Abstract
In this article, we consider the following family of random trigonometric polynomials \(p_n(t,Y)=\sum _{k=1}^n Y_{k}^1 \cos (kt)+Y_{k}^2\sin (kt)\) for a given sequence of i.i.d. random variables \(Y^i_{k}\), \(i\in \{1,2\}\), \(k\ge 1\), which are centered and standardized. We set \({\mathcal {N}}([0,\pi ],Y)\) the number of real roots over \([0,\pi ]\) and \({\mathcal {N}}([0,\pi ],G)\) the corresponding quantity when the coefficients follow a standard Gaussian distribution. We prove under a Doeblin’s condition on the distribution of the coefficients that
The latter establishes that the behavior of the variance is not universal and depends on the distribution of the underlying coefficients through their kurtosis. Actually, a more general result is proven in this article, which does not require that the coefficients are identically distributed. The proof mixes a recent result regarding Edgeworth expansions for distribution norms established in Bally et al. (Electron J Probab 23(45):1–51, 2018) with the celebrated Kac–Rice formula.
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Appendices
A Ergodic lemma
The following lemmas are based on the ergodic action of irrational rotations on the Torus.
Lemma A.1
Set \(\alpha \) a positive number such that \(\frac{\alpha }{\pi }\in \mathbb {R}/ \mathbb {Q}\), f a \(2\pi \)–periodic function and \(q\ge 1\) a positive integer. One gets
Proof
We denote by \(\mathcal {C}_{2\pi }^{0}(\mathbb {R})\) the space of continuous \(2\pi \) periodic functions. We set
and
Let us first prove that \(\mathcal {E}\) is dense in \(\mathcal {H}_{0}\). We take T a continuous linear form on \(\mathcal {H}_{0}\) and we extend it to \( \mathcal {C}_{2\pi }^{0}(\mathbb {R})\) by taking \(T(\phi )=T(\phi -m(\phi ))\) with \(m(\phi )=\int _{0}^{2\pi }\phi (t)dt\). We have to prove that if T vanishes on \(\mathcal {E}\) then \(T=0\) (in virtue of the Hahn-Banach Theorem, this implies that \(\mathcal {E}\) is dense in \(\mathcal {H}_{0})\). The Riesz Theorem ensures us that there exists a finite measure \(\mu \) on \( \mathbb {R}/2\pi \mathbb {Z}\) such that
Since \(Tf=0\) for every \(f\in \mathcal {E}\), for any integer \(n\ge 1\) one has
and since the sequence \(n\alpha \) is dense modulo \(2\pi \) one deduces that for any \(y\in \mathbb {R}{\setminus } \mathbb {Q}\):
By the continuity of \(\phi \), this is true for each \(y\in \mathbb {R}\). As a result, \(\mu \) is invariant under translations and necessarily it is the Lebesgue measure up to a multiplicative constant. Hence, we get that \(T=0\) over \(\mathcal {H}_0\) and that \(\mathcal {E}\) is dense for the uniform topology. Finally, this preliminary consideration enables us to consider that \(f(x)=\phi (x+\alpha )-\phi (x)\) in the statements (A.1) and (A.2). Then, the conclusion is immediate since an Abel transforms gives us
\(\square \)
In the following \(C_{n}(k,t)\) is the matrix introduced in (3.25).
