An asymptotic formula for the variance of the number of zeroes of a stationary Gaussian process

We study the variance of the number of zeroes of a stationary Gaussian process on a long interval. We give a simple asymptotic description under mild mixing conditions. This allows us to characterise minimal and maximal growth. We show that a small (symmetrised) atom in the spectral measure at a special frequency does not affect the asymptotic growth of the variance, while an atom at any other frequency results in maximal growth.


Introduction
Zeroes of Gaussian processes, and in particular stationary Gaussian processes (SGPs), have been widely studied, with diverse applications in physics and signal processing.The expected number of zeroes may be computed by the celebrated Kac-Rice formula.Estimating the fluctuations, however, proved to be a much more difficult task.For a comprehensive historical account see [25].
The aim of this paper is to give a simple expression which describes the growth of the variance of the number of zeroes in the interval [0, T ], as T → ∞.Following the ideas of Slud [36], it is easy to give a lower bound for this quantity.Our main contribution is a matching upper bound, which holds under a very mild hypothesis.In particular we give a sharp asymptotic expression for the variance for any process with decaying correlations, no matter how slow the decay.
An intriguing feature of our results is the emergence of a 'special frequency': adding an atom to the spectral measure at this frequency does not change the order of growth of the fluctuations.
1.1.Results.Let f : R → R be a stationary Gaussian process (SGP) with continuous covariance kernel r(t) = E[f (0)f (t)].Denote by ρ the spectral measure of the process, that is, the unique finite, symmetric measure on R such that We normalise the process so that r(0) = ρ(R) = 1.It is well-known (see, e.g., [7,Section 7.6]) that the distribution of f is determined by ρ, and further that any such ρ is the spectral measure of some SGP.
We study the number of zeroes of f in a long 'time' interval [0, T ], which we denote The expectation of N (T ) is given by the Kac-Rice formula (see [16,41]) where Throughout we assume that N (T ) has finite variance, which turns out to be equivalent to the Geman condition [13] ε 0 r (t) − r (0) t dt < ∞ for some ε > 0. (2) An SGP f is degenerate if its spectral measure consists of a single symmetrised atom ρ = δ * α = 1 2 (δ α + δ −α ), or equivalently if the covariance is r(t) = cos(αt).In this case the zero set is a random shift of the lattice π α Z, and the variance Var[N (T )] is bounded.We formulate our results in terms of the function The notation A(T ) B(T ) denotes that there exist C 1 , C 2 > 0 such that C 1 ≤ A(T ) B(T ) ≤ C 2 for all T > 0, while A(T ) ∼ B(T ) denotes that lim T →∞ A(T ) B(T ) = 1.Our main result is the following.Theorem 1.For any SGP satisfying we have where the implicit constants depend on ρ.
The condition (4) may be viewed as a very mild mixing condition, which in particular holds whenever the spectral measure is absolutely continuous.In fact, the condition r(t) |t|→∞ −→ 0 implies that ϕ(t) |t|→∞ −→ 0. Under some additional conditions, we are able to compute the leading constant in (5).
Theorem 2. Suppose that r + r σ 2 ∈ L 2 (R), and lim |t|→∞ ϕ(t) = 0. Then In fact, the lower bound in this theorem holds for any process.We state this separately.
Proposition 3.For any SGP, In particular, for any non-degenerate SGP there exists a constant C = C(ρ) > 0 such that The variance for non-degenerate SGPs is therefore always at least linear in T .By stationarity, it is at most quadratic.Our next result characterises these two extremes.The emergence of a special frequency σ in Theorem 4 (b) is new, and intriguing.One naturally asks what the effect of an atom at this frequency is.Notice that modifying a measure by adding an atom at frequency σ does not change E[N (T )].The following result follows from Theorem 1, and shows that the asymptotic growth of Var N (T ) remains unchanged as well -at least under some mild assumptions.
1.2.Discussion.It is evident from our results that the decay of r(t) + r (t) σ 2 determines the variance of N (T ).Observe that r + r σ 2 = F[µ] where the signed measure µ is defined by dµ(λ) = 1 − λ 2 σ 2 dρ(λ); this is crucial to some of our proofs.In fact, it follows from Parseval's identity that where S T (λ) = T 2π sinc 2 T λ 2 .For details, see Section 6.2.Cancellation between the terms of r(t) + r (t) σ 2 plays an important rôle in determining the variance.This is further emphasised in Section 2.7.One consequence of this cancellation is the emergence of the special atom (in the sense of Theorem 4 (b) and Corollary 5).This phenomenon is explained, in part, by the fact that the measure µ does not 'see' σ.
