Análisis de los puntos del Plan de aseguramiento de Calidad del proyecto SGDG

Recall that the autocovariance function γ : Z → R of a stationary process (Y_t)_t∈Z is given by

γ(h) = E(Yt+hYt) − E(Yt+h) E(Yt), h ∈ Z, with the properties

γ(0) ≥ 0, |γ(h)| ≤ γ(0), γ(h) = γ(−h), h ∈ Z. (5.1) The following result characterizes an autocovariance function in terms of positive semidefiniteness.

Theorem 5.1.1. A symmetric function K : Z → R is the autoco-variance function of a stationary process (Yt)_t∈Z iff K is a positive semidefinite function, i.e., K(−n) = K(n) and

X

1≤r,s≤n

x_rK(r − s)x_s ≥ 0 (5.2)

for arbitrary n ∈ N and x1, . . . , x_n ∈ R.

Proof. It is easy to see that (5.2) is a necessary condition for K to be the autocovariance function of a stationary process, see Exercise 5.19.

It remains to show that (5.2) is sufficient, i.e., we will construct a stationary process, whose autocovariance function is K.

We will define a family of finite-dimensional normal distributions, which satisfies the consistency condition of Kolmogorov’s theorem, cf.

Brockwell and Davis (1991, Theorem 1.2.1). This result implies the existence of a process (V_t)_t∈Z, whose finite dimensional distributions coincide with the given family.

Define the n × n-matrix

K⁽ⁿ⁾ := K(r − s)

1≤r,s≤n,

which is positive semidefinite. Consequently there exists an n-dimen-sional normal distributed random vector (V₁, . . . , V_n) with mean vector

5.1 Characterizations of Autocovariance Functions 161 zero and covariance matrix K⁽ⁿ⁾. Define now for each n ∈ N and t ∈ Z

a distribution function on Rⁿ by

Ft+1,...,t+n(v1, . . . , vn) := P {V1 ≤ v₁, . . . , Vn ≤ v_n}.

This defines a family of distribution functions indexed by consecutive integers. Let now t₁ < · · · < t_m be arbitrary integers. Choose t ∈ Z and n ∈ N such that tⁱ = t + ni, where 1 ≤ n1 < · · · < nm ≤ n. We define now

F_t1,...,tm((v_i)_1≤i≤m) := P {V_ni ≤ v_i, 1 ≤ i ≤ m}.

Note that Ft₁,...,t_m does not depend on the special choice of t and n and thus, we have defined a family of distribution functions indexed by t₁ < · · · < t_m on R^m for each m ∈ N, which obviously satisfies the consistency condition of Kolmogorov’s theorem. This result implies the existence of a process (V_t)_t∈Z, whose finite dimensional distribution at t1 < · · · < tm has distribution function Ft₁,...,t_m. This process has, therefore, mean vector zero and covariances E(V_t+hV_t) = K(h), h ∈ Z.

Spectral Distribution Function and Spectral Density

The preceding result provides a characterization of an autocovariance function in terms of positive semidefiniteness. The following char-acterization of positive semidefinite functions is known as Herglotz’s theorem. We use in the following the notation R1

0 g(λ) dF (λ) in place of R

(0,1]g(λ) dF (λ).

Theorem 5.1.2. A symmetric function γ : Z → R is positive semidef-inite iff it can be represented as an integral

γ(h) = Z 1

0

e^i2πλhdF (λ) = Z 1

0

cos(2πλh) dF (λ), h ∈ Z, (5.3) where F is a real valued measure generating function on [0, 1] with F (0) = 0. The function F is uniquely determined.

The uniquely determined function F , which is a right-continuous, in-creasing and bounded function, is called the spectral distribution func-tion of γ. If F has a derivative f and, thus, F (λ) = F (λ) − F (0) =

Rλ

0 f (x) dx for 0 ≤ λ ≤ 1, then f is called the spectral density of γ.

Note that the propertyP

h≥0|γ(h)| < ∞ already implies the existence of a spectral density of γ, cf. Theorem 4.2.5 and the proof of Corollary 5.1.5.

Recall that γ(0) = R1

0 dF (λ) = F (1) and thus, the autocorrelation function ρ(h) = γ(h)/γ(0) has the above integral representation, but with F replaced by the probability distribution function F/γ(0).

Proof of Theorem 5.1.2. We establish first the uniqueness of F . Let G be another measure generating function with G(λ) = 0 for λ ≤ 0 and constant for λ ≥ 1 such that

Let now ψ be a continuous function on [0, 1]. From calculus we know (cf. Rudin, 1986, Section 4.24) that we can find for arbitrary ε > 0 a trigonometric polynomial pε(λ) = PN

h=−N ahe^i2πλh, 0 ≤ λ ≤ 1, such that

sup

0≤λ≤1

|ψ(λ) − p_ε(λ)| ≤ ε.

As a consequence we obtain that Z 1

Since ψ was an arbitrary continuous function, this in turn together with F (0) = G(0) = 0 implies F = G.

5.1 Characterizations of Autocovariance Functions 163 Suppose now that γ has the representation (5.3). We have for

arbi-trary x_i ∈ R, i = 1, . . . , n i.e., γ is positive semidefinite.

