Persistence as a spectral property

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1 , Georgia Tech. Joint work with Naomi Feldheim and Ohad Feldheim. March 18, 2017

2 Advertisement. SEAM2018, March 23-25, Georgia Tech, Atlanta, GA. Organizing committee: Michael Lacey Wing Li Galyna Livshyts

3 Gaussian stationary processes (GSP) Let T =Z or R. Definition: Stationary Gaussian process A random sequence f : T R is said to be: Gaussian if all finite marginals are Gaussian, i.e. (f (x 1 ),...f (x N )) N R N (0,Σ k1,...,k N ), for all N IN and x 1,...,x N T. Stationary if it is shift invariant in distribution, i.e. (f (x 1 + y),...f (x N + y)) d (f (x 1 ),...f (x N )), for all N IN, x 1,...,x N T, and y T.

4 Gaussian stationary processes (GSP) Definition: The covariance function Given a Gaussian process f let r(x,y) =E(f (x)f (y)) x,y T. If the process is stationary then r(x,y) = r(x y,0) =: r(x y). We call r(x) the covariance function of the process f.

5 Gaussian stationary processes (GSP) Definition: The covariance function Given a Gaussian process f let r(x,y) =E(f (x)f (y)) x,y T. If the process is stationary then r(x,y) = r(x y,0) =: r(x y). We call r(x) the covariance function of the process f. Assumption: Over R we assume r(x) to be continuous.

6 Gaussian stationary processes (GSP) Definition: The covariance function Given a Gaussian process f let r(x,y) =E(f (x)f (y)) x,y T. If the process is stationary then r(x,y) = r(x y,0) =: r(x y). We call r(x) the covariance function of the process f. Assumption: Over R we assume r(x) to be continuous. Definition: The spectral measure By Bochner s theorem there exists a positive measure ρ f over [ π,π] or R s.t. ˆρ f (x) = r(x). ρ f is called the spectral measure of the process f.

7 Gaussian stationary processes (GSP) Definition: The covariance function Given a Gaussian process f let r(x,y) =E(f (x)f (y)) x,y T. If the process is stationary then r(x,y) = r(x y,0) =: r(x y). We call r(x) the covariance function of the process f. Assumption: Over R we assume r(x) to be continuous. Definition: The spectral measure By Bochner s theorem there exists a positive measure ρ f over [ π,π] or R s.t. ˆρ f (x) = r(x). ρ f is called the spectral measure of the process f. Proposition A stationary Gaussian process is determined uniquely by its covariance function and therefore also by its spectral measure. Moreover, there is a 1 1 correspondence between stationary gaussian processes f and symmetric, positive, Borel measures ρ.

8 Example - Gaussian wave ξ 1,ξ 2 i.i.d. N (0,1) f (x) = ξ 0 sin(x) + ξ 1 cos(x) r(x) = cos(x) ρ = 2 1 (δ 1 + δ 1 ) Covariance Kernel Three Sample Paths Spectral Measure

9 Example - Almost periodic wave ξ 1,ξ 2,ξ 3,ξ 4 i.i.d. N (0,1) f (x) = ξ 0 sin(x) + ξ 1 cos(x) + ξ 2 sin( 2x) + ξ 3 cos( 2x) r(x) = cos(x) + cos( 2x) ( ρ = 1 2 δ1 + δ 1 + δ 2 + δ ) Covariance Kernel Three Sample Paths Spectral Measure

10 Example - i.i.d. sequence ξ n i.i.d. N (0,1) f (n) = ξ n 1 Covariance Kernel r(n) = δ n,0 dρ(λ) = 1 2π 1I [ π,π](λ)dλ Three Sample Paths Spectral Measure

11 Example - Sinc kernel ξ n i.i.d. N (0,1) f (x) = ξ n sinc(x n) n IN r(x) = sin(πx) = sinc(x) πx dρ(λ) = 1 2π 1I [ π,π](λ)dλ Covariance Kernel 2 Three Sample Paths Spectral Measure

12 Toy Examples over Z f (k) are iid with f (k) N R (0,1)

13 Toy Examples over Z f (k) are iid with f (k) N R (0,1) (r(k)) k = e 0 := (...,0,1,0,...).

14 Toy Examples over Z f (k) are iid with f (k) N R (0,1) (r(k)) k = e 0 := (...,0,1,0,...). ρ f is the Lebesgue measure.