For crossings of non-zero levels, the presence of an atom at any frequency leads to quadratic variance.The existence of a special atom at a distinguished frequency is therefore unique to the zero level.Furthermore, this phenomenon is purely real.No such frequency exists for complex zeroes, see [12].
We remark that, following Arcones [3], many previous results were stated in terms of the function rather than the function ϕ that we introduced in (3).For purposes of comparison, our assumption (4) is implied by the stronger assumption lim sup |t|→∞ ψ(t) < 1 2 .While the condition (4) is a very mild mixing condition, there are some processes with singular spectral measure for which it does not hold.We believe that the result holds in greater generality.
Even the weak form of the conjecture would allow us to prove stronger results, e.g., improve Theorem 4 (a) to completely characterise linear variance and prove that Corollary 5 holds for any θ ∈ [0, 1).We provide further evidence for the conjectures in Section 5.4.
1.3.Background and motivation.The origins for the Kac-Rice method for computing the expected number of zeroes lie in the independent work of Kac [19,20] and of Rice [31,32].Applying this method to SGPs yields the formula (1), even when both sides are infinite, as was done by Ylvisaker [41] and Ito [16].Sufficiency of the Geman condition (2) for finite variance was proved by Cramér and Leadbetter [7, Equation 10.6.2 or 10.7.5], while necessity was established by Geman [13].Qualls [30,Lemma 1.3.4]showed that the Geman condition is equivalent to the spectral condition R log(1 + |λ|)λ2 dρ(λ) < ∞ (see also [4,Theorem 3]).An exact but somewhat inaccessible formula for the variance was rigorously derived 2 by Cramér and Leadbetter [7,.Little progress in understanding the asymptotic growth of the variance was made until Slud [35,36] introduced Multiple Wiener Integral techniques some decades later -these were in turn refined and extended by Kratz and Léon [26,27], using Wiener chaos expansions.
The formulas mentioned above were used to prove various properties of the zeroes, such as sufficient conditions for linearity of the variance and for a central limit theorem (see, e.g., [8,29]).However, extracting the asymptotic growth of the variance under reasonably general conditions has proved fruitless.For example, the only attempt at a systematic study of super-linear growth of the variance that we are aware of is a special family of examples due to Slud [36,Theorem 3.2].In Section 2.5 we use Theorem 2 to improve Slud's result.More generally, one can now analyse a large number of examples, as we do in Section 2, due to the simplicity of the quantity r + r /σ 2 which appears throughout our results.
The case of linear variance was historically of interest.Previously, the only condition for asymptotically linear variance (that we are aware of) was r, r ∈ L 2 (R), which follows from combining the results of Cuzick [8] and Slud [35].We show in Section 2.7 that the condition r + r σ 2 ∈ L 2 (R) is strictly weaker, therefore Theorem 4 (a) improves upon their result.It also follows from their work that r, r ∈ L 2 (R) implies that 1  T Var[N (T )] converges as T → ∞.Ancona and Letendre [1,Proposition 1.11] give an exact expression for this limit (see also [9,Proposition 3.1]), although their main focus is on the growth of the central moments of linear statistics (which generalise the zero count).The lower bound (6) also appears in the work of Lachièze-Rey [28], who studies rigidity and predictability of the zero set.
We finally mention that our work has parallels in different but related models.In the setting of complex zeroes of a random Gaussian analytic f : C → C an asymptotic formula for the variance, an L 2 -condition that guarantees linearity, and a characterisation of maximal (i.e., quadratic) growth were given in [12].Analogous results were then proved for the winding number of a Gaussian stationary f : R → C in [5].
1.4.Outline of our methods.Let us briefly outline our method.We write where π q denotes the projection onto the q'th Wiener chaos.Explicit expressions for this decomposition are well known, it turns out that only the even chaoses contribute, and so we have The diagram formula allows us to compute (see Lemma 8) where P q is a polynomial expression that involves r, r and r .Our lower bound comes from explicitly evaluating the term with q = 1.For the upper bound we establish that r + r σ 2 2 divides the polynomial 3 P q exactly, see Proposition 11.This yields for some C q .The remainder of our proof of the upper bound involves showing that this sequence C q is summable under the given hypothesis. Acknowledgements

Examples
In this section we give a number of examples that expand on or illustrate our results.
