Real zeros of random trigonometric polynomials with dependent coefficients
Authors: Angst Jürgen , Pautrel Thibault , Poly Guillaume 
Published: June 2022

We further investigate the relations between the large degree asymptotics of the number of real zeros of random trigonometric polynomials with dependent coefficients and the underlying correlation function. We consider trigonometric polynomials of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Subscript n Baseline left-parenthesis t right-parenthesis colon-equal StartFraction 1 Over StartRoot n EndRoot EndFraction sigma-summation Underscript k equals 1 Overscript n Endscripts a Subscript k Baseline cosine left-parenthesis k t right-parenthesis plus b Subscript k Baseline sine left-parenthesis k t right-parenthesis comma x element-of left-bracket 0 comma 2 pi right-bracket comma">

f
n

(
t
)
≔

1

n

∑

k
=
1

n

a
k

cos
⁡
(
k
t
)
+

b
k

sin
⁡
(
k
t
)
,
 
x
∈
[
0
,
2
π
]
,

f_n(t)≔\frac {1}{\sqrt {n}}\sum _{k=1}^{n}a_k \cos (kt)+b_k\sin (kt), ~x\in [0,2\pi ],

\]
where the sequences

(

a
k

)

k
≥
1

(a_k)_{k\geq 1}

and

(

b
k

)

k
≥
1

(b_k)_{k\geq 1}

are two independent copies of a stationary Gaussian process centered with variance one and correlation function

ρ
\rho

with associated spectral measure

μ

ρ

\mu _{\rho }

. We focus here on the case where

μ

ρ

\mu _{\rho }

is not purely singular and we denote by

ψ

ρ

\psi _{\rho }

its density component with respect to the Lebesgue measure

λ
\lambda

. Quite surprisingly, we show that the asymptotics of the number of real zeros

N

(

f
n

,
[
0
,
2
π
]
)

\mathcal {N}(f_n,[0,2\pi ])

of

f
n

f_n

in

[
0
,
2
π
]

[0,2\pi ]

is not related to the decay of the correlation function

ρ
\rho

but instead to the Lebesgue measure of the vanishing locus of

ψ

ρ

\psi _{\rho }

. Namely, assuming that

ψ

ρ

\psi _{\rho }

is

C

1

\mathcal {C}^1

with Hölder derivative on an open set of full measure, one establishes that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript n right-arrow plus normal infinity Endscripts StartFraction double-struck upper E left-bracket script upper N left-parenthesis f Subscript n Baseline comma left-bracket 0 comma 2 pi right-bracket right-parenthesis right-bracket Over n EndFraction equals StartFraction lamda left-parenthesis left-brace psi Subscript rho Baseline equals 0 right-brace right-parenthesis Over pi StartRoot 2 EndRoot EndFraction plus StartFraction 2 pi minus lamda left-parenthesis left-brace psi Subscript rho Baseline equals 0 right-brace right-parenthesis Over pi StartRoot 3 EndRoot EndFraction period">

lim

n
→
+
∞

E

[

N

(

f
n

,
[
0
,
2
π
]
)
]

n

=

λ
(
{

ψ

ρ

=
0
}
)

π

2

+

2
π
−
λ
(
{

ψ

ρ

=
0
}
)

π

3

.

\lim _{n \to +\infty } \frac {\mathbb {E}\left [\mathcal {N}(f_n,[0,2\pi ])\right ]}{n}= \frac {\lambda (\{\psi _{\rho }=0\})}{\pi \sqrt {2}} + \frac {2\pi - \lambda (\{\psi _{\rho }=0\})}{\pi \sqrt {3}}.

\]
On the other hand, assuming a sole log-integrability condition on

ψ

ρ

\psi _{\rho }

, which implies that it is positive almost everywhere, we recover the asymptotics of the independent case:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript n right-arrow plus normal infinity Endscripts StartFraction double-struck upper E left-bracket script upper N left-parenthesis f Subscript n Baseline comma left-bracket 0 comma 2 pi right-bracket right-parenthesis right-bracket Over n EndFraction equals StartFraction 2 Over StartRoot 3 EndRoot EndFraction period">

lim

n
→
+
∞

E

[

N

(

f
n

,
[
0
,
2
π
]
)
]

n

=

2

3

.

\lim _{n \to +\infty } \frac {\mathbb {E}\left [\mathcal {N}(f_n,[0,2\pi ])\right ]}{n}= \frac {2}{\sqrt {3}}.

\]
The latter asymptotics thus broadly generalizes the main result of Angst, Dalmao, and Poly [Proc. Amer. Math. Soc. 147 (2019), pp. 205–214] where the spectral density was assumed to be continuous and bounded from below. Besides, with further assumptions of regularity and existence of negative moment for

ψ

ρ

\psi _{\rho }

, which encompass e.g. the case of random coefficients being increments of fractional Brownian motion with any Hurst parameter, we moreover show that the above convergence in expectation can be strengthened to an almost sure convergence.