽  
pp. 783-794
Author(s):  
Søren Asmussen ◽  
Tatyana S. Turova
Keyword(s):  
Neural Networks ◽  
Random Walks ◽  
Renewal Process ◽  
Random Variable ◽  
Neural Model ◽  
Symmetric Case ◽  
Interarrival Time ◽  
Renewal Processes ◽  
Firing Time ◽  
Ladder Height

A neural model with N interacting neurons is considered. A firing of neuron i delays the firing times of all other neurons by the same random variable θ(i), and in isolation the firings of the neuron occur according to a renewal process with generic interarrival time Y (i). The stationary distribution of the N-vector of inhibitions at a firing time is computed, and involves waiting distributions of GI/G/1 queues and ladder height renewal processes. Further, the distribution of the period of activity of a neuron is studied for the symmetric case where θ(i) and Y (i) do not depend upon i. The tools are probabilistic and involve path decompositions, Palm theory and random walks.

Tail behaviour of ladder-height distributions in random walks

1985 ◽  
Vol 22 (3) ◽  
pp. 705-709 ◽  
Author(s):  
Rudolf Grübel
Keyword(s):  
Random Walks ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Ladder Height ◽  
Tail Behaviour ◽  
Necessary And Sufficient

We give necessary and sufficient conditions for various results connecting the tail behaviour of a distribution with that of its right Wiener–Hopf factor.

scholarly journals Asymptotic expansions on moments of the first ladder height in Markov random walks with small drift

2007 ◽  
Vol 39 (3) ◽  
pp. 826-852 ◽  
Author(s):  
Cheng-Der Fuh
Keyword(s):  
Random Walks ◽  
Asymptotic Expansions ◽  
First Passage Time ◽  
Passage Time ◽  
Boundary Crossing ◽  
Renewal Theorem ◽  
Ladder Height ◽  
Markov Random ◽  
Markov Random Walk ◽  
General State Space

Let {(Xn, Sn), n ≥ 0} be a Markov random walk in which Xn takes values in a general state space and Sn takes values on the real line R. In this paper we present some results that are useful in the study of asymptotic approximations of boundary crossing problems for Markov random walks. The main results are asymptotic expansions on moments of the first ladder height in Markov random walks with small positive drift. In order to establish the asymptotic expansions we study a uniform Markov renewal theorem, which relates to the rate of convergence for the distribution of overshoot, and present an analysis of the covariance between the first passage time and the overshoot.

scholarly journals Asymptotic expansions on moments of the first ladder height in Markov random walks with small drift

2007 ◽  
Vol 39 (03) ◽  
pp. 826-852 ◽  
Author(s):  
Cheng-Der Fuh
Keyword(s):  
Random Walks ◽  
Asymptotic Expansions ◽  
First Passage Time ◽  
Passage Time ◽  
Boundary Crossing ◽  
Renewal Theorem ◽  
Ladder Height ◽  
Markov Random ◽  
Markov Random Walk ◽  
General State Space

Let {(X n , S n ), n ≥ 0} be a Markov random walk in which X n takes values in a general state space and S n takes values on the real line R. In this paper we present some results that are useful in the study of asymptotic approximations of boundary crossing problems for Markov random walks. The main results are asymptotic expansions on moments of the first ladder height in Markov random walks with small positive drift. In order to establish the asymptotic expansions we study a uniform Markov renewal theorem, which relates to the rate of convergence for the distribution of overshoot, and present an analysis of the covariance between the first passage time and the overshoot.

Tail behaviour of ladder-height distributions in random walks

1985 ◽  
Vol 22 (03) ◽  
pp. 705-709 ◽  
Author(s):  
Rudolf Grübel
Keyword(s):  
Random Walks ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Ladder Height ◽  
Tail Behaviour ◽  
Necessary And Sufficient

We give necessary and sufficient conditions for various results connecting the tail behaviour of a distribution with that of its right Wiener–Hopf factor.

Stationarity properties of neural networks

1998 ◽  
Vol 35 (4) ◽  
pp. 783-794 ◽  
Author(s):  
Søren Asmussen ◽  
Tatyana S. Turova
Keyword(s):  
Neural Networks ◽  
Random Walks ◽  
Renewal Process ◽  
Random Variable ◽  
Neural Model ◽  
Symmetric Case ◽  
Interarrival Time ◽  
Renewal Processes ◽  
Firing Time ◽  
Ladder Height

A neural model with N interacting neurons is considered. A firing of neuron i delays the firing times of all other neurons by the same random variable θ(i), and in isolation the firings of the neuron occur according to a renewal process with generic interarrival time Y(i). The stationary distribution of the N-vector of inhibitions at a firing time is computed, and involves waiting distributions of GI/G/1 queues and ladder height renewal processes. Further, the distribution of the period of activity of a neuron is studied for the symmetric case where θ(i) and Y(i) do not depend upon i. The tools are probabilistic and involve path decompositions, Palm theory and random walks.

Non-homogeneous Random Walks

2016 ◽  
Author(s):  
Mikhail Menshikov ◽  
Serguei Popov ◽  
Andrew Wade
Keyword(s):  
Random Walks
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