Matrix-Exponential Distributions in Applied Probability by Mogens Bladt & Bo Friis Nielsen

Author:Mogens Bladt & Bo Friis Nielsen
Language: eng
Format: epub
Publisher: Springer US, Boston, MA

6. Random Walks

Mogens Bladt1 and Bo Friis Nielsen2

(1)Institute for Applied Mathematics (IIMAS), Universidad Nacional Autónoma de México, Coyoacan, Mexico

(2)Department of Applied Mathematics and Computer Science, Technical University of Denmark, Kgs. Lyngby, Denmark

By a random walk on we understand a stochastic process in discrete time for which S 0 = 0, and for n ≥ 1,

where X 1, X 2, … are i.i.d.  ∼ F for some distribution F on .

If F is concentrated on [0, ∞) (or (−∞, 0]), then S n (or − S n ) is simply the nth arrival epoch in a renewal process, which we have already studied in some detail in Chapter 5 We shall henceforth assume that F does not have a support contained in either [0, ∞) or (−∞, 0]. This also excludes the pathological case in which F is degenerate at zero and the random walk becomes trivial. We will assume, unless otherwise stated, that F is absolutely continuous.

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