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

Abstract

We prove an Erdős-Feller-Kolmogorov-Petrowsky law of the iterated logarithm for self-normalized martingales. Our proof is given in the framework of the game-theoretic probability of Shafer and Vovk. As many other game-theoretic proofs, our proof is self-contained and explicit.

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

“…A similar argument is also in [14]. One should notice that the equation (1) can be seen as one discretization of (4), where π is the uniform density on [−1/2, 1/2].…”

“…Then, in the fair-coin tossing game, Skeptic can force S n − √ nψ(n) ≥ 0 for infinitely many n. This is a corollary from the classical EFKP-LIL. This fact also follows from the main theorem in [14]. Thus, Reality can comply with this event with this restriction, which means that ψ does not belong to the upper class in OUFG.…”

“…Later, Wang [18] and Csörgő et al [3] eliminated some of the conditions. A similar argument in the self-normalized form can also be seen in game-theoretic probability as shown in [14]. Self-normalized processes including t-statistic are known for their statistical applications.…”

“…Here we generalize the prior (14) in view of EFKP-LIL. We give a more general and complete treatment of EFKP-LIL in Section 5. x.…”

“…In studying self-normalized processes, selfnormalized sums S n /A n are regarded as important and one of the reasons is that they have close relations to t-statistic; see, e.g., [5], [6]. For a survey of results concerning self-normalized processes, especially self-normalized sums, see [4], [16], and references in [14]. In the self-normalized form of SLLN we only consider paths of Reality's moves such that A ∞ = lim n A n = ∞.…”