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Explicit distributional results in pattern formation. (English) Zbl 0893.60005

The concept of runs, and more generally patterns, has been used in various areas. A new and unified approach is presented for constructing joint generating functions for quantities of interest associated with pattern formation in binary sequences. The methodology presented in this paper is based on first imbedding the problem into a more general one for an appropriate finite-state Markov chain with one absorbing state, and second, treating that chain by the tools of exponential families. The first step of imbedding the problem into a similar one is natural and it has been used earlier while the second step based on exponential families is new for this area. The technology presented in this article is general enough to cover all existing results in this direction for binary sequences as well as to provide explicit expressions for some patterns which were not available earlier.
Reviewer: S.Chukova (Flint)

MSC:

60E10 Characteristic functions; other transforms
60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
60G40 Stopping times; optimal stopping problems; gambling theory
62H10 Multivariate distribution of statistics
68R99 Discrete mathematics in relation to computer science
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