2.1.Purely atomic measure.If ρ consists of a single symmetrised atom ρ = δ * al , then f (t) is a degenerate process and thus Var[N (T )] is bounded.However, a superposition of such processes results in a random almost periodic function, with non-trivial behavior.Specifically let α i ∈ R and w i > 0 satisfy i w i = 1 and i w i α 2 i log(1 + |α i |) < +∞ 4 .We consider the measure Then the covariance function is r(t) = i w i cos(α i t) which yields r (t) = − i w i α 2 i cos(α i t) and σ 2 = −r (0) = i w i α 2 i .By Theorem 4 (b) we have quadratic growth Var[N (T )] T 2 .Using Proposition 3, we may give a concrete lower bound, that is, lim inf we omit the details.
2.2.Cuzick-Slud covariance functions.We have already discussed in detail the case r, r ∈ L 2 (R); in this case the limit lim T →∞

Var[N (T )]
T exists and we further have a CLT for N (T ).We mention that a number of classic kernels satisfy this hypothesis, for example, the Paley-Wiener kernel sinc t = sin t t or the Gaussian kernel (sometimes referred to as the Bargman-Fock kernel) e −t 2 .
3 Strictly speaking we first add a small computable quantity, which leads to the difference between Pq and Pq in Section 3. 4 There might only be finitely many αi in which case this second condition is redundant.
, where K α is the modified Bessel function of the second kind of order α.Alternatively 5 note that 5 This argument appears in [36] where X and X are independent Γ(b, 1) random variables.It is easy to check that the hypotheses of Theorem 2 are satisfied and so More generally let L be a function that 'varies slowly at infinity', that is, for every x > 0 we have as t → ∞ and suppose that we have Suppose further that there exist C > 0 and δ < b such that Then Slud [36,Theorem 3.2] showed that and moreover that N (T ) satisfies a non-CLT.Theorem 2 allows us to prove Slud's result, without imposing the hypothesis (9).Indeed, Slud uses this hypothesis to show that the higher order chaoses are negligible, that is, that6 But this is precisely the conclusion of Theorem 2, which applies here since r(t) → 0 as t → ∞.To compute the asymptotic growth of the variance we write, as in (7), Now [10, Proposition 1] implies that ρ T converges locally weakly to the measure ρ 0 with density 1 uniformly on bounded sets we have µ T → ρ 0 locally weakly also.Further, by [10, Equation 1.10], we have ρ0 (t) = |t| −2b .This yields 2.6.Singular continuous spectral measure.If the spectral measure is absolutely continuous with respect to Lebesgue measure, then (4) holds by the Riemann-Lebesgue lemma and we may apply Theorem 1.If there are atoms in the spectral measure then Theorem 4 (b) applies.Singular continuous measures fall between these two stools.Here we give a family of examples that show that there is no simple characterisation for this class of processes. Fix In this case the spectral measure ρ α is the infinite convolution of the atomic measures δ * αn and is usually referred to as a symmetric Bernoulli convolution.By the Jessen-Wintner theorem [18,Theorem 11] it is of pure type, being either absolutely continuous or singular continuous.(In particular it contains no atoms.)Moreover, the support of ρ α (the 'spectrum') is a perfect set and is either compact or all of R, according as ∞ n=1 α n converges or diverges.In the former case we write R n = ∞ i=n+1 α i and the spectrum is a subset of [−R 0 , R 0 ].There are two special cases that are particularly tractable: 1. α n > R n for every n ≥ 1, and 2. α n ≤ R n for every n ≥ 1.In the first case Kershner and Wintner [24, showed that the support of ρ α is nowhere dense (and so is a Cantor-type set) and has total length = 2 lim n→∞ 2 n R n .The measure ρ α is singular if and only if = 0; if > 0 then where Leb is the usual Lebesgue measure on the real line and S α is the spectrum.In Case 2 the spectrum is all of [−R 0 , R 0 ] [24, Page 547], and, as the examples below illustrate, the measure ρ α may be absolutely continuous or singular.
We now list some examples.One particularly simple choice is α n = a n for some a ∈ (0, 1) and we write ρ a and r a for the corresponding spectral measure and covariance function.
• If 0 < a < 1 2 we are in Case 1 and ρ a is singular.For rational a, by [23], r a (t) → 0 as |t| → ∞ if and only if 1 a is not an integer.In this case we have r a (t) = O(| log t| −γ ) for large |t| where γ = γ(a) > 0.
In the particular case a = 1 3 we get the usual Cantor middle-third set (shifted to be contained in [− 1 2 , 1 2 ]) and the distribution function 2 then we are in Case 2. If a = 1 2 then r 1/2 (t) = sinc t (this formula was discovered by Euler), which was covered in Section 2.2.
