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Independence and fair coin-tossing. (English) Zbl 0583.60031
We consider a one-parameter family of stationary stochastic processes \(\{X_ 1,X_ 2,\ldots \}\) with the properties that:
(a) \(P(X_ i=H)=P(X_ i=T)=1/2\) \((i=1,2,\ldots)\), where H and T stand for head and tail respectively,
(b) the \(\{X_ i\}\) are pairwise independent,
(c) \(\alpha =P[X_ i=X_{i+1}=X_{i+2}=H]\) \((i=1,2,\ldots)\) is a parameter such that 1/16\(\leq \alpha \leq 3/16\) (the only possible values for such a process).
The parameter \(\alpha\) completely determines the distribution of the processes considered, and when \(\alpha =1/8\) the process is mutually independent. We show how to construct sufficient statistics for this process and describe a simple test of the hypothesis that \(\alpha =1/8\).

60G10 Stationary stochastic processes
60A10 Probabilistic measure theory