Celebratio Mathematica

D. Blackwell and D. Freedman: “The tail \( \sigma \)-field of a Markov chain and a theorem of Orey,” Ann. Math. Stat. 35 : 3 (1964), pp. 1291–1295. MR 0164375 Zbl 0127.35204 article

D. Blackwell and D. Freedman: “A remark on the coin tossing game,” Ann. Math. Stat. 35 : 3 (1964), pp. 1345–1347. MR 0169257 Zbl 0129.31502 article

D. Blackwell, P. Deuel, and D. Freedman: “The last return to equilibrium in a coin-tossing game,” Ann. Math. Stat. 35 : 3 (1964), pp. 1344. MR 0169256 Zbl 0129.31501 article

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D. Blackwell and D. Freedman: “On the local behavior of Markov transition probabilities,” Ann. Math. Stat. 39 : 6 (1968), pp. 2123–2127. MR 0233429 Zbl 0175.46903 article

Let \( P(t) = P(t,i,j) \) be a semigroup of stochastic matrices on the countable set \( I = \{i,j,\dots\}. \) Suppose \[ \lim_{t\to 0}P(t,i,i) = 1 \quad\text{for each }i \in I .\] Fix one state \( a \in I \) and abbreviate \( f(t) = P(t,a,a) \).

Suppose \( 0 < \varepsilon < 1 \) and \( f(1) \leq 1 - \varepsilon \). Then \[ \int_0^1 f(t)\,dt < 1 - \tfrac12 \varepsilon.\]

Suppose \( 0 < \varepsilon < 1/4 \) and \( f(1) \geq 1 - \varepsilon \). Then, for all \( t \) in \( [0,1] \), \begin{align*} f(t) &\geq \tfrac12[1 + (1 - 4\varepsilon)^{1/2}]\\ & = 1 - \varepsilon - \varepsilon^2 - O(\varepsilon^3). \end{align*}

Suppose \( 0 < \varepsilon < 1/4 \) and \( \int_0^1 f(t)\, dt \geq 1 - \varepsilon \). Then \[ f(t) > 1 - 2\varepsilon \quad\textit{for }0 \leq t \leq 1 .\]

Note: The last statement can be restated (using algebra) in this more attractive form: if \( 0 < \delta < 1/2 \) and \( f(1) \geq 1 - \delta + \delta^2 \), then \( f(t) \geq 1 - \delta \) for \( 0\leq t \leq 1 \).

D. Blackwell and D. Freedman: “On the amount of variance needed to escape from a strip,” Ann. Probab. 1 : 5 (1973), pp. 772–787. MR 0356214 Zbl 0293.60041 article

D. Blackwell, D. Freedman, and M. Orkin: “The optimal reward operator in dynamic programming,” Ann. Probab. 2 : 5 (1974), pp. 926–941. MR 0359818 Zbl 0318.49021 article

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David H. Blackwell

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