Bibliographic Details
| Title: |
A law of large numbers concerning the distribution of critical points of random Fourier series. |
| Authors: |
"Brandon" Fu, Qiangang1 (AUTHOR) qfu3@nd.edu, Nicolaescu, Liviu I.1 (AUTHOR) lnicolae@nd.edu |
| Source: |
Stochastic Processes & Their Applications. May2026, Vol. 195, pN.PAG-N.PAG. 1p. |
| Subjects: |
Law of large numbers, Random functions (Mathematics), Stochastic processes, Torus, Gaussian distribution, Asymptotic distribution |
| Abstract: |
On the flat torus T m = R m / Z m we consider the Gaussian random function F a R defined as a random Fourier series (1.1). The Fourier coefficients are mean zero independent normal variables whose variances depend on the frequencies via an even Schwartz function a on R and large rescaling parameter R. For any open subset U of the torus denote by Z R (U) the number of critical points of F a R in U. We prove that if U is contained in a geodesic ball, then the variance of Z R (U) is asymptotic to const × Rmvol [ U ] as R → ∞. We use this to prove that if m ≥ 2, then as N → ∞, the random measures N − m Z N (−) converge a.s. to an explicit multiple of the volume measure on the flat torus. [ABSTRACT FROM AUTHOR] |
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| Database: |
Engineering Source |
