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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 $$\alpha =1/8$$.

##### MSC:
 60G10 Stationary stochastic processes 60A10 Probabilistic measure theory
##### Keywords:
coin-tossing; sufficient statistics