Estimating a probability using finite memory

Let
\{X_{i}\}_{i=1}^{\infty}
be a sequence of independent Bernoulli random variables with probability
p
that
X_{i} = 1
and probability
q=1-p
that
X_{i} = 0
for all
i \geq 1
. Time-invariant finite-memory (i.e., finite-state) estimation procedures for the parameter p are considered which take
X_{1}, \cdots
as an input sequence. In particular, an n-state deterministic estimation procedure is described which can estimate p with mean-square error
O(\log n/n)
and an
n
-state probabilistic estimation procedure which can estimate
p
with mean-square error
O(1/n)
. It is proved that the
O(1/n)
bound is optimal to within a constant factor. In addition, it is shown that linear estimation procedures are just as powerful (up to the measure of mean-square error) as arbitrary estimation procedures. The proofs are based on an analog of the well-known matrix tree theorem that is called the Markov chain tree theorem.