Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userA game-theoretic proof of Erdos-Feller-Kolmogorov-Petrowsky law of the iterated logarithm for fair-coin tossing
123
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We give a game-theoretic proof of the celebrated Erdos-Feller-Kolmogorov-Petrowsky law of the iterated logarithm for fair coin tossing. Our proof, based on Bayesian strategy, is explicit as many other game-theoretic proofs of the laws in probability theory.
rate research
Read More
We prove an Erdos-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.
In this note we prove the following law of the iterated logarithm for the Grenander estimator of a monotone decreasing density: If $f(t_0) > 0$, $f(t_0) < 0$, and $f$ is continuous in a neighborhood of $t_0$, then begin{eqnarray*} limsup_{nrightarrow infty} left ( frac{n}{2log log n} right )^{1/3} ( widehat{f}_n (t_0 ) - f(t_0) ) = left| f(t_0) f(t_0)/2 right|^{1/3} 2M end{eqnarray*} almost surely where $ M equiv sup_{g in {cal G}} T_g = (3/4)^{1/3}$ and $ T_g equiv mbox{argmax}_u { g(u) - u^2 } $; here ${cal G}$ is the two-sided Strassen limit set on $R$. The proof relies on laws of the iterated logarithm for local empirical processes, Groenebooms switching relation, and properties of Strassens limit set analogous to distributional properties of Brownian motion.
Given a sequence of numbers ${p_n}$ in $[0,1]$, consider the following experiment. First, we flip a fair coin and then, at step $n$, we turn the coin over to the other side with probability $p_n$, $nge 2$. What can we say about the distribution of the empirical frequency of heads as $ntoinfty$? We show that a number of phase transitions take place as the turning gets slower (i.e. $p_n$ is getting smaller), leading first to the breakdown of the Central Limit Theorem and then to that of the Law of Large Numbers. It turns out that the critical regime is $p_n=text{const}/n$. Among the scaling limits, we obtain Uniform, Gaussian, Semicircle and Arcsine laws.
We establish a Chung-type law of the iterated logarithm and the exact local and uniform moduli of continuity for a large class of anisotropic Gaussian random fields with a harmonizable-type integral representation and the property of strong local nondeterminism. Compared with the existing results in the literature, our results do not require the assumption of stationary increments and provide more precise upper and lower bounds for the limiting constants. The results are applicable to the solutions of a class of linear stochastic partial differential equations driven by a fractional-colored Gaussian noise, including the stochastic heat equation.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
