COVARIANCES OF ZERO CROSSINGS IN GAUSSIAN PROCESSES
MATHIEU SINN∗ AND KARSTEN KELLER†
Abstract. For a zero-mean Gaussian process, the covariances of zero crossings can be expressed as the sum of quadrivariate normal orthant probabilities. In this paper, we demonstrate
the evaluation of zero crossing covariances using one-dimensional integrals. Furthermore, we provide
asymptotics of zero crossing covariances for large time lags and derive bounds and approximations.
Based on these results, we analyze the variance of the empirical zero crossing rate. We illustrate
the applications of our results by autoregressive (AR), fractional Gaussian noise and fractionally
integrated autoregressive moving average (FARIMA) processes.
Key words. zero crossing, binary time series, quadrivariate normal orthant probability, AR(1)
process, fractional Gaussian noise, FARIMA(0,d,0) process
1. Introduction. Indicators of zero crossings are widely applied in various fields
of engineering and natural science, such as the analysis of vibrations, the detection of
signals in presence of noise and the modelling of binary time series. A large number of
literature has been contributed to the studies of zero crossing analysis. Dating back
to the 1940es, telephony engineers found that replacing the original speech signal with
rectangular waves having the same zero crossings retained high intelligibility [5]. Since
the beginning of digital processing of speech signals, empirical rates of zero crossings
have been used for the detection of pitch frequencies and to distinguish voiced and
unvoiced intervals [11, 19].
For a discrete-time stationary Gaussian process or a sampled random sinusoid,
the zero crossing rate is related to the first-order autocorrelation and to the dominant
spectral frequency. Kedem [14] has developed estimators for autocorrelations and
spectral frequencies by higher order zero crossings and shows diverse applications.
Ho and Sun [12] have proved that the empirical zero crossing rate is asymptotically
normally distributed if the autocorrelations of the Gaussian process decay faster than
1
k − 2 . Coeurjolly [7] has proposed to use zero crossings to estimate the Hurst parameter
in fractional Gaussian noise, which generally can be applied to the estimation of
monotonic functionals of the first-order autocorrelation. Coeurjolly’s estimator has
been used to the analysis of hydrological time series [16] and atmospheric turbulence
data [20].
Up to now, no closed-form expression is known for the variance of the empirical
zero crossing rate. Basically, covariances of zero crossings are sums and products of
four-dimensional normal orthant probabilities which can be evaluated only numerically in general. Abrahamson [1] derives an expression involving two-dimensional
integrals for the special case of orthoscheme probabilities and gives a representation of any four-dimensional normal orthant probability as the linear combinations
of six orthoscheme probabilities. For some simpler correlation structure, Cheng [6]
proposes expressions involving the dilogarithm function. Kedem [14], Damsleth and
El-Shaarawi [9] introduce approximations for processes with short memory. Most recent approaches apply Monte Carlo sampling for four dimensions and higher (see [8]
for an overview).
∗ Universität zu Lübeck, Institut für Mathematik, Wallstr. 40, D-23560 Lübeck, Germany
(sinn@math.uni-luebeck.de).
† Universität zu Lübeck, Institut für Mathematik, Wallstr. 40, D-23560 Lübeck, Germany
(keller@math.uni-luebeck.de).
1
2
M. SINN AND K. KELLER
In this paper, we propose a simple formula for the exact numerical evaluation
of zero crossing covariances and derive their asymptotics, bounds and approximations. The results are obtained by analyzing partial derivatives of four-dimensional
orthant probabilities with respect to correlation coefficients. In Theorem 3.4, we give
a representation of zero crossing covariances and four-dimensional normal orthant
probabilities by the sum of four one-dimensional integrals. By a Taylor expansion,
we derive Theorem 4.1, which gives asymptotics of zero crossing covariances for large
time lags. In particular, when the autocorrelation function of the underlying process
decreases to 0 with the same order of magnitude as a function f (k), the zero crossing
covariances decrease to 0 with the same order of magnitude as (f (k))2 .
Theorem 5.3 states sufficient conditions on the autocorrelation structure of the
underlying process to obtain lower and upper bounds by setting equal certain correlation coefficients in the orthant probabilities. Approximations of these expressions
given by Theorem 5.5.
In Theorem 6.1 we establish asymptotics of the variance of the empirical zero
crossing rate. Furthermore, we discuss how the previous results can be used for
a numerical evaluation of the variance. In Section 7, we apply the results to zero
crossings in AR(1) processes, fractional Gaussian noise and FARIMA(0,d,0) models.
2. Preliminaries. Let Y = (Yk )k∈Z be a stationary and non-degenerate zeromean Gaussian process on some probability space (Ω, A, P) with autocorrelations
ρk = Corr(Y0 , Yk ) for k ∈ Z. For k ∈ Z, let
Ck := 1{Yk >0,Yk+1 <0} + 1{Yk <0,Yk+1 >0}
be the indicator of a zero crossing at time k. Since Y is stationary,
P(Ck = 1) is
Pn
constant in k, and the empirical zero crossing rate ĉn := n1 k=1 Ck is an unbiased
estimator of P(C1 = 1), that is, E(ĉn ) = P(C0 = 1) for all n ∈ N. Denote the
covariance of zero crossings by
γk := Cov(C0 , Ck )
for k ∈ Z. The variance of ĉn is given by
n−1
X
1
Var(ĉn ) = 2 n γ0 + 2
(n − k) γk .
n
(2.1)
k=1
This paper investigates the evaluation of γ0 , γ1 , γ2 , . . . . Next, we give closed-form
expressions for γ0 and γ1 based on well-known formulas for the evaluation of bi- and
trivariate normal orthant probabilities.
2.1. Orthant probabilities. For a non-singular strictly positive definite and
symmetric matrix Σ ∈ Rn×n with n ∈ N, let φ(Σ, ·) denote the Lebesgue density of
the n-dimensional normal distribution with zero means and the covariance matrix Σ,
that is
− 1
1
φ(Σ, x) = (2π)n |Σ| 2 exp{− xT Σ−1 x}
2
for x ∈ Rn , where |Σ| denotes the determinant of Σ. The n-dimensional normal
orthant probability with respect to Σ is given by
Z
Φ(Σ) :=
φ(Σ, x) dx.
[0,∞)n
3
COVARIANCES OF ZERO CROSSINGS IN GAUSSIAN PROCESSES
If Z = (Z1 , Z2 , . . . , Zn ) is a non-degenerate zero-mean Gaussian random vector and
Σ = Cov(Z) = (Cov(Zi , Zj ))ni,j=1 is the covariance matrix of Z, then
P(Z1 > 0, Z2 > 0, . . . , Zn > 0) = Φ(Σ).
√
√ √
Furthermore, if a1 , a2 , . . . , an > 0 and A = diag( a1 , a2 , . . . , an ) is the n × n
√ √
√
diagonal matrix with entries a1 , a2 , . . . , an on the main diagonal, then A Σ A
√
√
√
is the covariance matrix of A Z = ( a1 Z1 , a2 Z2 , . . . , an Zn ). Consequently,
Φ(A Σ A) = Φ(Σ). By choosing a1 = a2 = . . . = an = a and ai = Var(Zi ) for
i = 1, 2, . . . , n, respectively, we obtain
Φ(Σ) = Φ(a · Σ) and Φ(Corr(Z)) = Φ(Cov(Z)).
(2.2)
The following closed-form expressions for two- and three-dimensional normal orthant
probabilities are well-known (see, e.g., [2]).