• More generally if a = ( 12 ) 1/k where k ∈ N then ρ a is absolutely continuous and r a (t) = O(|t| −k ) (see [40,Page 836]).Once more we are covered by Section 2.2.
• Erdős [11, Section 2] showed that if a = 1 b where b = 2 is a 'Pisot-Vijayaraghavan number' (a real algebraic integer whose conjugates lie in the unit disc) then lim sup t→∞ r a (t) > 0 and therefore ρ a is singular.Concrete values of a > 1 2 are a = (the Fibonacci number) and the positive root of the cubic a 3 + a 2 − 1.
• Conversely, Salem [33,Theorem II] showed that if r a (t) → 0 then 1 a is a Pisot-Vijayaraghavan number.This, of course, does not rule out the possibility that there are other values of a > 1 2 for which ρ a is singular but r a (t) → 0, but to the best of our knowledge there has been no progress on this question since.
More involved examples give more sophisticated behaviour, unless otherwise indicated the examples are taken from [18,.
• If the sequence α n consists of the numbers of the form 2 −m! repeated exactly 2 m! times for m = 1, 2, . . .then the spectrum is all of R and r α (t) → 0, which means that ρ α is singular.
We now present a general result which applies in Case 1.Its proof appears in Section 6.3.Proposition 6. Suppose that α n > R n for every n ≥ 1, and define Remark.This is of course a lower bound for Var[N (ρ α ; T )] in general, and gives the correct order of growth when r α (t) → 0.
Let us apply this result to some of the examples above.
is the Hausdorff dimension of the spectrum.It would be interesting to understand if there is any relation between the dimension of the spectrum and the behaviour of the variance for a general singular measure.
(See also the remarks after Lemma 19.) log log T .A less precise lower bound for the variance of this process is claimed in [28, Proposition 11], although the proof given there is not entirely convincing.Nonetheless, we do use some of the ideas found there in our proof of Proposition 6.
• By choosing a sequence R n decaying sufficiently fast, it is clear that we can make the term 2 −M decay arbitrarily slowly.We can therefore construct a process with non-atomic spectral measure whose variance grows faster than T 2−φ(T ) where φ(T ) → 0 arbitrarily slowly, that is, arbitrarily close to maximal growth.

2.7.
Cancellation in the quantity r + r σ 2 .As we indicated previously, an important message of this paper is that the behaviour of the variance is governed by the quantity r + r σ 2 .We wish to emphasise the important rôle of cancellation between the two terms here, and we have already presented some examples of this when the spectral measure has an atom at a 'special frequency'.However this cancellation phenomenon is not just about atoms, and as an illustrative example we will produce a 8 covariance function r such that: Writing dρ(λ) = φ(λ)dλ and applying the Fourier transform we see that it is equivalent to produce a function φ ≥ 0 satisfying: 7 It is actually stated there that (−1) k rα(πk!) → 1 but this is easily seen to be incorrect. 8In fact we produce a family of such covariance functions.
We proceed to produce such a function φ.
and let c 1 , c 2 ∈ R be the solution of the linear system We note that (10) ensures that the determinant of the matrix associated to (11) is positive, and since we also have Then: • Integration yields, by the first equation in (11) • Now the second equation in (11)

A formula for the variance
The goal of this section is to give an infinite series expansion for Var[N (T )], each coming from a different component of the Wiener chaos (or Hermite-Itô) expansion of N (T ).We begin with some notation.For q ∈ N and l, l and Next define the polynomials a q (l 1 )a q (l 2 ) and where We are now ready to state the expansion.
Proposition 7. We have Furthermore The starting point in our calculations is the following Hermite expansion for N (T ) given by Kratz and Léon [27, Proposition 1] assuming only the Geman condition (though they and other authors had considered it previously under more restrictive assumptions).We have (the sum converges in L 2 (P)) where 10   N q (T ) = q l=0 a q (l) and H l is the l'th Hermite polynomial.Further each N q (T ) belongs to the 2q'th Wiener chaos which yields and Furthermore The next lemma allows us to evaluate where P q is given by (14).
We now show how this lemma yields the desired expression.
Proof of Proposition 7, assuming Lemma 8. Lemma 8 yields 10 Under the Geman condition, one cannot assume that f is continuously differentiable, and 'conversely' a continuously differentiable process need not satisfy the Geman condition, see [13,Section 4].However the existence of r implies the existence of the derivative in quadratic mean of the process, and this is how the object f should be understood if the process is not differentiable.
