A tail inequality for suprema of unbounded empirical processes with applications to Markov chains

Abstract

We present a tail inequality for suprema of empirical processes generated by variables with finite ψ α norms and apply it to some geometrically ergodic Markov chains to derive similar estimates for empirical processes of such chains, generated by bounded functions. We also obtain a bounded difference inequality for symmetric statistics of such Markov chains.AMS 2000 Subject Classification: Primary 60E15, Secondary 60J05. * Research partially supported by MEiN Grant 1 PO3A 012 29. n i=1 Ef (X i ) 2 . Then for all η > 0 and t ≥ 0,where K 1 = 4, K 2 (η) = 2.5 + 32/η, K 3 = 5.4, K 4 (η) = 2.5 + 43.2/η. Similar, more refined results were obtained subsequently by Bousquet [2] and Klein and Rio [7]. The latter article contains an inequality for suprema of empirical processes with the best known constants.

Central limit theorems and bootstrap in high dimensions

“…Recall the definition of ρ n (A) in (1) and the definitions of D (1) n and D (2) n,q in (9). An extension of Proposition 2.1 leads to the following result.…”

“…Recalling that ϵ = a/n and B n ≥ 1, we have ϵ log 1/2 (pn) ≤ CD (1) n . Hence the assertions of the proposition follow if we prove…”

“…We shall apply Theorem 4.1 to prove the proposition. Observe that since n −1 log 1/2 (pn) ≤ CD (1) n (α), it suffices to construct an appropriate ∆ n such that P(∆ n (A) > ∆ n ) ≤ α and to bound ∆ 1/3 n log 2/3 (pn). We begin with noting that since (S) holds for all A ∈ A, ∆ n (A) ≤ C∆ n,r where ∆ n,r = max 1≤j,k≤p…”

“…Similar conditions are needed in other papers cited above. It is worthwhile to mention here that, when A is the class of all Borel measurable convex sets, it was proved by [29] that ρ n (A) ≥ cE[∥X 1 ∥ 3 ]/ √ n for some universal constant c > 0. In modern statistical applications, such as high dimensional estimation and multiple hypothesis testing, however, p is often larger or even much larger than n. It is therefore interesting to ask whether it is possible to provide a nontrivial class of sets A in R p for which we would have ρ n (A) → 0 even if p is potentially larger or much larger than n.…”

“…For α ∈ [1, ∞), ∥ · ∥ ψα is an Orlicz norm, while for α ∈ (0, 1), ∥ · ∥ ψα is not a norm but a quasi-norm, that is, there exists a constant K α depending only on α such that ∥ξ 1 …”