Self-Normalized Processes: Limit Theory and Statistical Applications

By Victor H. de la Peña, Tze Leung Lai, Qi-Man Shao. 2001

290 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/ed76544b6aad2af74c430f857a1171adf7301b2d

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Citations in the corpus

Regret Bounds for Kernel-Based Reinforcement Learning (2020, methodology): “They both follow from the so-called method of mixtures (e.g., Peña et al. (2008)), and can have applications beyond the scope of this work.”
Time-uniform Chernoff bounds via nonnegative supermartingales (2018, background): “20 from Theorem 4 of Delyon (2009) [B,D], Theorem 2.”
Bandit Algorithms (2020, methodology): “5 are called self-normalized bounds [de la Peña et al., 2008]. The method of mixtures goes back to the work by Robbins and Siegmund [1970]. In practice, the improvement provided by the method of mixtures relative to the covering arguments is quite large.”
Linear Programming for Large-Scale Markov Decision Problems (2014, background): “Let z 1 , z 2 , . . . , z T ∈ Z be deﬁned by z 1 = 0 and , where Π Z is the Euclidean projection onto Z , η > 0 is a learning rate, and By Theorem 1 of Zinkevich (2003), we get that Thus, Let S t = t − 1 s =1 ( z ∗ − z s ) η s , which is a self-normalized sum (de la Pe˜na et al., 2009).”
Efficient Exploration Through Bayesian Deep Q-Networks (2018, methodology): “[53] Peña, V.”
Estimating means of bounded random variables by betting (2020, methodology): “Composite test supermartingales for P have been used in past works on concentration inequalities and/or conﬁdence sequences (which are related but di ﬀ erent), from the initial series of works by Robbins and collaborators in the 1960s and 1970s, to de la Peña et al. [2007], to recent work by Jun and”
Self-normalized limit theorems: A survey (2013, background): “(2009) [52] for a systematic presentation on self-normalized processes and their statistical applications.”
Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors (2012, background): “The property (d) is a consequence of sub-Gaussian inequality of [19], Theorem 2.”
Efficient Sequential Decision Making (2017, unknown): “Let St = ∑t−1 s=1(z∗ − zs)ηs, which is a self-normalized sum [14].”
Cross-Validation With Confidence (2017, methodology): “…note that We can re-write ˆ σ 2 j /σ 2 j − 1 as Then the claim follows by applying Cramer’s moderate deviation (see, e.g., Theorem 2.13 of [10]) with centered and scaled random variable A − 2 n [( ξ j − µ j ) 2 /σ 20) follows from the diﬀerence between maxima of Gaussian random vectors…”
Improved Algorithms for Linear Stochastic Bandits (2011, methodology): “The new inequality is derived using techniques from the theory of self-normalized processes (de la Pe˜na et al., 2004, 2009).”
Robust Inference on Average Treatment Effects with Possibly More Covariates than Observations (2013, background): “The constants Ak, k=1, 2, 3, 4 are not given explicitly but are known to be absolute bounds (de la Peña, Lai, and Shao 2009).”