Lemma 2.1. Let (Z1 , Z2 , Z3 ) be a zero-mean non-degenerate Gaussian random
vector and ρij = Corr(Zi , Zj ) for i, j ∈ {1, 2, 3}. Then
1
1
+
arcsin ρ12 ,
4 2π
1
1
1
1
P(Z1 > 0, Z2 > 0, Z3 > 0) = +
arcsin ρ12 +
arcsin ρ13 +
arcsin ρ23 .
8 4π
4π
4π
P(Z1 > 0, Z2 > 0) =
Lemma 2.1 allows to derive a closed-form expression for the probability of a
change, namely,
P(C0 = 1) = 1 − P(Y0 > 0, Y1 > 0) − P(Y0 < 0, Y1 < 0)
1
1
= − arcsin ρ1 .
2 π
(2.3)
Furthermore,
γ0 = P(C0 = 1) (1 − P(C0 = 1))
= (1 − 2 P(Y0 > 0, Y1 > 0)) 2 P(Y0 > 0, Y1 > 0)
1
1
= − 2 (arcsin ρ1 )2
4 π
(2.4)
and
γ1 = P(C0 = 1, C1 = 1) − (P(C0 = 1))2
= 2 P(Y0 > 0, −Y1 > 0, Y2 > 0) − (1 − 2 P(Y0 > 0, Y1 > 0))2
1
1
arcsin ρ2 − 2 (arcsin ρ1 )2 .
=
2π
π
(2.5)
If k > 1, then γk can be expressed as the sum and product, respectively, of bi- and
quadrivariate normal orthant probabilities,
γk = Cov(1 − C0 , 1 − Ck ) = 2 P(Y0 > 0, Y1 > 0, Yk > 0, Yk+1 > 0)
+ 2 P(Y0 > 0, Y1 > 0, −Yk > 0, −Yk+1 > 0)
− 4 P(Y0 > 0, Y1 > 0) P(Yk > 0, Yk+1 > 0).
(2.6)
Note that, in general, no closed-form expression is available for normal orthant probabilities of dimension n ≥ 4.
4
M. SINN AND K. KELLER
2.2. Context of the investigations. We consider the problem of evaluating
γk for k > 1 in a more general context. For this, let R denote the set of r =
6
(r1 , r2 , r3 , r4 , r5 , r6 ) ∈ [−1, 1] such that the matrix


1 r 1 r 2 r3
 r 1 1 r 4 r5 

Σ(r) := 
 r 2 r 4 1 r6 
r3 r5 r6 1
is strictly positive definite, that is, xT Σ(r) x > 0 for x ∈ R4 \ {0}. Note that Σ(R) is
the set of 4 × 4-correlation matrices of non-degenerate Gaussian random vectors, and
r ∈ R implies that all components of r lie within (−1, 1).
For h ∈ [−1, 1] consider the diagonal matrix
Ih := diag(1, h, h, h, h, 1).
If r, s ∈ R, then xT Σ(h · r + (1 − h) · s) x = h xT Σ(r) x + (1 − h) xT Σ(s) x > 0 for all
x ∈ R4 \ {0} and h ∈ [0, 1], in other words, R is convex. Furthermore, r ∈ R implies
I−1 r ∈ R. This can be seen as follows: If r ∈ R, then there exists a zero-mean nondegenerate Gaussian random vector Z = (Z1 , Z2 , Z3 , Z4 ) such that Cov(Z) = Σ(r).
Since Z0 = (Z10 , Z20 , Z30 , Z40 ) := (Z1 , Z2 , −Z3 , −Z4 ) is also non-degenerate Gaussian,
the matrix Cov(Z0 ) = Σ(I−1 r) is strictly positive definite, too, and hence I−1 r ∈ R.
Now, since I1 r = r and
Ih r =
1−h
1+h
I1 r +
I−1 r
2
2
for all r ∈ R and h ∈ [−1, 1], the convexity of R implies that Ih r ∈ R for all r ∈ R
and h ∈ [−1, 1], a fact we will repeatedly use in the rest of the paper.
For r ∈ R, write
Φ(r) = Φ(Σ(r)),
and define
Ψ(r) := 2 Φ(r) + 2 Φ(I−1 r) − 4 Φ(I0 r).
(2.7)
Note that if Z = (Z1 , Z2 , Z3 , Z4 ) is a zero-mean non-degenerate Gaussian random
vector with covariance matrix Cov(Z) = Σ(r), then
Ψ(r) = 2 P(Z1 > 0, Z2 > 0, Z3 > 0, Z4 > 0)
+ 2 P(Z1 > 0, Z2 > 0, −Z3 > 0, −Z4 > 0)
− 4 P(Z1 > 0, Z2 > 0) P(Z3 > 0, Z4 > 0).
Thus, according to (2.6),
γk = Ψ(ρ1 , ρk , ρk+1 , ρk−1 , ρk , ρ1 )
(2.8)
for k > 1. The evaluation of Φ and Ψ is the main concern of this paper. In Sec. 3
and Sec. 4, we consider the general problem to evaluate Φ(r) and Ψ(r) for arbitrary
r = (r1 , r2 , r3 , r4 , r5 , r6 ) ∈ R. In Sec. 5, we focus on the special case where r1 = r6
and r2 = r5 .
5
COVARIANCES OF ZERO CROSSINGS IN GAUSSIAN PROCESSES
3. Numerical evaluation. The following lemma establishes basic equations
and closed-form expressions for Φ and Ψ in some special cases.
Lemma 3.1. For every r = (r1 , r2 , r3 , r4 , r5 , r6 ) ∈ R,
Ψ(r) = Ψ(I−1 r) = Ψ(−r1 , −r2 , r3 , r4 , −r5 , −r6 )
= Ψ(−r1 , r2 , −r3 , −r4 , r5 , −r6 ).
If r2 = r3 = r4 = r5 = 0, then Ψ(r) = 0 and
1
1
1
1
Φ(r) =
+
arcsin r1
+
arcsin r6 .
4 2π
4 2π
(3.1)
Proof. The first equation follows by the definition of Ψ and I0 I−1 = I0 . Now,
let Z = (Z1 , Z2 , Z3 , Z4 ) be zero-mean Gaussian with Cov(Z) = Σ(r). Define Z0 =
(Z10 , Z20 , Z30 , Z40 ) := (Z1 , −Z2 , −Z3 , Z4 ) and r0 := (−r1 , −r2 , r3 , r4 , −r5 , −r6 ). Since
Cov(Z0 ) = Σ(r0 ), the second equation follows because
Ψ(r) = Cov(1{Z1 >0,Z2 >0} + 1{Z1 <0,Z2 <0} , 1{Z3 >0,Z4 >0} + 1{Z3 <0,Z4 <0} )
= Cov(1{Z1 >0,−Z2 <0} + 1{Z1 <0,−Z2 >0} , 1{Z3 >0,−Z4 <0} + 1{Z3 <0,−Z4 >0} )
= Ψ(r0 ) ,
Applying Ψ(r) = Ψ(I−1 r) to r = (−r1 , −r2 , r3 , r4 , −r5 , −r6 ) yields the third equation.
Now, assume r2 = r3 = r4 = r5 = 0. Since r = I−1 r = I0 r, we obtain Ψ(r) =
0. Furthermore, if Z = (Z1 , Z2 , Z3 , Z4 ) is zero-mean non-degenerate Gaussian with
Cov(Z) = Σ(r), then (Z1 , Z2 ) and (Z3 , Z4 ) are independent. Thus, (3.1) follows from
Lemma 2.1.
Note that bounds for Ψ(r) can be obtained by the Berman-inequality, namely,
|Ψ(r)| ≤
5
2 X |rk |
√
.
π
1 − rk
k=2
(see [17]). In the remaining part of this section, we show how to compute Ψ(r) and
Φ(r) for any r ∈ R by the numerical evaluation of four one-dimensional integrals.