Note that r(t and so We therefore have We identify the last series as for all |x| ≤ 1 implying that where the last equality follows from arccos(x) = π 2 − arcsin(x).We now proceed to prove Lemma 8.
Proof of Lemma 8. Squaring the expression for N q (T ) given in (19) yields and so a q (l 1 )a q (l 2 ) Applying Lemma 9 below, and using the simple change of variables Noting that r is an even function and that only even powers of y appear in P q yields Lemma 8.
Lemma 9.For all q ∈ N and l 1 , l 2 ∈ N 0 such that 0 ≤ l 1 , l 2 ≤ q we have Before proving the lemma we first recall the diagram formula.• There are n i vertices labelled X i for each i (and each vertex has a single label).For a vertex a we write X (a) for the label of a.
• Each vertex has degree 1.
• No edge joins 2 vertices with the same label.Let D be the set of such diagrams.For γ ∈ D we define the value of γ to be where E(γ) is the set of edges of γ.Then Proof of Lemma 9. We apply the diagram formula to the random variables f (t), f (s), f (t)/σ and f (s)/σ and corresponding integers 2(q − l 1 ), 2(q − l 2 ), 2l 1 and 2l 2 and denote by D the collection of relevant Feynman diagrams.Since it is enough to consider diagrams whose edges do not join vertices labeled f (t) to f (t)/σ or vertices labeled f (s) to f (s)/σ.
Let n be the number of edges joining a vertex labeled f (t)/σ to a vertex labeled f (s)/σ, see Figure 1.Then 0 ≤ n ≤ min(2l 1 , 2l 2 ).Moreover, as the other vertices labeled f (t)/σ must be joined to vertices labeled f (s), we see that Further, every value of n in this range is attained by some diagram.
We compute the value of such a diagram to be Finally, we count the number of such diagrams.There are Counting the number of Feynman diagrams ways to choose n vertices labeled f (t)/σ, to choose n vertices labeled f (s)/σ and to pair them.There are ways to choose 2l 1 − n vertices labeled f (s) and to pair them with the remaining vertices labeled ways to choose 2q − 2l 1 − 2l 2 + n vertices labeled f (t) and to pair them with the remaining ones labeled f (s).There are ways to pair the remaining vertices labeled f (t) and f (s)/σ.Since these choices are independent, we multiply these counts to get that there are b q (l 1 , l 2 , n) such diagrams, where b q is given by (13).
Applying the diagram formula completes the proof.

Lower bound
In this section we prove Proposition 3. From Proposition 7 we have and the first statement of Proposition 3 follows simply by computing To deduce the second statement it is enough to find an interval I such that r + r σ 2 ≥ C > 0 on I.But this follows from the fact that r is continuous and r is not cosine.

Upper bound
In this section, we prove Theorems 1 and 2. Our method is to bound each V q (T ) by V 1 (T ) and apply Proposition 7. We achieve this by proving the following properties of the polynomials P q (recall (15)).Proposition 11.For all q ≥ 1 we have (x + z) 2 | P q (x, y, z).
Proving Proposition 11 amounts to proving some identities for the coefficients of the polynomials P q , which is deferred to Section 7 where we implement a general method due to Zeilberger [2].We proceed to prove Proposition 12.
5.1.Proof of Proposition 12.By Proposition 11, we may prove Proposition 12 by bounding the second derivative of P q .To achieve this we borrow the main idea from the proof of Arcones' Lemma [3, Lemma 1].
We now bound Algebraic manipulation of this last quantity yields Applying the Binomial Theorem to the last term gives where the last identity is due to Lemma 13 below, and we remind the reader of ( 16).We also have, from (15), that We next bound this final summand.Note that for q = 1 this term vanishes.Otherwise, on the domain D M = {|x| + |y| ≤ M, |y| + |z| ≤ M }, it attains its maximum on the boundary, and a calculation reveals the maximum is attained at |z| = |x| = M, y = 0. Therefore Combining these two estimates we obtain Using Sterling's bounds 11 we see that 2q √ q which yields By the mean value theorem, P q (x, y, z) = P q (−z, y, z) + ∂P q ∂x (−z, y, z)(x + z) + 1 2 for some t between x and −z.It follows from Proposition 11 that P q (−z, y, z) = ∂Pq ∂x (−z, y, z) = 0, so that P q (x, y, z) = 1 2 Note that |t| ≤ max(|x|, |z|) ≤ M − |y| and so by (26) we have In the course of the proof we used the following computation.
