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

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
.