How to Gamble If You Must: Inequalities for Stochastic Processes

according to Theorem applies Aryeh Dvoretzky binary rational bold strategy Borel measurable Borel sets bounded function Cartesian casino basement casino inequality casinoe maker chapter closed under composition condition conserving stakes continuous conventional convex convex combination COROLLARY countably additive decreasing defined distribution do(w doo(f Dubins e-conserves equivalent Eudoxus Example exponential houses fair casino family of optimal family of strategies Finetti finitary finite number finitely additive fortune f full house gambler gambler's problem gambling house implies incomplete stop rule increases in displacement indicator function inductively integrable infimum initial fortune interest interval least leavable Lebesgue measure Lemma lottery Mathematical miserly negative nondecreasing nonnegative optimal strategies partial history play Proof Q is excessive real numbers satisfies Section sequence shows stationary family stochastic processes stop rule subhouse subset sufficiently Suppose supremum Sw,r theory tion trivial U₁ upper bound utility