Lemma 13.For all q ∈ N we have Proof.For q ≥ 0, let us denote where We next compute φ.We have +2 arcsin(2x)+C for some constant C. Since all the functions in this equation are odd, it follows that C = 0, and so φ(x) = √ 1 − 4x 2 + 2x arcsin(2x).Therefore, using the Taylor series (22) once more, 11 The constants here are not asymptotically optimal, but this is irrelevant for our purposes.

Proof of Theorem 1.
Having Proposition 12 at our disposal, we are ready to prove Theorem 1.Let and choose M ∈ (M , 1).Then there exists some T 0 > 0 such that ϕ(t) ≤ M for all |t| > T 0 .We can rearrange (17) to obtain Proposition 12 yields see (23).Since M < 1 we see that By Proposition 7, since we are assuming the Geman condition, we have ∞ q=1 Vq(T ) 4 q < ∞ for every T > 0 and so we may write, from (28) Combining this with (29) we get where C 0 , C 1 and C 2 depend on T 0 and M .Recalling Proposition 7 we have where we have used the lower bound proved in Section 4 for the final bound.

Proof of Theorem 2. By (23) we need to show that Var[N (T )]
The lower bound follows immediately from Proposition 3 and so we focus on the upper bound.We proceed as in the previous section, but estimate more carefully.By Proposition 7 we have Now fix ε > 0 and choose T 0 = T 0 (ε) such that ϕ(t) < ε for all t > T 0 .As in the previous section we write and estimate This yields and we finally note that since r as T → ∞.This completes the proof.

Conjectural Bounds.
In this section we give some evidence in favor of the conjectures stated in the Introduction.The precise expression for the variance appearing in Proposition 7 establishes a way to prove even tighter upper bounds, by reducing to combinatorial statements about the polynomials P q , defined in (15).It is not difficult to see that the vector (r(t), r (t)/σ, r (t)/σ 2 ) always lies in the domain By Proposition 11, R q (x, y, z) = P q (x, y, z)/(x + z) 2 is a homogeneous polynomial and since D contains all segments to the origin, it follows that R q attains the maximum of its absolute value on the boundary.We expect that the maximum should be obtained at the points where |x| = |z|.When x = −z, the same techniques employed in this paper show the value to be and so on this boundary component the value of R q is 2 2q−1 .We believe that this bound is the one relevant to Gaussian processes, however numerical computations suggest that R q can be much larger at the points where x = z.We believe that there is some 'hidden' structure that prevents r(t) from being close to r (t)/σ 2 in certain subregions of D. For example, if r(t) is close to 1 then we should be close to a local maximum and so we would expect r (t) to be negative.Understanding the 'true domain' where the vector (r(t), r (t)/σ, r (t)/σ 2 ) 'lives' already appears to be a quite interesting question.
6. Singular spectral measure 6.1.Atoms in the spectral measure: proofs of Theorem 4 (b) and Corollary 5.In this section we consider the effect of atoms in the spectral measure, that is, we prove Theorem 4 (b) and Corollary 5. Our proof relies on the following proposition.
Proposition 14.Let µ be a signed-measure with R d|µ| < ∞.Then µ contains an atom if and only if there exists c > 0 such that We postpone the proof of Proposition 14 to Section 6.2.We will also need the following result.
Lemma 15.Let f be a SGP with covariance kernel r, spectral measure ρ and suppose that ρ has a continuous component.Let Write and notice that F is periodic with period π.This yields Moreover, since F is even we have Substituting u = |A| cos(y) we obtain Inserting this value into (30) yields the result.
Assume that ρ has an atom at a point different from σ.By ( 18) and ( 23), to show that Var[N (T )] But this follows from Proposition 14 if we define the signed measure µ by dµ(λ) = (1 − λ 2 σ 2 )dρ(λ) and notice that μ = r + r σ 2 and that µ has an atom.For the converse, notice that it is enough to check that for integer J we have since this implies that Var[N (T )] = o(T 2 ), by stationarity.Assume first that ρ has no atoms; we adapt the proof of [5,Thm 4].By the Fomin-Grenander-Maruyama theorem, f is an ergodic process (see, e.g., [15,Sec. 5.10]).By standard arguments, this also implies that the sequence is ergodic.Recall that we assume the Geman condition, which implies that the first and second moments of are finite.Thus, by von Neumann's ergodic theorem, we have where the convergence is both in L 1 and L 2 (see [39,Cor. 1.14.1]).We conclude that Finally suppose that ρ = θρ c + (1 − θ)δ * σ where 0 < θ < 1 and ρ c has no atoms.We may represent the corresponding process as where f c is a SGP with spectral measure ρ c , X ∼ χ 2 (2), Φ ∼ Unif([0, 2π]), and moreover f c , X and Φ are pairwise independent.By the law of total variance and Lemma 15 we have We define, for A ∈ R and α ∈ [0, 2π], As before the process f c is ergodic, and so is the sequence N A,α j for fixed A and α.This implies that (almost surely), exactly as before.Furthermore, using stationarity we have and using (31) we see that since we assume the Geman condition.It follows from dominated convergence that lim Proof of Corollary 5. Let M = lim sup |t|→∞ ϕ(t), where ϕ is defined in (3).By assumption we have M < 1 and we define We would like to apply Theorem 1 to the spectral measure ρ θ .Writing we have r θ (t) = (1 − θ)r(t) + θ cos(σt), and σ 2 θ = −r θ (0) = σ 2 .We accordingly compute and so lim sup |t|→∞ ϕ θ (t) < 1 for θ < θ 0 .Applying Theorem 1 to ρ θ and to ρ we obtain Var[N (ρ; T )].
