Game-theoretic versions of strong law of large numbers for unbounded variables

Stochastic Processes and their Applications

2012

Self Cite

We give a unified treatment of the convergence of random series and the rate of convergence of strong law of large numbers in the framework of game-theoretic probability of Shafer and Vovk [24]. We consider games with the quadratic hedge as well as more general weaker hedges. The latter corresponds to existence of an absolute moment of order smaller than two in the measure-theoretic framework. We prove some precise relations between the convergence of centered random series and the convergence of the series of prices of the hedges. When interpreted in measure-theoretic framework, these results characterize convergence of a martingale in terms of convergence of the series of conditional absolute moments. In order to prove these results we derive some fundamental results on deterministic strategies of Reality, who is a player in a protocol of game-theoretic probability. It is of particular interest, since Reality's strategies do not have any counterparts in measure-theoretic framework, ant yet they can be used to prove results which can be interpreted in measure-theoretic framework.

“…Notice that Theorem 5.1 with h(x) = x 2 implies Theorem 4.4. In contrast Theorem 5.1 with g(x) = h(x/ν), m n = 0 and v n = ν implies Theorem 3.1 of [13].…”

Section: The Case Of Convergencementioning

confidence: 93%

Section: Proof Of Theorem 31 (⇒) Consider a Capital Processmentioning

confidence: 99%

See 1 more Smart Citation

Convergence of random series and the rate of convergence of the strong law of large numbers in game-theoretic probability

Miyabe

Stochastic Processes and their Applications

2012

Self Cite

“…We do not present anything new for this step of the argument. An extension of the present paper to unbounded games is presented in Kumon et al (2006).…”

mentioning

confidence: 98%

On a simple strategy weakly forcing the strong law of large numbers in the bounded forecasting game

Kumon

2007

Ann Inst Stat Math

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“…For coin-tossing games clarified the structure of Skeptic's capital process in terms of the Kullback divergence. Kumon et al (2006) considered a minimal set of hedges for forcing the SLLN for the case that Reality's moves are unbounded.…”

Section: Introductionmentioning

confidence: 99%

Game-Theoretic Derivation of Discrete Distributions and Discrete Pricing Formulas

Suzuki

2007

JJSS

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In this expository paper, we illustrate the generality of the game-theoretic probability protocols of Shafer and Vovk (2001) in finite-horizon discrete games. By restricting ourselves to finite-horizon discrete games, we can explicitly describe how discrete distributions with finite support and discrete pricing formulas, such as the Cox-Ross-Rubinstein formula, are naturally derived from game-theoretic probability protocols. Corresponding to any discrete distribution with finite support, we construct a finite-horizon discrete game, a replicating strategy of Skeptic, and a neutral forecasting strategy of Forecaster, such that the discrete distribution is derived from the game. Construction of a replicating strategy is the same as in the standard arbitrage arguments of pricing European options in binomial tree models. However the game-theoretic framework is advantageous because it eliminates the need for any a priori probabilistic assumption.