Characterization of LIL behavior in Banach space

By U. Einmahl, Deli Li. 2006

74 citations recorded by the source corpus.

In a recent paper by the authors a general result characterizing two-sided LIL behavior for real valued random variables has been established. In this paper, we look at the corresponding problem in the Banach space setting. We show that there are analogous results in this more general setting. In particularly, we provide a necessary and sufficient condition for LIL behavior with respect to sequences of the form p nh(n), where h is from a suitable subclass of the positive, nondecreasing slowly varying functions. To prove these results we have to use a different method. One of our main tools is an improved Fuk-Nagaev type inequality in Banach space which should be of independent interest.

Subfields: Management Science and Operations Research, Numerical Analysis, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/365bfdc51ce40132ec104c438c0e613e54b0aa69

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In Management Science and Operations Research

In Numerical Analysis

In Statistics and Probability

Citations in the corpus

Inference on Causal and Structural Parameters using Many Moment Inequalities (2013, methodology): “Hence in this case Λ 2 n in Theorem 3.1 of Einmahl and Li (2008) is bounded by (and indeed equal to) σ2.”
Gaussian Approximation for High Dimensional Time Series (2015, methodology): “For s > 1, we apply Theorem 3.1 of Einmahl and Li (2008) Minkowski’s and H¨older’s inequalities, we have Hence, the Λ n therein is bounded by Let X i be a mean zero p -dimensional stationary process, and We are interested in bounding the tail probabilities holds for all 1 ≤ m ≤ n .”
Central limit theorems and bootstrap in high dimensions (2014, unknown): “See also [22].”
Gaussian and bootstrap approximations for high-dimensional U-statistics and their applications (2016, methodology): “An alternative simple data splitting approach by reducing the U-statistics to sums of iid random vectors can give the exact rate for bounding the moments in the non-degenerate case [69, 53, 43, 29].”
THE ASYMPTOTIC DISTRIBUTION AND BERRY-ESSEEN BOUND OF A NEW TEST FOR INDEPENDENCE IN HIGH DIMENSION WITH AN APPLICATION TO STOCHASTIC OPTIMIZATION (2008, unknown): “ kk≥(1−Mδ)x+Bn/2 ≤P max 1≤k≤n kSˆ k −ESˆkk≥(1−Mδ)x+ 3 2 EkSˆn −ESˆnk (6.4) ≤exp − ((1−Mδ)x)2 2(1+β)Λn +exp − (1−Mδ)a 2Dβ , where the last inequality follows from (3.4) in Einmahl and Li [10]. We now estimate P(max 1≤k≤n kS˘kk ≥Mδx +Bn/2). It follows from the Hoeﬀding–Bennett inequality that P”
Uniform post-selection inference for least absolute deviation regression and other Z-estimation problems (2013, unknown): “See also [20].”
Testing Many Moment Inequalities (2013, methodology): “1 in Einmahl and Li (2008). Note that Einmahl and Li (2008) assumed that s > 2 but their proof applies to the case where s > 1.”
Gaussian approximation of suprema of empirical processes (2012, background): “he supremum but requires uniform boundedness of F, which is violated in our applications. Another known deviation inequality similar in nature to Theorem 5.1 is a Fuk-Nagaev type inequality proved in [20] (see their Theorem 3.1). For the purpose of this paper, however, Theorem 5.1 is more convenient”
A tail inequality for suprema of unbounded empirical processes with applications to Markov chains (2007, background): “Let X1, . . . ,Xn be independent random variables with values in a measurable space (S,B) and let F be a countable class of measurable functions f : S → R. Assume that for every f ∈ F and every i, Ef(Xi) = 0 and for some α ∈ (0, 1] and all i, ‖ supf |f(Xi)|‖ψα ∞.”