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Estimation of the mean with time-varying finite memory (Corresp.)

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Let X_1,X_2,\cdots be a sequence of independent identically distributed observations with a common mean \mu . Assume that 0 \leq X_i \leq 1 with probability 1. We show that for each \varepsilon > 0 there exists an integer m , a finite-valued statistic T_n = T_n(X_1, \cdots , X_n) \in {t_1,\cdots ,t_m} and a real-valued function d defined on {t_1,\cdots ,t_m} such that i ) T_{n+1} = f_n(T_n,X_{n+1}) ; ii) P[\lim \sup \mid d(T_n) - \mu \mid \leq \varepsilon ] = 1 . Thus we have a recursive-like estimate of \mu , for which the data are summarized for each n by one of m states and which converges to within \varepsilon of \mu with probability 1. The constraint on memory here is time varying as contrasted to the time-invariant constraint that would have T_{n+1} = f(T_n, X_{n+1}) for all n .

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IEEE Transactions on Information Theory  (Volume:18 ,  Issue: 4 )