15 Toy Examples over Z f (k) are iid with f (k) N R (0,1) (r(k)) k = e 0 := (...,0,1,0,...). ρ f is the Lebesgue measure. f (k) = g(k + 1) g(k) where g(k) are iid with g(k) N R (0,1)

16 Toy Examples over Z f (k) are iid with f (k) N R (0,1) (r(k)) k = e 0 := (...,0,1,0,...). ρ f is the Lebesgue measure. f (k) = g(k + 1) g(k) where g(k) are iid with g(k) N R (0,1) (r(k)) k = 1/2e 1 + e 0 1/2e 1 := (...,0, 1/2,1, 1/2,0,...).

17 Toy Examples over Z f (k) are iid with f (k) N R (0,1) (r(k)) k = e 0 := (...,0,1,0,...). ρ f is the Lebesgue measure. f (k) = g(k + 1) g(k) where g(k) are iid with g(k) N R (0,1) (r(k)) k = 1/2e 1 + e 0 1/2e 1 := (...,0, 1/2,1, 1/2,0,...). dρ f (t) = (1 cost)dt.

18 Toy Examples over Z f (k) are iid with f (k) N R (0,1) (r(k)) k = e 0 := (...,0,1,0,...). ρ f is the Lebesgue measure. f (k) = g(k + 1) g(k) where g(k) are iid with g(k) N R (0,1) (r(k)) k = 1/2e 1 + e 0 1/2e 1 := (...,0, 1/2,1, 1/2,0,...). dρ f (t) = (1 cost)dt. f (k) = f (0) where g(0) N R (0,1)

19 Toy Examples over Z f (k) are iid with f (k) N R (0,1) (r(k)) k = e 0 := (...,0,1,0,...). ρ f is the Lebesgue measure. f (k) = g(k + 1) g(k) where g(k) are iid with g(k) N R (0,1) (r(k)) k = 1/2e 1 + e 0 1/2e 1 := (...,0, 1/2,1, 1/2,0,...). dρ f (t) = (1 cost)dt. f (k) = f (0) where g(0) N R (0,1) (r(k)) k = (...,1,1,1,...).

20 Toy Examples over Z f (k) are iid with f (k) N R (0,1) (r(k)) k = e 0 := (...,0,1,0,...). ρ f is the Lebesgue measure. f (k) = g(k + 1) g(k) where g(k) are iid with g(k) N R (0,1) (r(k)) k = 1/2e 1 + e 0 1/2e 1 := (...,0, 1/2,1, 1/2,0,...). dρ f (t) = (1 cost)dt. f (k) = f (0) where g(0) N R (0,1) (r(k)) k = (...,1,1,1,...). ρ f = δ 0.

21 Persistence Definition: Persistence The persistence of a shift invariant random sequence f :Z R is defined by P f (N) := Prob(f (k) 0, k [0,...,N])

22 Persistence Definition: Persistence The persistence of a shift invariant random sequence f :Z R is defined by P f (N) := Prob(f (k) 0, k [0,...,N]) One of the main questions in this area: Question: Estimate the size of P f (N) as N, i.e. P f (N)? as N.

23 Motivation Engineering: f is a random noise or wave transition. ( : Rice, Slepian, Newell, Rosenblatt, Piterbarg, Kolmogorov and others. These results are applicable mainly when r is non-negative or absolutely summable. ). Physics: A toy model in statistical mechanics for electrons in matter, diffusion, spin systems. (1990 ) Related to random zeroes and random nodal lines of polynomials and GAF s. What are possible asymptotic behaviors of the persistence probability of a GSP on large intervals? What features of the covariance function determine this behavior? (Slepian, 1962)

24 Persistency-Returning to the toy examples f (k) are iid with f (k) N R (0,1) (r(k)) k = e 0 := (...,0,1,0,...). ρ f is the Lebesgue measure.

