The law of the iterated logarithm in game-theoretic probability with quadratic and stronger hedges

Abstract

We prove both the validity and the sharpness of the law of the iterated logarithm in game-theoretic probability with quadratic and stronger hedges.O( A n / ln ln A n ) and which does not obey the LIL. A number of other sufficient conditions for the LIL (1) were given in the literature such as [2,8]. For instance, Egorov [1] showed that the following is a sufficient condition:for any ǫ > 0. Our result gives a new sufficient condition (2) for the LIL (1).In the case of independent, identically distributed (i.i.d.) random variables, Hartman and Wintner [3] proved that existence of a second moment suffices for the LIL and Strassen [10] proved conversely that existence of a second moment is necessary. The topic of this paper is the LIL in game-theoretic probability, which was studied in Shafer and Vovk [9] under two protocols. The first protocol "unbounded forecasting" only contains a quadratic hedge.

Erdős–Feller–Kolmogorov–Petrowsky law of the iterated logarithm for self-normalized martingales: A game-theoretic approach

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“…Hence, we have Then, it suffices to show (17) if ∞ k=1 (ln k)ψ(e 5k ln k )e −ψ(e 5k ln k ) 2 /2 = ∞. As in Chapter 5 of [17] and [14], we divide the time axis into "cycles". However, unlike in Chapter 5 of [17] and [14], our cycles are based on stopping times.…”

“…As in Chapter 5 of [17] and [14], we divide the time axis into "cycles". However, unlike in Chapter 5 of [17] and [14], our cycles are based on stopping times. Let n k := k 5k , k = 1, 2, .…”

Relation between the rate of convergence of strong law of large numbers and the rate of concentration of Bayesian prior in game-theoretic probability

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Itô Calculus without Probability in Idealized Financial Markets*

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