Lemma A.2
For every \(i,j,l,l^{\prime }\in \{1,2\}\) and every t, s such that \( t,s,t+s,t-s\) are irrational one has
Proof
We treat just two examples: take \(i=1,j=2,l=1,l^{\prime }=2\). Then
Then, the ergodic lemma (with \(q=2)\) gives
Take now \(i=2,j=2,l=1,l^{\prime }=2\). Then
Then, the ergodic lemma (with \(q=4)\) gives
\(\square \)
Lemma A.3
For every \(j,i,l\in \{1,2\}\) and every t, s such that \(t,s,t+s,t-s\) are irrational one has
Proof
All the computations are analogous so we treat just an example: \(l=i=j=1\). So we have
and using the ergodic lemma with \(q=0\) we get
\(\square \)
Lemma A.4
For every \(i_{1},i_{2},i_{3},l_{1},l_{2},l_{3}\in \{1,2\}\) and every t, s such that \(t,s,t+s,t-s\) are irrational one has
Proof
The poof is similar in all cases so we treat just an example: \(i_{1}=1,i_{2}=2,i_{3}=2,l_{1}=l_{2}=l_{3}=1\). Then we deal with
And using the ergodic lemma with \(q=2\) we get
\(\square \)
B Estimates of some trigonometric sums
For \(n\in \mathbb {N},\)\(i=0,1,2\) and \(b\in \mathbb {R}_{+}\backprime \{2\pi p;p\in \mathbb {N}\}\) we put
We also denote
The aim of this section is to prove the following lemma:
Lemma B.1
There exists a universal constant \(C\ge 1\) such that for every \(n\in \mathbb {N}\)\( i=0,1,2\) and \(\in \mathbb {R}_{+}\backprime \{2\pi p;p\in \mathbb {N}\}\)
The first step is the following abstract estimate:
Lemma B.2
A. Let \(f\in L^{2}(0,1)\) and let \(\phi (x)=\sum _{k=0}^{\infty }f(x-k)1_{[k,k+1)}(x)\). There exists a universal constant such that for every \(k<n\)
B. Moreover there exists a universal constant C such that, for \( i=0,1,2\)
and in particular, taking \(f=1,\)
The same estimates hold if we replace \(\cos \) by \(\sin \).
Proof of A
We denote \(\alpha _{0}=\int _{0}^{1}f(x)dx\) and
Then using the development in Fourier series of \(\phi \) we obtain
We write
and we use a similar decomposition for \(\sin (2\pi px)\cos (bx)\).
Notice that for every \(\theta >0\) one has
Using these inequalities we obtain
B. We just treat the case \(i=1\) (the other ones are similar). We write
with \(\psi \) associated to g( \(x)=xf(x)\). Using (B.3) (notice that \( \left\| g\right\| _{2}\le \left\| f\right\| _{2})\)
Moreover
so that, by (B.3) we upper bound the above term by
And (B.5) is a particular case of (B.4) with \(f=1\) (so that \(\phi =1)\). \(\square \)
We recall that
We also denote
We also define \(S_{b,i}(s)\) and \(J_{b,i}(s)\) by replacing the \(\cos \) by \( \sin \).
We will prove the following estimates:
Lemma B.3
Let \(a=bn\) with \(0<b\). There exists a universal constant C such that, for \(i=0,1,2\)
with \(\left| \varepsilon _{n}\right| \le C/n\).
Proof
Let us prove (B.8) for \(i=1\) (the proof is analogous for \(i=0\) and \(i=2)\). We write
Moreover
with \(\left| \delta _{n,k}\right| \le 1/n^{2}\) so that \( \sum _{k=1}^{n}\delta _{n,k}=\varepsilon _{n},\) with \(\left| \varepsilon _{n}\right| \le C/n\). We write now (recall that \(a=nb)\)
Summing over k this gives (with \(\varepsilon _{n}\) of order \(\frac{1}{n}\) and which changes from a line to another)
The same computations give
We insert this in the previous estimate and we get
and we are done. \(\square \)
Proof of (B.2)
so (B.2) follows. \(\square \)
C Non degeneracy
In this section we discuss the non degeneracy of the matrix \(\Sigma _{n}(t,s) \) which is the covariance matrix of \(S_{n}(t,s,Y)\). Direct computations show that:
We define \(\Sigma (t,s)\) just by passing to the limit (for fixed t and s):
Then it is easy to check that there exists a universal constant \(C\ge 1\) such that for every \(i,j=1,\ldots ,4\) and every \(0<s<t\)
Notice however that, if \(t-s\approx n\) the above inequality says nothing. So our strategy will be the following: we consider a first case, when \(t-s\le \sqrt{n}\) and then we use the non degeneracy of \(\Sigma (t,s)\) (which we prove in the following lemma) in order to obtain the non degeneracy of \( \Sigma _{n}(t,s)\). And in the case \(\sqrt{n}\le t-s\le n\pi \) we use the estimates from the previous section in order to obtain directly the non degeneracy of \(\Sigma _{n}(t,s)\).