According to a formula first given by David [10], this also allows to evaluate normal
orthant probabilities of dimension n = 5. Next, we derive explicit formulas for the
partial derivatives of Φ and Ψ with respect to r2 , r3 , r4 and r5 .
0
3.1. Partial derivatives. For r ∈ R and i, j ∈ {1, 2, 3, 4}, let σij
(r) denote
−1
the (i, j)-th component of the inverse (Σ(r))
of Σ(r). It is well-known that the
inverse and any principal submatrix of a symmetric strictly positive definite matrix
is symmetric and strictly positive definite (see [13]). Now, for fixed k ∈ {1, 2, . . . , 6},
let {i, j} with i 6= j be the unique subset of {1, 2, 3, 4} such that rk does not lie in the
i-th row and j-th column of Σ(r). Using the so-called reduction formula for normal
orthant probabilities (see [18], [4]), we obtain the first partial derivative of Φ with
respect to rk ,
0
−1
0
∂Φ
1
σii (r) σij
(r)
p
(r) =
Φ(
)
0
0
σij
(r) σjj
(r)
∂rk
2π 1 − r2
k
for r = (r1 , r2 , r3 , r4 , r5 , r6 ) ∈ R. Note that the argument of Φ is a principal submatrix
of (Σ(r))−1 and thus strictly positive definite. By the first equation in (2.2),
0
0
∂Φ
1
σii
(r) −σij
(r)
p
(r) =
Φ(
).
0
0
−σij
(r)
σjj
(r)
∂rk
2π 1 − rk2
6
M. SINN AND K. KELLER
0
0
Now, let σij (r) = −|Σ(r)| σij
(r) if i 6= j, and σij (r) = |Σ(r)| σij
(r) if i = j for
i, j ∈ {1, 2, 3, 4}. By the second equation in (2.2) and the formula for two-dimensional
normal orthant probabilities in Lemma 2.1,
1
∂Φ
1
p
(r) =
∂r2
2π 1 − r22 4
1
∂Φ
1
p
(r) =
∂r3
2π 1 − r32 4
1
∂Φ
1
p
(r) =
∂r4
2π 1 − r42 4
1
∂Φ
1
p
(r) =
∂r5
2π 1 − r52 4
1
σ24 (r)
,
arcsin p
2π
σ22 (r)σ44 (r)
1
σ23 (r)
,
+
arcsin p
2π
( σ22 (r)σ33 (r)
1
σ14 (r)
,
+
arcsin p
2π
σ11 (r)σ44 (r)
1
σ13 (r)
+
arcsin p
2π
σ11 (r)σ33 (r)
+
(3.2)
(3.3)
(3.4)
(3.5)
for r = (r1 , r2 , r3 , r4 , r5 , r6 ) ∈ R. Note that σij (r) is equal to the determinant of the
matrix obtained by deleting the ith row and the jth column of Σ(r), multiplied with
(−1)i+j+1 if i 6= j (see [13]). We obtain
σ11 (r) = 1 − r42 − r52 − r62 + 2r4 r5 r6 ,
σ22 (r) = 1 − r22 − r32 − r62 + 2r2 r3 r6 ,
σ33 (r) = 1 − r12 − r32 − r52 + 2r1 r3 r5 ,
σ44 (r) = 1 − r12 − r22 − r42 + 2r1 r2 r4 ,
σ13 (r) = r2 − r1 r4 + r3 r4 r5 − r2 r52 − r3 r6 + r1 r5 r6 ,
σ14 (r) = r3 − r1 r5 + r2 r4 r5 − r3 r42 − r2 r6 + r1 r4 r6 ,
σ23 (r) = r4 − r1 r2 + r2 r3 r5 − r4 r32 − r5 r6 + r1 r3 r6 ,
σ24 (r) = r5 − r1 r3 + r2 r3 r4 − r5 r22 − r4 r6 + r1 r2 r6 .
(3.6)
(3.7)
(3.8)
(3.9)
(3.10)
(3.11)
(3.12)
(3.13)
The following Corollary is an immediate consequence of (3.2)-(3.5) and (3.6)(3.13).
Corollary 3.2. For every r ∈ R, the partial derivatives of Φ of any order exist
and are continuous at r.
The next Lemma gives the partial derivatives of Ψ with respect to ri for i =
2, 3, 4, 5. For r ∈ R, let
ψi (r) :=
∂Ψ
(r).
∂ri
Lemma 3.3. For every r = (r1 , r2 , r3 , r4 , r5 , r6 ) ∈ R,
1
p
1 − r22
1
ψ3 (r) = p
π 2 1 − r32
1
ψ4 (r) = p
2
π 1 − r42
1
ψ5 (r) = p
2
π 1 − r52
ψ2 (r) =
π2
σ24 (r)
arcsin p
,
σ22 (r)σ44 (r)
σ23 (r)
arcsin p
,
σ22 (r)σ33 (r)
σ14 (r)
arcsin p
,
σ11 (r)σ44 (r)
σ13 (r)
arcsin p
.
σ11 (r)σ33 (r)
(3.14)
(3.15)
(3.16)
(3.17)
7
COVARIANCES OF ZERO CROSSINGS IN GAUSSIAN PROCESSES
Proof. Let i ∈ {2, 3, 4, 5}. Here we denote by Ih the mapping r 7→ Ih r from R
onto itself. By the definition of Ψ,
∂Φ
∂(Φ ◦ I−1 )
∂(Φ ◦ I0 )
(3.18)
(r) + 2
(r) − 4
(r).
∂ri
∂ri
∂ri
1
1
With (3.1) we obtain (Φ ◦ I0 )(r) = 14 + 2π
arcsin r1 14 + 2π
arcsin r6 . Thus, r 7→
(Φ ◦ I0 )(r) is constant in ri and, consequently, the last term on the right side of (3.18)
−1
is equal to 0. Furthermore, because ∂I
∂ri (r) = −1, the chain rule of differentiation
yields
ψi (r) = 2
∂(Φ ◦ I−1 )
∂Φ
(r) = −
(I−1 (r)).
∂ri
∂ri
According to (3.6)-(3.13), σij (I−1 (r)) = σij (r) if (i, j) ∈ {(1, 1), (2, 2), (3, 3), (4, 4)},
and σij (I−1 (r)) = −σij (r) if (i, j) ∈ {(1, 3), (1, 4), (2, 3), (2, 4)}. Since x 7→ arcsin x is
an odd function, inserting I−1 (r) instead of r into (3.2)-(3.5) yields (3.14)-(3.17).
3.2. Integral representation. Next we state the main result of this section.
Note that a similar representation of Ψ(r) as in (3.19) is used for the proof of the
Berman inequality (see above).
Theorem 3.4. For every r = (r1 , r2 , r3 , r4 , r5 , r6 ) ∈ R,
Ψ(r) =
5
X
Z
1
4
(3.19)
ψi (Ih r) dh,
0
i=2
Φ(r) =
1
ri
+
5
1
Ψ(r)
1
1
1 X
arcsin ri +
arcsin r1
+
arcsin r6 +
,
2π
4 2π
8π i=2
4
5
1 X
arcsin ri .
4π i=2
Φ(r) − Φ(I−1 r) =
(3.20)
(3.21)
Proof. Let r ∈ R. Since Ih r ∈ R for all h ∈ [0, 1], the mapping
[0, 1] 3 h 7→ u(h) := Ψ(Ih r)
is well-defined, being the concatenation of h 7→ Ih r and Ψ. Clearly, u(1) = Ψ(r) and,
by Lemma 3.1, u(0) = 0. Since Ψ has continuous partial derivatives (see Lemma 3.3),
u is differentiable at h for all h ∈ [0, 1], and hence, by the Fundamental Theorem of
Calculus and the chain rule of differentiation,
Z
Ψ(r) = u(0) +
0
1
u0 (h) dh =
Z
0
1
5
X
i=2
ri
Z 1
5
X
∂Ψ
(Ih r) dh =
ri
ψi (Ih r) dh.
