Celebratio Mathematica

D. Blackwell and L. Dubins: “Merging of opinions with increasing information,” Ann. Math. Stat. 33 : 3 (1962), pp. 882–886. MR 0149577 Zbl 0109.35704 article

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.

D. Blackwell and L. E. Dubins: “A converse to the dominated convergence theorem,” Illinois J. Math. 7 : 3 (1963), pp. 508–514. MR 0151572 Zbl 0146.37503 article

On a probability space \( (\Omega, \mathcal{B}, P) \), let \( \{f_n \), \( n=1 \), \( 2,\dots\} \) be a sequence of nonnegative random variables in \( L_1 \) such that \( f_n \to f\in L_1 \) with probability 1 and define \( g=\sup_n f_n \). If \( g\in L_1 \), the Lebesgue dominated convergence theorem asserts that \( E(f_n)\to E(f) \). More generally, as noted by [Doob 1953, p. 23], if \( g\in L_1 \), then for any Borel field \( \mathcal{B}_0 \) contained in \( \mathcal{B} \), \[ E(f_n\mid\mathcal{B}_0) \to E(f\mid\mathcal{B}_0) \quad\text{a.e.} \] If one extends this result in a minor manner, Lebesgue’s condition \( g\in L_1 \) is not only sufficient but necessary, as the following converse to the dominated convergence theorem asserts.

If \( f_n \geq 0 \), \( f_n \to f \) a.e., \( f_n\in L_1 \), and \( g = \sup_n f_n \notin L_1 \), there are, on a suitable probability space, random variables \( \{f_n^* \), \( n=1 \), \( 2,\dots\} \), \( f^* \), and a Borel field \( \mathcal{C} \) such that \( f^* \), \( f_1^* \), \( f_2^*,\dots \) have the same joint distribution as \( f \), \( f_1 \), \( f_2,\dots \), and \[ P\bigl\{E(f_n^*\mid\mathcal{C})\to E(f^*\mid\mathcal{C})\bigr\} = 0. \]

D. Blackwell and L. E. Dubins: “Sharp bounds on the distribution of the Hardy–Littlewood maximal function,” Proc. Am. Math. Soc. 14 : 3 (1963), pp. 450–453. MR 0148842 Zbl 0118.05401 article

D. Blackwell and L. Dubins: “An elementary proof of an identity of Gould’s,” Bol. Soc. Mat. Mexicana (2) 11 (1966), pp. 108–110. MR 0233713 Zbl 0196.29001 article

D. Blackwell and L. E. Dubins: “On existence and non-existence of proper, regular, conditional distributions,” Ann. Probab. 3 : 5 (1975), pp. 741–752. MR 0400320 Zbl 0348.60003 article

If \( \mathscr{A} \) is the tail, invariant, or symmetric field for discrete-time processes, or a field of the form \( \mathscr{F}_{t+} \) for continuous-time processes, then no countably additive, regular, conditional distribution given \( \mathscr{A} \) is proper. A notion of normal conditional distributions is given, and there always exist countably additive normal conditional distributions if \( \mathscr{A} \) is a countably generated sub \( \sigma \)-field of a standard space. The study incidentally shows that the Borel-measurable axiom of choice is false. Classically interesting subfields \( \mathscr{A} \) of \( \mathscr{B} \) possess certain desirable properties which are the defining properties for \( \mathscr{A} \) to be “regular” in \( \mathscr{B} \).

D. Blackwell and L. E. Dubins: “An extension of Skorohod’s almost sure representation theorem,” Proc. Am. Math. Soc. 89 : 4 (1983), pp. 691–692. MR 718998 Zbl 0542.60005 article

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