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.