On the rate of concentration of maxima in Gaussian arrays
- PDF / 558,818 Bytes
- 29 Pages / 439.642 x 666.49 pts Page_size
- 56 Downloads / 197 Views
On the rate of concentration of maxima in Gaussian arrays Rafail Kartsioukas1 · Zheng Gao1 · Stilian Stoev1 Received: 10 October 2019 / Revised: 8 October 2020 / Accepted: 6 November 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Recently in Gao and Stoev (2020) it was established that the concentration of maxima phenomenon is the key to solving the exact sparse support recovery problem in high dimensions. This phenomenon, known also as relative stability, has been little studied in the context of dependence. Here, we obtain bounds on the rate of concentration of maxima in Gaussian triangular arrays. These results are used to establish sufficient conditions for the uniform relative stability of functions of Gaussian arrays, leading to new models that exhibit phase transitions in the exact support recovery problem. Finally, the optimal rate of concentration for Gaussian arrays is studied under general assumptions implied by the classic condition of Berman (1964). Keywords Rate of relative stability · Concentration of maxima · Exact support recovery · Phase transitions · Functions of Gaussian arrays AMS 2000 Subject Classifications MSC 62G32 · 62G20 · 62G10 · 60G15 · 60G70
1 Introduction Let Zi , i = 1, 2, . . . be independent and identically distributed (iid) standard Normal random variables. It is well known that their maxima under affine normalization converge to the Gumbel extreme value distribution. If, however, one chooses to Stilian Stoev
[email protected] Rafail Kartsioukas [email protected] Zheng Gao [email protected] 1
Department of Statistics, University of Michigan, Ann Arbor, MI, USA
R. Kartsioukas et al.
standardize the maxima by only dividing by a sequence of positive numbers, then the only possible limits are constants. Specifically, for all ap ∼ 2 log(p), we have 1 P max Zi −→ 1, ap i∈[p]
as p → ∞,
(1.1)
where [p] := {1, · · · , p} and in fact the convergence is valid almost surely. This property, known as relative stability, dates back to the seminal work of Gnedenko (1943) who has characterized it in terms of rapid variation of the law of the Zi ’s (see Section 2.2 below, as well as Barndorff-Nielsen 1963; Resnick and Tomkins 1973; Kinoshita and Resnick 1991). In contrast, if the Zi ’s are iid and heavy-tailed, i.e., P[Zi > x] ∝ x −α , for some α > 0, with ap ∝ p1/α , we have 1 d maxZi −→ ξ, ap i∈[p]
(1.2)
where ξ is a random variable with the α-Fr´echet distribution. Comparing (1.1) and (1.2), we see that the maxima have fundamentally different asymptotic behavior relative to rescaling with constant sequences. In the light-tailed regime, they concentrate around a constant in the sense of (1.1), whereas in the heavy-tailed regime they disperse according to a probability distribution viz (1.2). Although this concentration of maxima phenomenon may be well-known under independence, we found that it is virtually unexplored under dependence. In this paper, we will focus on Gaussian sequences, and in fact, more generally, Gaussian triangular arrays E =
Data Loading...