25 Persistency-Returning to the toy examples f (k) are iid with f (k) N R (0,1) (r(k)) k = e 0 := (...,0,1,0,...). ρ f is the Lebesgue measure. P f (N) = 1 N log2 = e 2 N

26 Persistency-Returning to the toy examples f (k) are iid with f (k) N R (0,1) (r(k)) k = e 0 := (...,0,1,0,...). ρ f is the Lebesgue measure. P f (N) = 1 N log2 = e 2 N f (k) = g(k + 1) g(k) where g(k) are iid with g(k) N R (0,1) (r(k)) k = 1/2e 1 + e 0 1/2e 1 := (...,0, 1/2,1, 1/2,0,...). dρ f (t) = (1 cost)dt.

27 Persistency-Returning to the toy examples f (k) are iid with f (k) N R (0,1) (r(k)) k = e 0 := (...,0,1,0,...). ρ f is the Lebesgue measure. P f (N) = 1 N log2 = e 2 N f (k) = g(k + 1) g(k) where g(k) are iid with g(k) N R (0,1) (r(k)) k = 1/2e 1 + e 0 1/2e 1 := (...,0, 1/2,1, 1/2,0,...). dρ f (t) = (1 cost)dt. P f (N) = N! 1 log N e N

28 Persistency-Returning to the toy examples f (k) are iid with f (k) N R (0,1) (r(k)) k = e 0 := (...,0,1,0,...). ρ f is the Lebesgue measure. P f (N) = 1 N log2 = e 2 N f (k) = g(k + 1) g(k) where g(k) are iid with g(k) N R (0,1) (r(k)) k = 1/2e 1 + e 0 1/2e 1 := (...,0, 1/2,1, 1/2,0,...). dρ f (t) = (1 cost)dt. P f (N) = N! 1 log N e N f (k) = f (0) where g(0) N R (0,1) (r(k)) k = (...,1,1,1,...). ρ f = δ 0.

29 Persistency-Returning to the toy examples f (k) are iid with f (k) N R (0,1) (r(k)) k = e 0 := (...,0,1,0,...). ρ f is the Lebesgue measure. P f (N) = 1 N log2 = e 2 N f (k) = g(k + 1) g(k) where g(k) are iid with g(k) N R (0,1) (r(k)) k = 1/2e 1 + e 0 1/2e 1 := (...,0, 1/2,1, 1/2,0,...). dρ f (t) = (1 cost)dt. P f (N) = N! 1 log N e N f (k) = f (0) where g(0) N R (0,1) (r(k)) k = (...,1,1,1,...). ρ f = δ 0. P f (N) = 1 2

30 Persistence: Progress from recent years Theorem A [Antezana, Buckley, Marzo and Olsen, 12] For the sinc-kernel (dρ f = 11 [ π,π] ) : e cn < P(N) < 2 N

31 Persistence: Progress from recent years Theorem A [Antezana, Buckley, Marzo and Olsen, 12] For the sinc-kernel (dρ f = 11 [ π,π] ) : e cn < P(N) < 2 N Theorem B [Feldheim and Feldheim, 13] Assume that dρ f = φ(x)dx and 0 < δ φ(x) M on some interval [ a,a] then Ae αn P f (N) Be βn N IN.

32 Persistence: Progress from recent years Theorem A [Antezana, Buckley, Marzo and Olsen, 12] For the sinc-kernel (dρ f = 11 [ π,π] ) : e cn < P(N) < 2 N Theorem B [Feldheim and Feldheim, 13] Assume that dρ f = φ(x)dx and 0 < δ φ(x) M on some interval [ a,a] then Ae αn P f (N) Be βn N IN. Theorem D [Dembo and Mukherjee, 15] Roughly speaking: If r(x) is positive and dρ f = φ(x) is unbounded near zero then P f (N) >> e βn.

33 Persistence: Progress from recent years Theorem A [Antezana, Buckley, Marzo and Olsen, 12] For the sinc-kernel (dρ f = 11 [ π,π] ) : e cn < P(N) < 2 N Theorem B [Feldheim and Feldheim, 13] Assume that dρ f = φ(x)dx and 0 < δ φ(x) M on some interval [ a,a] then Ae αn P f (N) Be βn N IN. Theorem D [Dembo and Mukherjee, 15] Roughly speaking: If r(x) is positive and dρ f = φ(x) is unbounded near zero then P f (N) >> e βn. Theorem C [ Krishnapur and Maddaly, 15] Over Z. If ρ f is absolutely continuous (or has a non vanishing absolutely continuous part), then P f (N) Ce γn2 N IN.