∂ri
0
i=2
Analogously, define v(h) := Φ(Ih r) for h ∈ [0, 1]. According to (3.2)-(3.5) and (3.14)(3.17),
∂Φ
1
ψi (r)
p
(r) =
+
2
∂ri
4
8π 1 − ri
8
M. SINN AND K. KELLER
for i = 2, 3, 4, 5. Consequently,
Z
1
5
X
Z
1
∂Φ
(Ih r) dh
∂r
i
0
i=2
Z 1
Z 1
5
5
1 X
1 X
1
p
dh
+
=
ri
r
ψi (Ih r) dh
i
8π i=2
4 i=2
1 − ri2 h2
0
0
0
v (h) dh =
0
=
ri
5
1 X
Ψ(r)
arcsin ri +
.
8π i=2
4
According to (3.1),
v(0) = Φ(I0 r) =
1
4
+
1
1
1
arcsin r1
+
arcsin r6 ,
2π
4 2π
and hence (3.20) follows. (3.21) is an immediate consequence of (3.20), Lemma 3.1
and the fact that x 7→ arcsin x is an odd function.
Note that for i = 2, 3, 4, 5, the mappings
h 7→ ψi (Ih r)
have bounded derivatives on [0, 1]. (The derivatives are easily obtained from (3.14)(3.17).) Moreover for fixed r ∈ R, upper and lower bounds can be given in a closed
form which allows to evaluate the integrals in Theorem 3.4 numerically to any desired
precision.
4. Asymptotically equivalent expressions. For fixed n ∈ N, let (r(k))k∈N ,
(s(k))k∈N be sequences of vectors in Rn with r(k) = (r1 (k), r2 (k), . . . , rn (k)) and
s(k) = (s1 (k), s2 (k), . . . , sn (k)) for k ∈ N. We write
r(k) ∼ s(k)
and say that (r(k))k∈N and (s(k))k∈N are asymptotically equivalent iff ri (k) ∼ si (k)
0
for all i ∈ {1, 2, . . . , n}, that is, limk→∞ srii (k)
(k) = 1 where 0 := 1.
The following theorem relates asymptotics of special sequences in R and asymptotics of the corresponding values of Ψ. In Corollary 5.1, this result is used for deriving
asymptotics of zero crossing covariances.
Theorem 4.1. Let (r(k))k∈N be a sequence in R. If there exist an f : N → R
with limk→∞ f (k) = 0 and an α = (α1 , α2 , α3 , α4 , α5 , α6 ) ∈ R6 with |α1 |, |α6 | < 1
such that r(k) ∼ (α1 , α2 f (k), α3 f (k), α4 f (k), α5 f (k), α6 ), then
Ψ(r(k)) ∼
(f (k))2 q(α)
p
+ O (f (k))4 ,
2
2
2
2π (1 − α1 )(1 − α6 )
P5
where q(α) = α1 α6 i=2 αi2 −2α1 (α2 α3 +α4 α5 )−2α6 (α2 α4 +α3 α5 )+2(α2 α5 +α3 α4 ).
Proof. Let r(k) = (r1 (k), r2 (k), r3 (k), r4 (k), r5 (k), r6 (k)) for k ∈ N. According
to Corollary 3.2, Taylor’s Theorem asserts for each k ∈ N the existence of h1 (k) ∈ [0, 1]
9
COVARIANCES OF ZERO CROSSINGS IN GAUSSIAN PROCESSES
and h2 (k) ∈ [−1, 0] such that
5
X
5
∂Φ
1 X
∂2Φ
Φ(r(k)) = Φ(I0 r(k)) +
ri (k)
(I0 r(k)) +
ri (k)rj (k)
(I0 r(k))
∂ri
2 i,j=2
∂ri ∂rj
i=2
5
1 X
∂3Φ
+
ri (k)rj (k)rl (k)
(I0 r(k))
6
∂ri ∂rj ∂rl
i,j,l=2
+
1
24
5
X
ri (k)rj (k)rl (k)rm (k)
i,j,l,m=2
∂4Φ
(I0 + h1 (k)(I1 − I0 )) r(k)
∂ri ∂rj ∂rl ∂rm
and, using the fact that I0 I−1 = I0 ,
Φ(I−1 r(k)) = Φ(I0 r(k)) −
5
X
ri (k)
i=2
5
∂Φ
1 X
∂2Φ
(I0 r(k)) +
ri (k)rj (k)
(I0 r(k))
∂ri
2 i,j=2
∂ri ∂rj
−
5
1 X
∂3Φ
ri (k)rj (k)rl (k)
(I0 r(k))
6
∂ri ∂rj ∂rl
i,j,l=2
+
1
24
5
X
ri (k)rj (k)rl (k)rm (k)
i,j,l,m=2
∂4Φ
(I0 − h2 (k)(I−1 − I0 )) r(k) .
∂ri ∂rj ∂rl ∂rm
Since I0 + h(I1 − I0 ) = Ih and I0 + h(I−1 − I0 ) = I−h for h ∈ [0, 1],
2 Φ(r(k)) + 2 Φ(I−1 r(k)) = 4 Φ(I0 r(k)) + 2
5
X
ri (k)rj (k)
i,j=2
+
+
1
12
1
12
5
X
ri (k)rj (k)rl (k)rm (k)
∂4Φ
(Ih (k) r(k))
∂ri ∂rj ∂rl ∂rm 1
ri (k)rj (k)rl (k)rm (k)
∂4Φ
(Ih (k) r(k)). (4.1)
∂ri ∂rj ∂rl ∂rm 2
i,j,l,m=2
5
X
∂2Φ
(I0 r(k))
∂ri ∂rj
i,j,l,m=2
Under the assumptions, ri1 (k) ri2 (k) . . . rin (k) ∼ αi1 αi2 . . . αin (f (k))n for all i1 , i2 , . . . ,
in ∈ {2, 3, 4, 5} with n ∈ N. According to the definition of Ψ, inserting these asymptotically equivalent expressions into (4.1) yields
5
X
2
Ψ(r(k)) ∼ 2 (f (k))
i,j=2
αi αj
∂2Φ
(I0 r(k)) + (f (k))4 R(k)
∂ri ∂rj
(4.2)
with
R(k) =
1
12
5
X
αi αj αl αm
i,j,l,m=2
∂4Φ
∂4Φ
(Ih1 (k) r(k)) +
(Ih2 (k) r(k)) .
∂ri ∂rj ∂rl ∂rm
∂ri ∂rj ∂rl ∂rm
Note that |α1 |, |α6 | < 1 implies I0 α ∈ R. Furthermore, according to Corollary 3.2,
the second derivatives of Φ are continuous in I0 α. Since limk→∞ I0 r(k) = I0 α, we
10
M. SINN AND K. KELLER
obtain
∂2Φ
∂2Φ
(I0 r(k)) =
(I0 α)
k→∞ ∂ri ∂rj
∂ri ∂rj
lim
for all i, j ∈ {2, 3, 4, 5}. Inserting (4.3)-(4.6) from Lemma 4.2 below with r = α into
(4.2), we obtain
Ψ(r(k)) ∼
2(f (k))2 q(α)
p
+ (f (k))4 R(k),
2
2
2
4π (1 − α1 )(1 − α6 )
with q(α) as given above.