Suppose conversely that γ : Z → R is a positive semidefinite function.

This implies that for 0 ≤ λ ≤ 1 and N ∈ N (Exercise 5.2) selec-tion theorem (cf. Billingsley, 1968, page 226ff) to deduce the existence of a measure generating function ˜F and a subsequence (F_Nk)_k such that F_Nk converges weakly to ˜F i.e.,

for every continuous and bounded function g : [0, 1] → R (cf. Billings-ley, 1968, Theorem 2.1). Put now F (λ) := ˜F (λ) − ˜F (0). Then F is a

measure generating function with F (0) = 0 and Z 1

0

g(λ) d ˜F (λ) = Z 1

0

g(λ) dF (λ).

If we replace N in (5.4) by N_k and let k tend to infinity, we now obtain representation (5.3).

Example 5.1.3. A white noise (ε_t)_t∈Z has the autocovariance function γ(h) =

(σ², h = 0

0, h ∈ Z \ {0}.

Since

Z 1 0

σ²e^i2πλhdλ =

(σ², h = 0

0, h ∈ Z \ {0},

the process (ε_t) has by Theorem 5.1.2 the constant spectral density f (λ) = σ², 0 ≤ λ ≤ 1. This is the name giving property of the white noise process: As the white light is characteristically perceived to belong to objects that reflect nearly all incident energy throughout the visible spectrum, a white noise process weighs all possible frequencies equally.

Corollary 5.1.4. A symmetric function γ : Z → R is the autocovari-ance function of a stationary process (Y_t)_t∈Z, iff it satisfies one of the following two (equivalent) conditions:

(i) γ(h) = R1

0 e^i2πλhdF (λ), h ∈ Z, where F is a measure generating function on [0, 1] with F (0) = 0.

(ii) P

1≤r,s≤nx_rγ(r − s)x_s ≥ 0 for each n ∈ N and x¹, . . . , x_n ∈ R.

Proof. Theorem 5.1.2 shows that (i) and (ii) are equivalent. The assertion is then a consequence of Theorem 5.1.1.

Corollary 5.1.5. A symmetric function γ : Z → R withP

t∈Z|γ(t)| <

∞ is the autocovariance function of a stationary process iff f (λ) := X

t∈Z

γ(t)e^−i2πλt ≥ 0, λ ∈ [0, 1].

The function f is in this case the spectral density of γ.

5.1 Characterizations of Autocovariance Functions 165 Proof. Suppose first that γ is an autocovariance function. Since γ is in

this case positive semidefinite by Theorem 5.1.1, andP

t∈Z|γ(t)| < ∞ by assumption, we have (Exercise 5.2)

0 ≤ f_N(λ) : = 1 see Exercise 5.8. The function f is consequently nonnegative. The in-verse Fourier transform in Theorem 4.2.5 implies γ(t) =R1

0 f (λ)e^i2πλtdλ, t ∈ Z i.e., f is the spectral density of γ.

Suppose on the other hand that f (λ) = P

t∈Zγ(t)e^i2πλt ≥ 0, 0 ≤ λ ≤ 1. The inverse Fourier transform implies γ(t) = R1

0 f (λ)e^i2πλt dλ

= R1

0 e^i2πλtdF (λ), where F (λ) = Rλ

0 f (x) dx, 0 ≤ λ ≤ 1. Thus we have established representation (5.3), which implies that γ is positive semidefinite, and, consequently, γ is by Corollary 5.1.4 the autoco-variance function of a stationary process.

Example 5.1.6. Choose a number ρ ∈ R. The function

γ(h) =

is the autocovariance function of a stationary process iff |ρ| ≤ 0.5.

This follows from autocorre-lation function of an MA(1)-process, cf. Example 2.2.2.

The spectral distribution function of a stationary process satisfies (Ex-ercise 5.10)

F (0.5 + λ) − F (0.5⁻) = F (0.5) − F ((0.5 − λ)⁻), 0 ≤ λ < 0.5,

where F (x⁻) := lim_ε↓0F (x − ε) is the left-hand limit of F at x ∈ (0, 1]. If F has a derivative f , we obtain from the above symmetry f (0.5 + λ) = f (0.5 − λ) or, equivalently, f (1 − λ) = f (λ) and, hence,

γ(h) = Z 1

0

cos(2πλh) dF (λ) = 2 Z 0.5

0

cos(2πλh)f (λ) dλ.

The autocovariance function of a stationary process is, therefore, de-termined by the values f (λ), 0 ≤ λ ≤ 0.5, if the spectral density exists. Recall, moreover, that the smallest nonconstant period P₀ visible through observations evaluated at time points t = 1, 2, . . . is P₀ = 2 i.e., the largest observable frequency is the Nyquist frequency λ₀ = 1/P₀ = 0.5, cf. the end of Section 4.2. Hence, the spectral density f (λ) matters only for λ ∈ [0, 0.5].

Remark 5.1.7. The preceding discussion shows that a function f : [0, 1] → R is the spectral density of a stationary process iff f satisfies the following three conditions

(i) f (λ) ≥ 0,

(ii) f (λ) = f (1 − λ), (iii) R1

0 f (λ) dλ < ∞.