Iterated logarithm inequalities.
D. A. Darling,Herbert Robbins +1 more
TLDR
A sequence of independent, identically distributed random variables with mean 0, variance 1, and moment generating function ϕ(t) = E(etz) finite in some neighborhood of t= 0 is introduced.Abstract:
1. Introduction—Let x,x 1, x 2 … be a sequence of independent, identically distributed random variables with mean 0, variance 1, and moment generating function ϕ(t) = E(etz) finite in some neighborhood of t= 0, and put S n = x 1, + … + x n, \(\bar x\) n = S n/n. For any sequence of positive constants a n, n ≥ 1, let P m = P(|\(\bar x\) n| ≥ a n for some n ≥ m).read more
Citations
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Exponential estimate for the law of the iterated logarithm in banach space
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Sharp Uniform Martingale Concentration Bounds.
TL;DR: Upper concentration bounds for martingales that are uniform over finite times are given and the relationship between the central limit theorem and the law of the iterated logarithm in finite time is shed on.
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A unified recipe for deriving (time-uniform) PAC-Bayes bounds
TL;DR: In this article , a unified framework for deriving PAC-Bayesian generalization bounds is presented, which combines four tools in the following order: (a) nonnegative supermartingale or reverse submartingales, (b) the method of mixtures, (c) the Donsker-Varadhan formula (or other convex duality principles), and (d) Ville's inequality.
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An Optimal Policy for Dynamic Assortment Planning Under Uncapacitated Multinomial Logit Models
Xi Chen,Yining Wang,Yuan Zhou +2 more
TL;DR: A trisection based policy combined with adaptive confidence bound construction is developed, which achieves an {item-independent} regret bound of $O(\sqrt{T})$, where $T$ is the length of selling horizon and the matching lower bound result is established to show the optimality of the policy.
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Estimating means of bounded random variables by betting
TL;DR: In this article , the authors derived confidence intervals (CI) and time-uniform confidence sequences (CS) for the classical problem of estimating an unknown mean from bounded observations, which can be seen as a generalization and improvement of the celebrated Chernoff method.
References
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Diffusion Processes and their Sample Paths.
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A martingale inequality and the law of large numbers
TL;DR: The original Kolmogorov's inequality has been extended to a martingale inequality by Levy [8] and Ville [12] and later to a semimartingale equality by Doob [3] as discussed by the authors.