In order to prove that (f (k))4 R(k) = O (f (k))4 , we show supk∈N R(k) < ∞.
Because limk→∞ r(k) = I0 α = limk→∞ I−1 r(k), the set
[
S := {I0 α} ∪
{r(k), I−1 r(k)}
k∈N
6
6
is closed in R . Since S ⊂ [−1, 1] , the convex hull of S is compact. Now, because
[
S̃ :=
{Ih r(k) : h ∈ [−1, 1]}
k∈N
is a subset of the convex hull of S and the fourth partial derivatives of Φ are continuous
at every point of R (see Corollary 3.2),
sup
∂4Φ
(S̃) < ∞
∂ri ∂rj ∂rl ∂rm
for all i, j, l, m ∈ {2, 3, 4, 5}, and hence the result follows.
Lemma 4.2. For r = (r1 , r2 , r3 , r4 , r5 , r6 ) ∈ R, the second partial derivatives of
Φ with respect to r2 , r3 , r4 , r5 at I0 r are given by
∂2Φ
r1 r6
p
(I0 r) =
for i = 2, 3, 4, 5, (4.3)
∂ 2 ri
4π 2 (1 − r12 )(1 − r62 )
∂2Φ
∂2Φ
−r1
p
(I0 r) =
(I0 r) =
,
2
∂r2 ∂r3
∂r4 ∂r5
4π (1 − r12 )(1 − r62 )
(4.4)
∂2Φ
∂2Φ
−r6
p
,
(I0 r) =
(I0 r) =
∂r2 ∂r4
∂r3 ∂r5
4π 2 (1 − r12 )(1 − r62 )
(4.5)
∂2Φ
∂2Φ
1
p
(I0 r) =
(I0 r) =
.
2
∂r2 ∂r5
∂r3 ∂r4
4π (1 − r12 )(1 − r62 )
(4.6)
Proof. Let k ∈ {2, 3, 4, 5}. According to (3.2)-(3.5), there exist unique numbers
∂Φ
1
1
i, j ∈ {1, 2, 3, 4} such that ∂r
(r) = 2π
f (r) 14 + 2π
g(r) with
k
1
1 − rk2
f (r) = p
σij (r)
.
and g(r) = arcsin p
σii (r)σjj (r)
2
∂f
Φ
Clearly, f (I0 r) = 1 and ∂r
(I0 r) = 0 for l = 2, 3, 4, 5, and hence ∂r∂k ∂r
(I0 r) =
l
l
1 ∂g
(I
r).
Since
σ
(I
r)
=
0
and
the
first
derivative
of
arcsin
in
0
is
1,
we
obtain
ij 0
4π 2 ∂rl 0
∂2Φ
1
∂σij
p
(I0 r) =
(I0 r).
2
∂rk ∂rl
4π σii (I0 r)σjj (I0 r) ∂rl
Now, the result follows by (3.10)-(3.13).
COVARIANCES OF ZERO CROSSINGS IN GAUSSIAN PROCESSES
11
5. Bounds and approximations. Next, we apply the previous results to vectors r = (r1 , r2 , r3 , r4 , r5 , r6 ) ∈ R with r1 = r6 and r2 = r5 . Let π ∗ (r) := (r1 , r2 , r3 , r4 , r2 , r1 )
4
for r = (r1 , r2 , r3 , r4 ) ∈ (−1, 1) , and
4
R∗ := {r ∈ (−1, 1) | π ∗ (r) ∈ R}.
Clearly, r = (r1 , r2 , r3 , r4 ) ∈ R∗ if and only

1
 r1
∗
Σ(π (r)) = 
 r2
r3
if
r1
1
r4
r2
r2
r4
1
r1

r3
r2 

r1 
1
is strictly positive definite. Because R∗ ⊂ R and R is convex, r, s ∈ R∗ implies
π ∗ ((1 − h) r + h s) ∈ R and hence (1 − h) r + h s ∈ R∗ for all h ∈ [0, 1]. Thus, R∗ is
convex. Now, define
Ψ∗ (r) := Ψ(π ∗ (r))
for r ∈ R∗ . According to (2.8),
γk = Ψ∗ (ρ1 , ρk , ρk+1 , ρk−1 )
(5.1)
for k > 1. The following corollary is a special case of Theorem 4.1.
Corollary 5.1. Let (r(k))k∈N be a sequence in R∗ and assume f : N → R is
a function with limk→∞ f (k) = 0.
(i) If r(k) ∼ (α1 , α2 f (k), α3 f (k), α4 f (k)), for some vector α = (α1 , α2 , α3 , α4 )
with |α1 | < 1, then
Ψ∗ (r(k)) ∼
(f (k))2 q(α)
+ O (f (k))4 ,
2
2
2π (1 − α1 )
where q(α) = α12 (2α22 + α32 + α42 ) − 4α1 α2 (α3 + α4 ) + 2(α22 + α3 α4 ).
(ii) If f (k + 1) ∼ βf (k) for some β 6= 0 and there exists an α with |α| < 1 such
that r(k) ∼ (α, f (k), f (k + 1), f (k − 1)), then
Ψ∗ (r(k)) ∼
(f (k))2 (2 − α(β + β −1 ))2
+ O (f (k))4 .
2
2
2π (1 − α )
(iii) If the assumptions of (ii) hold with β = 1, then
Ψ∗ (r(k)) ∼
2 (f (k))2 (1 − α)
+ O (f (k))4 .
2
π (1 + α)
Proof. (i) follows by Theorem 4.1 and the fact that π ∗ (r(k)) is asymptotically
equivalent to (α1 , α2 f (k), α3 f (k), α4 f (k), α2 f (k), α1 ).
(ii) is a special case of (i) where r(k) ∼ (α, f (k), β f (k), f (k)/β) and thus
q(α, 1, β, 1/β) = α2 (2 + β 2 + β −2 ) − 4α(β + β −1 ) + 4
= (2 − α(β + β −1 ))2 .
Now, (iii) is obvious.
12
M. SINN AND K. KELLER
5.1. Lower and upper bounds. Theorem 5.3 below gives sufficient conditions
on r ∈ R∗ to obtain lower and upper bounds for Ψ∗ (r) by setting r2 , r3 , r4 equal to
r3 and r4 , respectively. We first prove the following lemma.
Lemma 5.2. For every r = (r1 , r2 , r3 , r4 ) ∈ R∗ ,
∂Ψ∗
2
σ13 (π ∗ (r))
p
(r) = p
arcsin
,
∂r2
σ11 (π ∗ (r))σ22 (π ∗ (r))
π 2 1 − r22
σ23 (π ∗ (r))
∂Ψ∗
1
arcsin
(r) = p
,
2
∂r3
σ22 (π ∗ (r))
π 2 1 − r3
σ14 (π ∗ (r))
1
∂Ψ∗
arcsin
(r) = p
,
∂r4
σ11 (π ∗ (r))
π 2 1 − r42
(5.2)
(5.3)
(5.4)
and
σ13 (π ∗ (r)) = r2 − r1 r3 + r2 r3 r4 − r23 − r1 r4 + r2 r12 ,
σ14 (π ∗ (r)) = r3 − 2r1 r2 + r4 r22 − r3 r42 + r4 r12 ,
σ23 (π ∗ (r)) = r4 − 2r1 r2 + r3 r22 − r4 r32 + r3 r12 .
(5.5)
(5.6)
(5.7)
Proof. The validity of (5.5)-(5.7) directly follows from (3.10)-(3.12). Furthermore,
∂Ψ∗
∂Ψ ∗
∂Ψ ∗
∂Ψ∗
∂Ψ ∗
(r) =
(π (r)) +
(π (r)) and
(r) =
(π (r)) for i = 3, 4.