34 Our main result over Z a particular case Theorem [Feldheim, Feldheim, N, 2016] Let b,a > 0 and γ > 1. Suppose f is a GSP over Z with spectral measure dρ f = φ f (x)dx. 1. If φ f (x) b x γ for almost every x [ a,a], then Ce cn1+γ logn γ < 0 P f (N) Ce cn γ = 0 Ce c(γ)nlogn γ > 0

35 Our main result over Z a particular case Theorem [Feldheim, Feldheim, N, 2016] Let b,a > 0 and γ > 1. Suppose f is a GSP over Z with spectral measure dρ f = φ f (x)dx. 1. If φ f (x) b x γ for almost every x [ a,a], then Ce cn1+γ logn γ < 0 P f (N) Ce cn γ = 0 Ce c(γ)nlogn γ > 0 2. If φ f (x) b x γ for almost every x [ a,a], then Ce cn1+γ logn γ < 0 P f (N) Ce cn γ = 0 Ce c(γ)nlogn γ > 0

36 Our main result over Z a particular case Theorem [Feldheim, Feldheim, N, 2016] Let b,a > 0 and γ > 1. Suppose f is a GSP over Z with spectral measure dρ f = φ f (x)dx. 1. If φ f (x) b x γ for almost every x [ a,a], then Ce cn1+γ logn γ < 0 P f (N) Ce cn γ = 0 Ce c(γ)nlogn γ > 0 2. If φ f (x) b x γ for almost every x [ a,a], then Ce cn1+γ logn γ < 0 P f (N) Ce cn γ = 0 Ce c(γ)nlogn γ > 0 3. If φ f (x) = 0 for almost every x [ a,a], then (C 1 e c1n2 ) P f (N) C 2 e c2n2

37 Our main result over Z a particular case Theorem [Feldheim, Feldheim, N, 2016] Let b,a > 0 and γ > 1. Suppose f is a GSP over Z with spectral measure dρ f = φ f (x)dx. 1. If φ f (x) b x γ for almost every x [ a,a], then Ce cn1+γ logn γ < 0 P f (N) Ce cn γ = 0 Ce c(γ)nlogn γ > 0 2. If φ f (x) b x γ for almost every x [ a,a], then Ce cn1+γ logn γ < 0 P f (N) Ce cn γ = 0 Ce c(γ)nlogn γ > 0 3. If φ f (x) = 0 for almost every x [ a,a], then (C 1 e c1n2 ) P f (N) C 2 e c2n2 Remarks: "The same theorem", "Absolute continuity", "Estimates in between".

38 Our main result over R a particular case Theorem [Feldheim, Feldheim, N, 2016] Let b,a > 0 and γ > 1. Suppose f is a GSP over R with spectral measure dρ f = φ f (x)dx. 1. If φ f (x) b x γ for almost every x [ a,a], then Ce cn1+γ logn γ < 0 P f (N) Ce cn γ = 0 Ce c(γ)nlogn γ > 0

39 Our main result over R a particular case Theorem [Feldheim, Feldheim, N, 2016] Let b,a > 0 and γ > 1. Suppose f is a GSP over R with spectral measure dρ f = φ f (x)dx. 1. If φ f (x) b x γ for almost every x [ a,a], then Ce cn1+γ logn γ < 0 P f (N) Ce cn γ = 0 Ce c(γ)nlogn γ > 0 2. If φ f (x) b x γ for almost every x [ a,a], then Ce cn1+γ logn γ < 0 P f (N) Ce cn γ = 0 γ > 0

40 Our main result over R a particular case Theorem [Feldheim, Feldheim, N, 2016] Let b,a > 0 and γ > 1. Suppose f is a GSP over R with spectral measure dρ f = φ f (x)dx. 1. If φ f (x) b x γ for almost every x [ a,a], then Ce cn1+γ logn γ < 0 P f (N) Ce cn γ = 0 Ce c(γ)nlogn γ > 0 2. If φ f (x) b x γ for almost every x [ a,a], then Ce cn1+γ logn γ < 0 P f (N) Ce cn γ = 0 γ > 0 3. If the spectral measure vanishes on [ a,a], and on [a, ) it satisfies φ(x) x 100, then P f (N) Ce cecn.

