Local universality for real roots of random trigonometric polynomials
Abstract
Consider a random trigonometric polynomial
X
n
:R→R
of the form
X
n
(t)=
∑
k=1
n
(
ξ
k
sin(kt)+
η
k
cos(kt)),
where
(
ξ
1
,
η
1
),(
ξ
2
,
η
2
),…
are independent identically distributed bivariate real random vectors with zero mean and unit covariance matrix. Let
(
s
n
)
n∈N
be any sequence of real numbers. We prove that as
n→∞
, the number of real zeros of
X
n
in the interval
[
s
n
+a/n,
s
n
+b/n]
converges in distribution to the number of zeros in the interval
[a,b]
of a stationary, zero-mean Gaussian process with correlation function
(sint)/t
. We also establish similar local universality results for the centered random vectors
(
ξ
k
,
η
k
)
having an arbitrary covariance matrix or belonging to the domain of attraction of a two-dimensional
α
-stable law.