∂r2
∂r2
∂r5
∂ri
∂ri
Since σ11 (π ∗ (r)) = σ44 (π ∗ (r)), σ22 (π ∗ (r)) = σ33 (π ∗ (r)) and σ13 (π ∗ (r)) = σ24 (π ∗ (r))
(compare to (3.6)-(3.9), (3.10) and (3.13)), we obtain (5.2)-(5.4) by equations (3.14)(3.17) in Lemma 3.3.
Theorem 5.3. Let r = (r1 , r2 , r3 , r4 ) ∈ R∗ with r4 , r2 ≥ r3 ≥ 0. For h ∈ [0, 1],
define sh := (1 − h) · r + h · (r1 , r3 , r3 , r3 ) and th := (1 − h) · r + h · (r1 , r4 , r4 , r4 ).
1. If 1 + r1 − 2r3 > 0 and σ13 (π ∗ (sh )), σ14 (π ∗ (sh )) ≥ 0 for all h ∈ [0, 1], then
Ψ∗ (r) ≥ Ψ∗ (r1 , r3 , r3 , r3 ).
(5.8)
2. If 1 + r1 − 2r4 > 0 and σ13 (π ∗ (th )), σ23 (π ∗ (th )) ≥ 0 for all h ∈ [0, 1], then
Ψ∗ (r1 , r4 , r4 , r4 ) ≥ Ψ∗ (r).
(5.9)
Proof. 1. First, note that the set of eigenvalues of Σ(π ∗ (r1 , r3 , r3 , r3 )) is given by
{1 − r1 , 1 + r1 − 2r3 , 1 + r1 + 2r3 }. Under the assumptions, each eigenvalue is strictly
larger than 0, so (r1 , r3 , r3 , r3 ) ∈ R∗ . Because R∗ is convex, we have sh ∈ R∗ for all
h ∈ [0, 1]. Hence, f (h) := Ψ∗ (sh ) is well-defined for all h ∈ [0, 1]. Since f (0) = Ψ∗ (r)
and f (1) = Ψ∗ (r1 , r3 , r3 , r3 ), it is sufficient to show that h 7→ f (h) is monotonically
decreasing on [0, 1], or, equivalently,
f 0 (h) = (r3 − r2 )
∂Ψ∗ (sh )
∂Ψ∗ (sh )
+ (r3 − r4 )
≤ 0
∂r2
∂r4
for all h ∈ [0, 1]. With the assumptions r3 − r2 ≤ 0 and r3 − r4 ≤ 0, a sufficient
condition for this inequality to be satisfied is σ13 (π ∗ (sh )) ≥ 0 and σ14 (π ∗ (sh )) ≥ 0 for
all h ∈ [0, 1] (compare to (5.2) and (5.4)).
13
COVARIANCES OF ZERO CROSSINGS IN GAUSSIAN PROCESSES
2. Analogously, define g(h) := Ψ∗ (th ), and note that a sufficient condition for
g 0 (h) = (r4 − r2 )
∂Ψ(th )
∂Ψ(th )
+ (r4 − r3 )
≥ 0
∂r2
∂r3
is given by σ13 (π ∗ (th )) ≥ 0 and σ23 (π ∗ (th )) ≥ 0 for all h ∈ [0, 1].
As the proof of Theorem 5.3 shows, a sufficient condition for strict inequality in
(5.8) is given by r4 > r3 and σ14 (π ∗ (sh )) > 0 for some h ∈ [0, 1], or r2 > r3 and
σ13 (π ∗ (sh )) > 0 for some h ∈ [0, 1]. Analogously, a sufficient condition for strict
inequality in (5.9) is given by r4 > r3 and σ23 (π ∗ (th )) > 0 for some h ∈ [0, 1], or
r4 > r2 and σ13 (π ∗ (th )) > 0 for some h ∈ [0, 1].
The next lemma gives easily verifiable conditions for the assumptions of Theorem
5.3.
Lemma 5.4. Let r = (r1 , r2 , r3 , r4 ) ∈ R∗ with r1 ≤ 0 and r2 , r3 , r4 ≥ 0. Then
σ13 (sh ), σ14 (sh ) > 0 and σ13 (th ), σ23 (th ) > 0 for all h ∈ [0, 1].
Proof. For fixed h ∈ [0, 1], let sh = (s1 , s2 , s3 , s4 ). Clearly, s1 ≤ 0 and s2 ,
s3 , s4 ∈ [0, 1). Because σ13 (π ∗ (sh )) ≥ s2 − s32 and σ14 (π ∗ (sh )) ≥ s3 − s3 s24 , we
obtain σ13 (sh ), σ14 (sh ) > 0. Analogously, let th = (t1 , t2 , t3 , t4 ), and note that
σ23 (π ∗ (th )) ≥ t4 − t4 t23 .
5.2. Approximations of the bounds. Next, we analyze approximations of
the lower and upper bounds of Ψ∗ (r) given by Theorem 5.3. Let R∗∗ be the set of
2
r = (r1 , r2 ) ∈ (−1, 1) such that π ∗∗ (r) := (r1 , r2 , r2 , r2 ) ∈ R∗ or, equivalently,


1 r 1 r 2 r2
 r 1 1 r 2 r2 

Σ(π ∗∗ (r)) = 
 r 2 r 2 1 r1 
r 2 r 2 r1 1
is strictly positive definite. Since the set of eigenvalues of Σ(π ∗∗ (r)) is given by
{1 − r1 , 1 + r1 + 2r2 , 1 + r1 − 2r2 },
2
R∗∗ = { (r1 , r2 ) ∈ (−1, 1) | 2 |r2 | < 1 + r1 }.
(5.10)
For r ∈ R∗∗ , define
Φ∗∗ (r) := Φ(π ∗∗ (r))
and
Ψ∗∗ (r) := Ψ(π ∗∗ (r)).
Note that, σii (π ∗∗ (r)) = (1 − r1 )(1 + r1 − 2r22 ) for i = 1, 2, 3, 4 and σ13 (π ∗∗ (r)) =
σ14 (π ∗∗ (r)) = σ23 (π ∗∗ (r)) = σ24 (π ∗∗ (r)) = r2 (1 − r1 )2 (compare to (3.6)-(3.13)).
Hence, according to (3.2)-(3.5),
∂Φ ∗∗
∂Φ ∗∗
∂Φ ∗∗
∂Φ ∗∗
∂Φ∗∗
(r) =
(π (r)) +
(π (r)) +
(π (r)) +
(π (r))
∂r2
∂r2
∂r3
∂r4
∂r5
2
1
1
r2 (1 − r1 )
= p
+
arcsin
.
2
1 + r1 − 2r22
π 1 − r2 4 2π
By formula (3.19), we obtain the integral representation
Z
4r2 1
1
r2 (1 − r1 )h
∗∗
p
Ψ (r) = 2
arcsin
dh
2
2
π 0
1
+ r1 − 2r22 h2
1 − r2 h
Z r2
1
(1 − r1 )t
4
√
arcsin
dt.
= 2
2
π 0
1 + r1 − 2t2
1−t
(5.11)
14
M. SINN AND K. KELLER
As the following theorem shows, Ψ∗∗ (r) can be approximated monotonically from
below by successively adding further terms of the Taylor expansion of Ψ∗∗ (r) in (r1 , 0).
Theorem 5.5. For every r = (r1 , r2 ) ∈ R∗∗ ,
∂ l Φ∗∗
((r1 , 0)T ) ≥ 0 for l ∈ N0 ,
∂ l r2
∞
X
r22l ∂ 2l Φ∗∗
Ψ∗∗ (r) = 4
(r1 , 0).