41 Intuition: Spectral perspective Both the covariance and the spectral measure are linear under sums of independent GSPs. Observation If f = f 1 f 2 where f 1, f 2 are independent GSPs, then r = r 1 + r 2, ρ = ρ 1 + ρ 2.

42 Intuition: Spectral perspective Both the covariance and the spectral measure are linear under sums of independent GSPs. Observation If f = f 1 f 2 where f 1, f 2 are independent GSPs, then r = r 1 + r 2, ρ = ρ 1 + ρ 2. A decomposition ρ = ρ 1 + ρ 2 defines f = f 1 f 2 iff ρ 1 0, ρ 2 0 and are both symmetric

43 Intuition: Spectral perspective Both the covariance and the spectral measure are linear under sums of independent GSPs. Observation If f = f 1 f 2 where f 1, f 2 are independent GSPs, then r = r 1 + r 2, ρ = ρ 1 + ρ 2. A decomposition ρ = ρ 1 + ρ 2 defines f = f 1 f 2 iff ρ 1 0, ρ 2 0 and are both symmetric

44 Intuition: Spectral perspective Both the covariance and the spectral measure are linear under sums of independent GSPs. Observation If f = f 1 f 2 where f 1, f 2 are independent GSPs, then r = r 1 + r 2, ρ = ρ 1 + ρ 2. A decomposition ρ = ρ 1 + ρ 2 defines f = f 1 f 2 iff ρ 1 0, ρ 2 0 and are both symmetric

45 Intuition: Spectral perspective

46 Intuition: Spectral perspective

47 Intuition: Spectral perspective Example - Exponential Covariance (Ornstein-Uhlenbeck) r(x) = e x dρ(λ) = 2 λ dλ Covariance Kernel Three Sample Paths Spectral Measure

48 Ideas from the proof.

49 Further discussion: ρ f "small" near zero, over Z Remark: We restrict ourself to the case T =Z and dρ f = φ(x)dx with φ(x) cx 2 on [ a,a].

50 Further discussion: ρ f "small" near zero, over Z Remark: We restrict ourself to the case T =Z and dρ f = φ(x)dx with φ(x) cx 2 on [ a,a]. Theorem [N. Feldheim, O. Feldheim, S. Nitzan] 1. Fix k IN. If dρ f = φ(x)dx with m k := T and E = suppφ, then φ(x)dx x 2k < P f (N) Ce E 16 kn log N N IN. 2. If dρ f = φ(x)dx and φ(x) = 0 on [ a,a] for some a > 0 then: P f (N) Ce αn2 N IN.

51 Further discussion: ρ f "small" near zero, over Z Remark: We restrict ourself to the case T =Z and dρ f = φ(x)dx with φ(x) cx 2 on [ a,a]. Theorem [N. Feldheim, O. Feldheim, S. Nitzan] 1. Fix k IN. If dρ f = φ(x)dx with m k := T and E = suppφ, then φ(x)dx x 2k < P f (N) Ce E 16 kn log N N IN. 2. If dρ f = φ(x)dx and φ(x) = 0 on [ a,a] for some a > 0 then: P f (N) Ce αn2 N IN. The main idea: Under these conditions f has a k-times (discrete) anti-derivative which is also a GSP, but it is hard for a GSP to be monotone on a long interval.

52 A sketch of the proof over Z. Step 1: The GSP f has a k th antiderivative which is also a GSP. For a = {a(n)} n Z denote ( a)(n) = a(n + 1) a(n). We say that a is a discrete antiderivative of a.

53 A sketch of the proof over Z. Step 1: The GSP f has a k th antiderivative which is also a GSP. For a = {a(n)} n Z denote ( a)(n) = a(n + 1) a(n). We say that a is a discrete antiderivative of a. Process integration dρ t 2k If T dρ F = < then a GSP F who s spectral measure satisfies 1 dρ (1 cos t) k f exists and satisfies k F = d f.

54 A sketch of the proof over Z. Step 1: The GSP f has a k th antiderivative which is also a GSP. For a = {a(n)} n Z denote ( a)(n) = a(n + 1) a(n). We say that a is a discrete antiderivative of a. Process integration If dρ < then a GSP F who s spectral measure satisfies T t 2k 1 dρ F = dρ (1 cos t) k f exists and satisfies k F = d f. Note: varf(0) = r F (0) = µ ˆ F (0) = T φ(x)dx x 2k = m k.

