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Runs in coin tossing: a general approach for deriving distributions for functionals

Part of: Stochastic processes Distribution theory - Probability Limit theorems

Published online by Cambridge University Press:  30 March 2016

Lars Holst  and
Takis Konstantopoulos
Lars Holst*
Affiliation:
Royal Institute of Technology
Takis Konstantopoulos*
Affiliation:
Uppsala University
*
∗ Postal address: Department of Mathematics, Royal Institute of Technology, SE-10044 Stockholm, Sweden. Email address: [email protected]
∗∗ Postal address: Department of Mathematics, Uppsala University, SE-75106 Uppsala, Sweden. Email address: [email protected]
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Abstract

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We take a fresh look at the classical problem of runs in a sequence of independent and identically distributed coin tosses and derive a general identity/recursion which can be used to compute (joint) distributions of functionals of run types. This generalizes and unifies already existing approaches. We give several examples, derive asymptotics, and pose some further questions.

Keywords

Regenerationcoin tossingrunslongest runPoisson approximationLaplace transformRouché's theorem

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 60G50: Sums of independent random variables; random walks 60F99: None of the above, but in this section 60E10: Characteristic functions; other transforms
Type
Research Papers
Information
Journal of Applied Probability , Volume 52 , Issue 3 , September 2015 , pp. 752 - 770
Copyright
Copyright © 2015 by the Applied Probability Trust 

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