(2l)! ∂ 2l r2
(5.12)
(5.13)
l=1
1
Proof. Let r = (r1 , r2 ) ∈ R∗∗ . We define f (x) := 2π
arcsin x for x ∈ (−1, 1), and
1
g1 (x) := x(1 − r1 ), g2 (x) := 1+r1 −2x2 , g(x) := g1 (x) · g2 (x) and h(x) := f (g(x)) for
1 1+r1
x ∈ − 1+r
. Clearly, Φ∗∗ (r1 , 0) ≥ 0, hence (5.12) is true for l = 0. According
2 , 2
∂Φ∗∗
0
0
1
to (5.10), |r2 | < 1+r
2 , so (5.11) yields ∂r2 (r) = f (r2 ) + 4f (r2 )h(r2 ). Applying
Leibniz’s rule gives
l−1 X
∂ l Φ∗∗
l − 1 (k+1)
T
(l)
((r1 , 0) ) = f (0) + 4
f
(0) h(l−1−k) (0)
∂ l r2
k
k=0
l X
l − 1 (k)
(l)
= f (0) + 4
f (0) h(l−k) (0)
k−1
k=1
for l ∈ N. Note that arcsin x =
P∞
3·5·...·(2n−1)
2n+1
,
n=0 2·4·...·(2n)·(2n+1) x
(l)
so f (l) (0) ≥ 0. Therefore,
in order to prove (5.12), it is sufficient to show that h (0) ≥ 0 for all l ∈ N.
(l)
Let g2 (x) = f2 (f1 (x)) with f1 (x) := 1+r1 −2x2 , f2 (x) := x1 . Note that f1 (0) 6= 0
(l)
only if l ∈ {0, 2}. For each l ∈ N, we can write g2 (0) = (f2 ◦ f1 )(l) (0) as the sum
(k)
(i )
(i )
(i )
of terms f2 (f1 (0)) · f1 1 (0) · f1 2 (0) · . . . · f1 k (0) with k, i1 , i2 , . . . , ik ∈ N which
satisfy i1 + i2 + . . . + ik = l. Each term can only be non-zero if i1 = i2 = . . . = ik = 2,
(l)
(l)
hence a necessary condition for g2 (0) 6= 0 is that l is even. Moreover, g2 (0) > 0
(k)
(2)
in this case, since f2 (f1 (0)) = (−1)k k!(1 + r1 )−(k+1) and f1 (0) = −4. Note that
(k)
(1)
(l−1)
g1 (0) 6= 0 only if k = 1, consequently, by Leibniz’s rule, g (l) (0) = l·g1 (0)·g2
(0) =
(l−1)
l · (1 − r1 ) · g2
(0) for all l ∈ N, and hence g (l) (0) ≥ 0 for all l ∈ N.
Now, similarly as above, we can write h(l) (0) = (f ◦ g)(l) (0) for each l ∈ N as
the sum of products consisting of factors of the form f (k) (g(0)) = f (k) (0) and g (m) (0)
with k, m ∈ N, implying h(l) (0) ≥ 0.
In order to prove (5.13), first note that g2 and g = g1 g2 have power
series expan
1+r1 1+r1
1
sions at 0 with the radius of convergence 1+r
,
and
g(
−
,
)
⊂
(−1,
1). Since
2
2
2
f has a power series expansion at 0 with the radius of convergence 1, according to ele∗∗
0
0
mentary properties of power series, the mapping · 7→ ∂Φ
∂r2 (r1 , ·) = f (·) + 4f (·)f (g(·))
1
has a power series expansion at 0 with the radius of convergence 1+r
2 , and hence it
∗∗
also holds for the mapping · 7→ Φ (r1 , ·).
1 1+r1
Now, note that r2 ∈ − 1+r
(see (5.10)). Therefore, according to the
2 , 2
COVARIANCES OF ZERO CROSSINGS IN GAUSSIAN PROCESSES
15
definition of Ψ,
Ψ∗∗ (r) = 2 Φ∗∗ (r) + 2 Φ∗∗ (r1 , −r2 ) − 4 Φ∗∗ (r1 , 0)
∞
∞
X
X
r2l ∂ l Φ∗∗
(−r2 )l ∂ l Φ∗∗
=2
(r
,
0)
+
2
(r1 , 0) − 4 Φ∗∗ (r1 , 0)
1
l! ∂ l r2
l!
∂ l r2
l=0
l=0
∞
X
r22l ∂ 2l Φ∗∗
=4
(r1 , 0).
(2l)! ∂ 2l r2
l=1
The proof is complete.
6. The variance of the empirical zero crossing rate. In this section, we
apply the previous results to the analysis of the variance of empirical zero crossing
rates. Recall formula (2.1),
n−1
X
1
(n − k) γk .
Var(ĉn ) = 2 n γ0 + 2
n
k=1
In order to evaluate Var(ĉn ) numerically, we can use formulas (2.4) and (2.5) for the
computation of γ0 and γ1 . For k > 1, formula (5.1) yields
γk = Ψ∗ (ρ1 , ρk , ρk+1 , ρk−1 ) ,
and the right hand side can be evaluated numerically using the integral representation
of Ψ given in (3.19).
When n is large, an “exact” numerical evaluation of γk for every k = 0, 1, . . . , n−1
is time-consuming. A quick way for getting approximate values of Var(ĉn ) is to use
approximations of γk in terms of the function Ψ∗∗ . If the assumptions of Theorem
5.3 are satisfied, this yields upper and lower bounds for γk . A further speed-up can
be achieved by using the finite-order approximations of Ψ∗∗ provided by Theorem
5.5. For instance, when the autocorrelations of Y are not too large, one can use the
first-order approximation
γϑ (k) ≈
2(1 − ρϑ (1))
(ρϑ (k))2 .
π 2 (1 + ρϑ (1))
An alternative method for computing approximate values of Var(ĉn ) is to use the
exact values of γk for k = 2, 3, . . . until the relative error of the approximations falls
below a given threshold > 0, and then to use the approximations of γk . If the
relative error does not get larger than anymore, then also the relative error of the
resulting approximation of Varϑ (ĉn ) is not larger than . For the calculations behind
Figures 7.1-7.3, we have used this method with the threshold = 0.001.
The following theorem establishes asymptotics of Var(ĉn ).
Theorem 6.1. Suppose there exists a mapping f : NP
→ R such that ρk ∼ f (k).
∞
(i) If |f (k)| = o(k −β ) with β > 12 , then σ 2 := γ0 + 2 k=1 γk < ∞ and
Var(ĉn ) ∼ σ 2 n−1 .
1
(ii) If f (k) = αk − 2 for some α ∈ (−1, 1) \ {0}, then
Var(ĉn ) ∼
4 α2 (1 − ρ1 ) ln n
.
π 2 (1 + ρ1 ) n
16
M. SINN AND K. KELLER
(iii) If f (k) = αk −β for some α ∈ (−1, 1) \ {0} and β ∈ (0, 12 ), then
Var(ĉn ) ∼
4 α2 (1 − ρ1 )
n−2β .
π 2 (1 + ρ1 )(1 − 2β)
2
Proof.
P∞ (i) According to Corollary 5.1 (i), we have γk = O((f (k)) ), which shows
that k=1 |γk | < ∞. By the Dominated Convergence Theorem, we obtain
∞
X
n − k n−k
γk ∼ lim
max
, 0 γk
n→∞
n
n
n−1
X
k=1
k=1
=
∞
X
k=1
lim max
n→∞
∞
X
n − k , 0 γk =
γk .
n
k=1
Now, with formula (2.1), the result follows.