55 A sketch of the proof over Z. Step 1: The GSP f has a k th antiderivative which is also a GSP. For a = {a(n)} n Z denote ( a)(n) = a(n + 1) a(n). We say that a is a discrete antiderivative of a. Process integration If dρ < then a GSP F who s spectral measure satisfies T t 2k 1 dρ F = dρ (1 cos t) k f exists and satisfies k F = d f. Note: varf(0) = r F (0) = µ ˆ F (0) = T φ(x)dx x 2k = m k. Step 2: In high probability F, the k th antiderivative of f, is not "too big". (A particular case of) Borell-TIS inequality [Borell, Tsirelson, Ibragimov, Sudakov For a GSP F : P( sup F > L) e [0,N] L2 2varF(0).

56 A sketch of the proof over Z. Step 3: If F is small, there is a large set S on which f is "very very small".

57 A sketch of the proof over Z. Step 3: If F is small, there is a large set S on which f is "very very small". Deterministic Lemma Let F = {F(n)} n Z. If k F 0 on [0,N] then there exists a set S [0,N], S > N 2, such that sup k F sup F ( Ck ) k S [0,N] N

58 A sketch of the proof over Z. Step 3: If F is small, there is a large set S on which f is "very very small". Deterministic Lemma Let F = {F(n)} n Z. If k F 0 on [0,N] then there exists a set S [0,N], S > N 2, such that sup k F sup F ( Ck ) k S [0,N] N Step 4: If f is "very very small" then there are "many" Gaussian "almost i.i.d" which are "very very small". Existence of "almost i.i.d" Example: If dρ f = 11 [ π, π/2] [π/2,π] dt then the random variables f (k) with k 2Z are Gaussian i.i.d.

59 A sketch of the proof over Z. Step 3: If F is small, there is a large set S on which f is "very very small". Deterministic Lemma Let F = {F(n)} n Z. If k F 0 on [0,N] then there exists a set S [0,N], S > N 2, such that sup k F sup F ( Ck ) k S [0,N] N Step 4: If f is "very very small" then there are "many" Gaussian "almost i.i.d" which are "very very small". Existence of "almost i.i.d" Example: If dρ f = 11 [ π, π/2] [π/2,π] dt then the random variables f (k) with k 2Z are Gaussian i.i.d. If ρ f = 11 E E [ π,π] then there exists Λ Z (with positive density) such that the random variables f (k) with k Λ are Gaussian "almost i.i.d". (Restricted Invertibility Theorem by Bourgain-Tzafriri)

60 A sketch of the proof over Z. Step 5: More generally, the GSP f is of the form described in the previous slide + "noise" Recall: spectral decomposition If ρ f = ρ 1 + ρ 2 where ρ 1 and ρ 2 are positive symmetric Borel measures then f = d g + h where. i. g and h are independent processes ii. ρ g = ρ 1 and ρ h = ρ 2.

61 A sketch of the proof over Z. Step 5: More generally, the GSP f is of the form described in the previous slide + "noise" Recall: spectral decomposition If ρ f = ρ 1 + ρ 2 where ρ 1 and ρ 2 are positive symmetric Borel measures then f = d g + h where i. g and h are independent processes ii. ρ g = ρ 1 and ρ h = ρ 2... We take ρ g = ρ 1 = 11 E where E suppρ.

62 A sketch of the proof over Z. Step 5: More generally, the GSP f is of the form described in the previous slide + "noise" Recall: spectral decomposition If ρ f = ρ 1 + ρ 2 where ρ 1 and ρ 2 are positive symmetric Borel measures then f = d g + h where i. g and h are independent processes ii. ρ g = ρ 1 and ρ h = ρ 2... We take ρ g = ρ 1 = 11 E where E suppρ. Anderson s lemma With the notations above: Prob( f (n) < A,n S) Prob( g(n) < A,n S)

Persistence of Gaussian stationary processes: a spectral perspective

Persistence of Gaussian stationary processes: a spectral perspective Naomi Feldheim Ohad Feldheim Shahaf Nitzan Abstract We study the persistence probability of a centered stationary Gaussian process on