6.2.Proof of Proposition 14.We begin with a review of some elementary harmonic analysis that we will need, for more details and proofs see, e.g., Katznelson's book [21,Ch. VI].Let M(R) denote the space of all finite signed measures on R endowed with the total mass norm µ 1 = R d|µ|.Recall that the convolution of two measures µ, ν ∈ M(R) is given by (µ * ν)(E) = µ(E − λ)dν(λ) for any measurable set E and satisfies µ * We identify a function f ∈ L 1 with the measure whose density is f .
The following lemma is a version of Parseval's identity, see [21,VI 2.2].
A simple application of Parseval's identity proves our next lemma.
Lemma 17. Suppose that µ, ν ∈ M(R) and S, F[S] ∈ L 1 (R).Then Proof.Note that S * µ is a function and further that A simple application of Lemma 16 finishes the proof.
We will also use the so-called 'triangle function' Notice that applying Lemma 17 to these functions, we obtain which is (7).
We are now ready to prove Proposition 14.First suppose that µ contains an atom at α. Write µ = µ 1 + µ 2 where µ 1 = cδ α for some c = 0 and µ 2 ({α}) = 0. Note that We have Using this and Lemma 17 we obtain It is therefore enough to show that R (S T * µ 1 ) dµ 2 = o(T ).We bound . By (32) we have This concludes the first part of the proof.
Conversely, suppose that µ contains no atoms.Recall that We will show that |(S T * µ)(λ)| = o(T ), uniformly in λ, which will conclude the proof.As before, denoting It therefore suffices to prove the following claim.
Claim 18.Let ν be a non-negative, finite measure on R that contains no atoms.Then . It is clear that m(ε) decreases with ε so m(ε) must converge as ε ↓ 0 to some non-negative limit, 2δ ≥ 0. Suppose that δ > 0 and choose Recall that by definition of δ we have m N 2 n ≥ 2δ.We conclude that for every n ∈ N we can find ) Next we shall construct a sequence of nested dyadic intervals {J n } ∞ n=0 such that, for all n, J n ∈ D n , J n+1 ⊆ J n , ν(J n ) ≥ δ.This will imply, by Cantor's lemma, that n J n = {x} for some x ∈ R, and further that ν({x}) = lim n→∞ ν(J n ) ≥ δ > 0. This contradicts the assumption that ν has no atoms, which will end our proof.
We start by setting J 0 = [−N, N ].Suppose that we have constructed J 0 ⊃ J 1 ⊃ J 2 ⊃ • • • ⊃ J m such that for every n > m we can find I n ∈ D n that satisfies that is, the interval J m has a descendant of any generation whose ν-measure is at least δ.Notice that this holds for m = 0 by (33).Notice that if (34) fails for both descendants of J m in the generation D m+1 , then it also fails for J m , since ν(J) ≥ ν(J ) for every descendant J ⊆ J.This completes the inductive construction of J m and consequently the proof.
for µ-almost every λ.It follows from this lemma that the measure ρ a is exact dimensional for 0 < a < 1 2 , with d = log 2/ log 1 a , and moreover log ρ a ((λ − δ, λ + δ)) uniformly for every λ in the support of ρ a .2. If 1 2 ≤ a < 1 then the measure ρ a is also exact dimensional, but understanding more detailed properties seems to be a difficult question, related to certain notions of entropy.We refer the reader to the surveys [14,37] for a more thorough discussion.3.For any compactly supported exact dimensional spectral measure ρ, of dimension d, such that (35) converges uniformly for ρ-almost every λ, we may imitate the proof of Proposition 6 to yield for any ε > 0 and some c, C > 0.
