One of us [Blackwell 1957] has shown that if \( Z_n \), \( n = 1 \), \( 2,\dots \) is a stochastic process with \( D \) states, \( {}0 \), \( 1,\dots \), \( D-1 \) such that
\[ X=\sum_{n=1}^{\infty} \frac{Z_n}{D^n} \]
has an absolutely continuous distribution with respect to Lebesgue measure, then the conditional distribution of
\[ R_k = \sum_{n=1}^{\infty} \frac{Z_{k+n}}{D^n} \]
given \( Z_1,\dots \), \( Z_k \) converges with probability one as \( k\to\infty \) to the uniform distribution on the unit interval, in the sense that for each \( \lambda \), \( 0 < \lambda \leq 1 \),
\[ P(R_k < \lambda\mid Z_1,\dots,Z_k)\to \lambda \]
with probability 1 as \( k\to \infty \). It follows that the unconditional distribution of \( R_k \) converges to the uniform distribution as \( k\to\infty \). If \( \{Z_n\} \) is stationary, the distribution of \( R_k \) is independent of \( k \), and hence uniform, a result obtained earlier by Harris [1955]. Earlier work relevant to convergence of opinion can be found in [Savage, 1954, chap. 3, sect. 6].
Here we generalize these results and also show that the conditional distribution of \( R_k \) given \( Z_1,\dots \), \( Z_k \) converges in a much stronger sense. All probabilities in this paper are countably additive.