(ii) Note that f (k) ∼ f (k + 1) and thus, according to Corollary 5.1 (iii),
γk ∼
Using the fact that
Pn−1
k=1
2 α2 (1 − ρ1 ) −1
k .
π 2 (1 + ρ1 )
k −1 ∼ ln n, we obtain
n−1
X
γk ∼
k=1
2 α2 (1 − ρ1 )
ln n .
π 2 (1 + ρ1 )
(6.1)
Furthermore, we have
n−1
X
k=1
γk −
n−1
X
k=1
n−1
Xk
n−k
γk =
γk
n
n
k=1
∼
n−1
1 X 2 α2 (1 − ρ1 )
= o(ln n),
n
π 2 (1 + ρ1 )
k=1
which shows that
n−1
X
γk ∼
k=1
n−1
X
k=1
n−k
γk .
n
According to formula (2.1), we obtain
Var(ĉn ) ∼
n−1
2 X
γk ,
n
k=1
and together with (6.1) the statement follows.
Pn−1
(iii) The proof is similar to (ii), using the fact k=1 k −2β ∼
1
1−2β
n1−2β .
7. Examples. In this section, we apply the previous results to empirical zero
crossing rates in AR(1) processes, fractional Gaussian noise and ARFIMA(0,d,0) processes.
COVARIANCES OF ZERO CROSSINGS IN GAUSSIAN PROCESSES
17
7.1. AR(1) processes. Assume that Y is an AR(1) process with autoregressive coefficient a ∈ (−1, 1), that is, Y is stationary, non-degenerate and zero-mean
Gaussian with the autocorrelations ρk = ak for k ∈ N0 (where 00 := 1). According
to formula (2.3),
P(C0 = 1) =
1
1
− arcsin a,
2 π
(7.1)
hence the higher the autoregressive coefficient, the lower the probability of a zero
crossing.
Fig. 7.1. Variance of ĉn in AR(1) processes for a ∈ (−1, 1) and n = 10, 11, . . . , 100.
By using the method explained in Sec. 6, we can evaluate the variance of ĉn .
Figure 7.1 displays the values of Var(ĉn ) for n = 10, 11, . . . , 100 and a ranging in
(−1, 1). For fixed n, the variance of ĉn tends to 0 as a tends to −1 and 1, respectively.
According to (7.1), the probability of a zero crossing is equal to 1 and 0 in these limit
cases, and thus ĉn is P-almost surely equal to 1 and 0, respectively.
For fixed a, the variance of ĉn is decreasing in n. In particular, according to
Theorem 6.1 (i),
Var(ĉn ) ∼ σ 2 n−1
P∞
where σ 2 := γ0 + 2 k=1 γk < ∞. In the case a = 0, formulas (2.4) and (2.5) yield
γ0 = 41 and γ1 = 0. Furthermore, according to Lemma 3.1,
γk = Ψ∗ (0, 0, 0, 0) = 0
1
for all k > 1. Therefore, Var(ĉn ) = 4n
in this case.
Remarkably, Var(ĉn ) is always identical for a and −a. In fact, one can show that
γk is identical for a and −a for all k ∈ Z. For k = 0 and k = 1, this is an immediate
consequence of formulas (2.4) and (2.5) and the fact that (arcsin a)2 = (arcsin(−a))2 .
For k > 1, this is true because, according to Lemma 3.1,
Ψ∗ (a, ak , ak+1 , ak−1 ) = Ψ∗ (−a, (−a)k , (−a)k+1 , (−a)k−1 ).
18
M. SINN AND K. KELLER
7.2. Fractional Gaussian noise. Assume that Y is fractional Gaussian noise
(fGn) with the Hurst parameter H ∈ (0, 1), that is, Y is stationary, non-degenerate
and zero-mean Gaussian with the autocorrelations
ρk =
1
|k + 1|2H − 2|k|2H + |k − 1|2H
2
for k ∈ Z. With ρ1 = 22H−1 − 1, we obtain that the probability of a zero crossing is
given by
1
1
− arcsin(22H−1 − 1)
2 π
2
(7.2)
= 1 − arcsin 2H−1 ,
π
p
where the second equation follows from arcsin x = 2 arcsin (1 + x)/2 − π2 . Thus, the
larger the Hurst parameter, the lower the probability of a zero crossing.
P(C0 = 1) =
Fig. 7.2. Variance of ĉn in fGn for H ∈ (0, 1) and n = 10, 11, . . . , 100.
Figure 7.2 displays Var(ĉn ) for n = 10, 11, . . . , 100 and H ranging in (0, 1). For
fixed n, the variance tends to 0 as H tends to 1. Note that the probability of a zero
crossing is 0 in the limit case (see (7.2)), and thus ĉn is almost surely equal to 0.
Next, we derive asymptotics of Var(ĉn ). It is well-known that ρk ∼ H(2H −
1)k 2H−2 as k → ∞ (see [3]). According to Theorem 6.1 (i), we obtain
Var(ĉn ) ∼ σ 2 n−1
P∞
for H < 43 , where σ 2 := γ0 + 2 k=1 γk < ∞. In the case H = 12 , where ρk = 0 for all
1
k > 0, we obtain Var(ĉn ) = 4n
by the same argument as in the case a = 0 for AR(1)
3
processes. If H = 4 , then Theorem 6.1 (ii) yields
√
9 ( 2 − 1) ln n
Var(ĉn ) ∼
16 π 2
n
COVARIANCES OF ZERO CROSSINGS IN GAUSSIAN PROCESSES
(in particular, H 2 (2H − 1)2 (22−2H − 1) =
then Theorem 6.1 (iii) yields
Var(ĉn ) ∼
√
9
64 (
19
2 − 1) in this case). Finally, if H > 34 ,
4 H 2 (2H − 1)2 (22−2H − 1) 4H−4
n
.
π 2 (4H − 3)
7.3. ARFIMA(0,d,0) processes. If Yis an ARFIMA(0,d,0) process with the
fractional differencing parameter d ∈ − 21 , 12 , then Y is stationary, non-degenerate
and zero-mean Gaussian with the autocorrelations
ρk =
for k ∈ Z. With ρ1 =
d
1−d ,
Γ(1 − d) Γ(k + d)
Γ(d) Γ(k + 1 − d)
we obtain
P(C0 = 1) =
1
1
d
− arcsin
2 π
1−d
for the probability of a zero crossing.
Fig. 7.3. Variance of ĉn in ARFIMA(0,d,0) for d ∈
− 12 ,
1
2
and n = 10, 11, . . . , 100.
Figure 7.3 displays the variance of ĉn for n = 10, 11, . . . , 100 and d ∈ − 21 , 12 .
The picture is very similar to Figure 7.2. In particular, the variance is only slowly
decreasing for large parameter values and tends to 0 as d tends to 21 .
2d−1
Next, we derive asymptotics of Var(ĉn ). It is well-known that ρk ∼ Γ(1−d)
Γ(d) k
as k → ∞ (see [3]). According to Theorem 6.1 (i), we obtain
Var(ĉn ) ∼ σ 2 n−1
P∞
for d < 14 , where σ 2 := γ0 + 2 k=1 γk < ∞. In the case d = 0, where ρk = 0 for all
1
k > 0, we obtain Var(ĉn ) = 4n
. If d = 14 , then Theorem 6.1 (ii) yields
2
2 Γ( 34 )
ln n
Var(ĉn ) ∼
2
1
n
π 2 Γ( )
4
20
M. SINN AND K. KELLER
In the case d > 14 , Theorem 6.1 (iii) yields
Var(ĉn ) ∼
4 (Γ(1 − d))2 (1 − 2d) 4d−2
n
.
π 2 (Γ(d))2 (4d − 1)
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