Lemma C.1
For every \(\varepsilon >0\) there exists \(\lambda (\varepsilon )>0\) such that for every t and s such that \(\left| t-s\right| >\varepsilon \) one has
Proof
We first prove that \(\det \Sigma (t,s)>0\) for every \(t\ne s\). Let \((X_t)_{t\ge 0}\) denote a centred stationary Gaussian process whose covariance function is \(r(t,s)=\Gamma (t-s)\), with \(\Gamma (\tau )=\sin \tau /\tau \). Recall that X has smooth paths (see e.g. [4], Ch. 1, Sect. 4.3). It is known (see Ex. 3.5 in [4]) that if the spectral measure of X has at least an accumulation point (in our case the spectral measure is \(1_{|x|<1}dx\)) then, for \(t\ne s\), the law of \(\xi =(X_t,X'_t, X_s, X'_s)\) is non degenerated—consequently the covariance matrix is non degenerated. Straightforward computations give that \(\Sigma (t,s)\) is the covariance matrix of \(\xi \), so \(\det \Sigma (t,s)>0\) when \(t\ne s\).
Now, if \(i\ne j\) one has \(\lim _{t-s\rightarrow \infty } \Sigma ^{i,j}(t,s)=0\), so that \(\lim _{t-s\rightarrow \infty } {\mathrm {det}\,}\Sigma (t,s)= 1/9\). As a consequence, for some \(M>0\) one gets \({\mathrm {det}\,}\Sigma ^{i,j}(t,s)\ge 1/18\) for \(|t-s|>M\). If \(|t-s|\in [\varepsilon ,M]\) the function \((t,s)\mapsto {\mathrm {det}\,}\Sigma (t,s)\) is continuous, so it achieves a strictly positive minimum. The statement now follows. \(\square \)
Corollary C.2
Let \(b_{*}<2\pi \). For \(\varepsilon >0\) let \(\lambda (\varepsilon )\) be given as in Lemma C.1. Then there exists \( n(\varepsilon )\) such that for \(n\ge n(\varepsilon )\) one has
Proof
Suppose first that \(\varepsilon <t-s\le n^{1/2}\). Then
for sufficiently large n.
We consider now the case \(t-s>n^{1/2}\). We will use (B.2) with \(b= \frac{t-s}{n}\) in order to prove that all the terms out of the diagonal are very small, so the determinant will be close to the product of the terms of the diagonal which is (almost) \(\frac{1}{9}\). We look to
Since \(t-s\le b_{*}n\) it follows that \(b=\frac{t-s}{n}\le b_{*}<2\pi \) and this guarantees that \(\overline{b}=\min \{b,2\pi -b_{*}\}\). Since \(nb=t-s\ge \sqrt{n},\) for sufficiently large n we have \(\overline{b} n\ge \sqrt{n}\) and so, by (B.2)
The same is true for the other terms out of the diagonal. \(\square \)
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Bally, V., Caramellino, L. & Poly, G. Non universality for the variance of the number of real roots of random trigonometric polynomials. Probab. Theory Relat. Fields 174, 887–927 (2019). https://doi.org/10.1007/s00440-018-0869-2
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DOI: https://doi.org/10.1007/s00440-018-0869-2
Keywords
- Random trigonometric polynomials
- Edgeworth expansion for non smooth functions
- Kac–Rice formula
- Small balls estimates
Mathematics Subject Classification
- 60G50
- 60F05