scholarly journals Spectral Analysis of Markov Kernels and Application to the Convergence Rate Of Discrete Random Walks

2014 ◽  
Vol 46 (4) ◽  
pp. 1036-1058
Author(s):  
Loïc Hervé ◽  
James Ledoux
Keyword(s):  
Spectral Analysis ◽  
Random Walks ◽  
General Procedure ◽  
Bounded Operator ◽  
Invariant Probability Measure ◽  
Measurable Space ◽  
Transition Kernel ◽  
Linear Bounded Operator ◽  
Markov Random ◽  
Precise Analysis

Let {Xn}n∈ℕ be a Markov chain on a measurable space with transition kernel P, and let The Markov kernel P is here considered as a linear bounded operator on the weighted-supremum space associated with V. Then the combination of quasicompactness arguments with precise analysis of eigenelements of P allows us to estimate the geometric rate of convergence ρV(P) of {Xn}n∈ℕ to its invariant probability measure in operator norm on A general procedure to compute ρV(P) for discrete Markov random walks with identically distributed bounded increments is specified.

scholarly journals Spectral Analysis of Markov Kernels and Application to the Convergence Rate Of Discrete Random Walks

2014 ◽  
Vol 46 (04) ◽  
pp. 1036-1058 ◽  
Author(s):  
Loïc Hervé ◽  
James Ledoux
Keyword(s):  
Spectral Analysis ◽  
Random Walks ◽  
General Procedure ◽  
Bounded Operator ◽  
Invariant Probability Measure ◽  
Measurable Space ◽  
Transition Kernel ◽  
Linear Bounded Operator ◽  
Markov Random ◽  
Precise Analysis

Let {Xn}n∈ℕbe a Markov chain on a measurable spacewith transition kernelP, and letThe Markov kernelPis here considered as a linear bounded operator on the weighted-supremum spaceassociated withV. Then the combination of quasicompactness arguments with precise analysis of eigenelements ofPallows us to estimate the geometric rate of convergence ρV(P) of {Xn}n∈ℕto its invariant probability measure in operator norm onA general procedure to compute ρV(P) for discrete Markov random walks with identically distributed bounded increments is specified.

Certain boundary functionals for Markov random walks

1978 ◽  
Vol 23 (2) ◽  
pp. 169-175
Author(s):  
V. A. Ivanov ◽  
G. I. Ivchenko
Keyword(s):  
Random Walks ◽  
Markov Random ◽  
Boundary Functionals

scholarly journals Rare-Event Simulation of Heavy-Tailed Random Walks by Sequential Importance Sampling and Resampling

2012 ◽  
Vol 44 (04) ◽  
pp. 1173-1196
Author(s):  
Hock Peng Chan ◽  
Shaojie Deng ◽  
Tze-Leung Lai
Keyword(s):  
Random Walks ◽  
Importance Sampling ◽  
Rare Event ◽  
Sequential Importance Sampling ◽  
New Approach ◽  
Martingale Representation ◽  
Event Simulation ◽  
Markov Random ◽  
Heavy Tailed ◽  
Asymptotically Efficient

We introduce a new approach to simulating rare events for Markov random walks with heavy-tailed increments. This approach involves sequential importance sampling and resampling, and uses a martingale representation of the corresponding estimate of the rare-event probability to show that it is unbiased and to bound its variance. By choosing the importance measures and resampling weights suitably, it is shown how this approach can yield asymptotically efficient Monte Carlo estimates.

scholarly journals More on Inequalities for Weaving Frames in Hilbert Spaces

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 141 ◽  
Author(s):  
Zhong-Qi Xiang
Keyword(s):  
Hilbert Spaces ◽  
Operator Theory ◽  
Triangle Inequality ◽  
Bounded Operator ◽  
Point Of View ◽  
Real Numbers ◽  
Linear Bounded Operator ◽  
Bessel Sequences ◽  
New Inequalities

In this paper, we present several new inequalities for weaving frames in Hilbert spaces from the point of view of operator theory, which are related to a linear bounded operator induced by three Bessel sequences and a scalar in the set of real numbers. It is indicated that our results are more general and cover the corresponding results recently obtained by Li and Leng. We also give a triangle inequality for weaving frames in Hilbert spaces, which is structurally different from previous ones.

Asymptotic expansions in multidimensional Markov renewal theory and first passage times for Markov random walks

2001 ◽  
Vol 33 (3) ◽  
pp. 652-673 ◽  
Author(s):  
Cheng-Der Fuh ◽  
Tze Leung Lai
Keyword(s):  
Random Walks ◽  
First Passage Time ◽  
Renewal Theory ◽  
Passage Time ◽  
Boundary Crossing ◽  
High Threshold ◽  
Renewal Theorem ◽  
First Passage ◽  
Markov Random ◽  
General State Space

We prove a d-dimensional renewal theorem, with an estimate on the rate of convergence, for Markov random walks. This result is applied to a variety of boundary crossing problems for a Markov random walk (Xn,Sn), n ≥0, in which Xn takes values in a general state space and Sn takes values in ℝd. In particular, for the case d = 1, we use this result to derive an asymptotic formula for the variance of the first passage time when Sn exceeds a high threshold b, generalizing Smith's classical formula in the case of i.i.d. positive increments for Sn. For d > 1, we apply this result to derive an asymptotic expansion of the distribution of (XT,ST), where T = inf { n : Sn,1 > b } and Sn,1 denotes the first component of Sn.

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