THE ASYMPTOTIC APPROXIMATION OF THE TRANSITION PROBABILITY FOR THE RANDOM WALKS WITH HEAVY TAILS.

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Agbor, A. (2013). THE ASYMPTOTIC APPROXIMATION OF THE TRANSITION PROBABILITY FOR THE RANDOM WALKS WITH HEAVY TAILS. Unc Charlotte Electronic Theses And Dissertations.
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Abstract

The main result of this dissertation concerns the asymptotics, uniform in t and x, of the probability distribution of a random walk with heavy tails. The random walk is a Markov process and thus can be characterized in terms of their generators. We impose certain conditions on the Fourier transform of the kernel of the generator, which still allow us to consider rather general class of processes on Zd. The process we consider can be viewed as a generalization of the simple symmetric walk (in continuous time) for which both the central limit theorem and large deviation results are well-known. For problems with heavy tails, the analogue of the central limit theorem is the convergence of the properly normalized process to the stable laws. In terms of probability densities, these limit theorems give the asymptotics of p(t,x,0) when x is of order t1/á. For the class of random walk under consideration, we obtain the asymptotics of p(t,x,0) uniformly in t and x for all t>1, x ϵ Rd, covering, in particular, the regime of the central limit theorem and large deviations.

Details
Author

Agbor, Andu

Title

THE ASYMPTOTIC APPROXIMATION OF THE TRANSITION PROBABILITY FOR THE RANDOM WALKS WITH HEAVY TAILS.

Physical Description

1 online resource (71 pages) : PDF

Date

2013

Degree Granting Institution

University of North Carolina at Charlotte

Abstract

The main result of this dissertation concerns the asymptotics, uniform in t and x, of the probability distribution of a random walk with heavy tails. The random walk is a Markov process and thus can be characterized in terms of their generators. We impose certain conditions on the Fourier transform of the kernel of the generator, which still allow us to consider rather general class of processes on Zd. The process we consider can be viewed as a generalization of the simple symmetric walk (in continuous time) for which both the central limit theorem and large deviation results are well-known. For problems with heavy tails, the analogue of the central limit theorem is the convergence of the properly normalized process to the stable laws. In terms of probability densities, these limit theorems give the asymptotics of p(t,x,0) when x is of order t1/á. For the class of random walk under consideration, we obtain the asymptotics of p(t,x,0) uniformly in t and x for all t>1, x ϵ Rd, covering, in particular, the regime of the central limit theorem and large deviations.

Genre

doctoral dissertations

Subjects--Topics

Statistics
Mathematics

Degree

Ph.D.

Keywords

Asymptotics Approximation
Central Limit Theorem
Heavy Tails
Random Walk
Stable Laws
Transition Probability

Subject Area

Applied Mathematics

Advisor(s)

Vainberg, Boris

Committee Members

Molchanov, Stanislav
Safronov, Oleg
Steck, Todd

Degree Note

Thesis (Ph.D.)--University of North Carolina at Charlotte, 2013.

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Rights Holder Information

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Identifier

Agbor_uncc_0694D_10430

Permalink

http://hdl.handle.net/20.500.13093/etd:1342