Convergence of the centered maximum of log-correlated Gaussian fields

By Jian Ding, Rishideep Roy, Ofer Zeitouni. 2015

97 citations recorded by the source corpus.

We show that the centered maximum of a sequence of log-correlated Gaussian fields in any dimension converges in distribution, under the assumption that the covariances of the fields converge in a suitable sense. We identify the limit as a randomly shifted Gumbel distribution, and characterize the random shift as the limit in distribution of a sequence of random variables, reminiscent of the derivative martingale in the theory of Branching Random Walk and Gaussian Chaos. We also discuss applications of the main convergence theorem and discuss examples that show that for logarithmically correlated fields, some additional structural assumptions of the type we make are needed for convergence of the centered maximum.

Subfields: Finance, Management Science and Operations Research, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6a2361f725f8e8337f4896203bf75666a2ac4eea

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Finance

1. Zhan Shi
2. L. Arguin
3. Joseph Najnudel
4. David Belius
5. Ofer Zeitouni
6. P. Bourgade
7. Elliot Paquette
8. Jian Ding

In Management Science and Operations Research

1. Zhan Shi
2. Joseph Najnudel
3. Elliot Paquette
4. Ofer Zeitouni
5. David Belius
6. L. Arguin

In Mathematical Physics

1. Joseph Najnudel
2. L. Arguin
3. Zhan Shi
4. David Belius
5. P. Bourgade
6. Elliot Paquette
7. Ofer Zeitouni
8. Thomas Madaule

Citations in the corpus

• The maximum of the CUE field (2016, methodology): “Using existing machinery, such as [DRZ15], it is relatively straightforward to show that Conjecture 1.”
• Branching random walks (2015, background): “Weak convergence for the minimum of general log-correlated Gaussian fields (in any dimension), centered at the mean, is established by Ding, Roy and Zeitouni [96].”
• On the extreme values of the Riemann zeta function on random intervals of the critical line (2016, methodology): “This conjecture is not fully proven, but Chhaibi, Madaule and Najnudel in [13] have recently shown that the family (1.2) of random variables is tight, improving successive results by Arguin, Belius and Bourgade [3] and by Paquette and Zeitouni [26].”
• Maximum of the Characteristic Polynomial of Random Unitary Matrices (2015, background): “A general theorem for the convergence of the maximum of log-correlated Gaussian fields was proved in [31].”