Proof.(i) We begin by recalling Kershner and Wintner's proof that the spectrum is a Cantor type We inductively construct C n to consist of 2 n (closed) intervals formed by deleting an interval of length 2(α n − R n ) from the centre of each of the intervals in the previous generation.
and assigns a mass of 2 −n to each of the points of S n .Furthermore, the elements of S n are the midpoints of the intervals that make up C n , and if s, s ∈ S n are distinct then |s − s | ≥ 2α n > 2R n .
We conclude that if |I| < R n then I contains at most one element of S n and so ρ n (I) ≤ 2 −n .Moreover I can only intersect with the descendants of one of the intervals of C n , and so ρ m (I) ≤ 2 −n for every m ≥ n.
It remains to note that ρ m converges weakly to ρ, and that an interval is always a continuity set of the measure ρ, whence ρ(I) = lim m→∞ ρ m (I) ≤ 2 −n .
(ii) In fact the measure ρ n introduced above is the law of the random variable where ε j denotes a sequence of i.i.d.Rademacher 13 random variables, while ρ is the law of the infinite sum For λ in the support of ρ we can write λ = ∞ j=1 ε j α j for some fixed sequence ε j ∈ {−1, 1} and we write We now proceed to prove Proposition 6.
Proof of Proposition 6.Using (7) we see that we wish to prove that where, as before, dµ(λ) = 1 − λ 2 σ 2 dρ(λ).Fix A > 4 (to be chosen large) and 1 2 < β < 1 and define M by R M < 2A T ≤ R M −1 .We claim that 14 for some constant η > 0 (depending only on ρ) and that Let us first see how the claims imply the proposition.
Inserting this into our claims and combining them we get 13 That is, P[εj = 1] = P[εj = −1] = 1 2 . 14Throughout this proof c, c , C and C denote positive constants whose exact value is irrelevant and which may vary from one occurrence to the next.They may depend on the measure ρ but are independent of A and T .
We now fix A so large that the lower bound in this expression is at least η for sufficiently large T .We finally bound since β > 1 2 and A has been fixed.This yields (37).It remains only to establish the two claims.We begin with the terms near the diagonal.Since the spectrum is compact we note that 1 − λ 2 σ 2 is bounded on the support of ρ, and using part (i) of Lemma 19 we see that where η > 0, as desired.
We finally estimate the off-diagonal contribution.We have where the second equality holds for z not a non-positive integer.We next reformulate Proposition 21 in terms of the purely hypergeometric terms H q (l 1 , l 2 , k) = (2q − l 1 − l 2 − k)H q (l 1 , l 2 , k), in order to be able to apply Zeilberger's algorithm in Section 7.3.We note that H q , H q are defined for every k, l 1 , l 2 ∈ Z, by expressing everything in terms of the Gamma function.
First, we will handle the case where k = q.
Proof of Proposition 22 (a).We proceed by induction on q.For the base case note that , for k = 0, which is exactly the case q = 1.Similarly, one verifies the formula for q = 2.
the number of crossings of the curve ψ by the process.Then E[N J (ψ)] = J.Proof.Denote the Gaussian density function by ϕ and by Φ the corresponding distribution function.The generalised Rice formula [7, Equation 13.2.1]gives

P 1 (
x, y, z) = 2(x + z) 2 whence, recalling(40) and the relation S q (k) = 2 −4q d q (k), . Mikhail Sodin first suggested the project to us.We had a number of interesting and fruitful discussions with Yan Fyodorov, Marie Kratz and Igor Wigman on various topics related to this article.Eugene Shargorodsky contributed the main idea in the proof of Claim 18.We are thankful to Ohad Feldheim for suggesting improvements to the presentation of this paper.
2.3.Exponential kernel and approximations.Consider the Ornstein-Uhlenbeck (OU) process, defined by the covariance function r(t) = e −|t| .This process has attracted considerable attention since it arises as a time-space change of Brownian motion.Since the covariance is not differentiable at the origin, none of our results may be directly applied.However, one may approximate the OU process by differentiable processes.
Lemma 10 (The diagram formula [6, Page 432; 17, Theorem 1.36]).Let X 1 , . . ., X k be jointly Gaussian random variables, and n 1 , . . ., n k ∈ N. A Feynman diagram is a graph with n 1 + . . .+ n k vertices such that 6.3.Singular continuous measures: Proof of Proposition 6.Throughout this section we assume the notation of Section 2.6 and that α n > R n for every n ≥ 1.We begin with the following observation.