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On asymptotic behavior of the prediction error for a class of

deterministic stationary sequences

N. M. Babayan∗ and M. S. Ginovyan†

November 23, 2021

Abstract

One of the main problem in prediction theory of stationary processes X(t) is to describe theasymptotic behavior of the best linear mean squared prediction error in predicting X(0) givenX(t),−n ≤ t ≤ −1, as n goes to infinity. This behavior depends on the regularity (deterministic ornon-deterministic) of the processX(t). In his seminal paper ’Some purely deterministic processes’(J. of Math. and Mech., 6(6), 801-810, 1957), for a specific spectral density that has a very highorder contact with zero M. Rosenblatt showed that the prediction error behaves like a poweras n → ∞. In the paper Babayan et al. ’Extensions of Rosenblatt’s results on the asymptoticbehavior of the prediction error for deterministic stationary sequences’ (J. Time Ser. Anal. 42,622-652, 2021), Rosenblatt’s result was extended to the class of spectral densities of the formf = fdg, where fd is the spectral density of a deterministic process that has a very high ordercontact with zero, while g is a function that can have polynomial type singularities. In this paper,we describe new extensions of the above quoted results in the case where the function g can havearbitrary power type singularities. Examples illustrate the obtained results.

Key words and phrases. Prediction problem, deterministic stationary process, singular spectraldensity, Rosenblatt’s theorem, weakly varying sequence.

2010 Mathematics Subject Classification. 60G10, 60G25, 62M15, 62M20.

1 Introduction

Let X(t), t ∈ Z := 0,±1, . . ., be a centered discrete-time second-order stationary process. Theprocess is assumed to have an absolutely continuous spectrum with spectral density function f(λ),λ ∈ [−π, π]. The ’finite’ linear prediction problem is as follows.

Suppose we observe a finite realization of the process X(t): X(t), −n ≤ t ≤ −1, n ∈ N :=1, 2, . . .. We want to make an one-step ahead prediction, that is, to predict the unobserved randomvariable X(0), using the linear predictor Y =

∑nk=1 ckX(−k).

The coefficients ck, k = 1, 2, . . . , n, are chosen so as to minimize the mean-squared error:IE |X(0)−Y|2 , where IE[·] stands for the expectation operator. If such minimizing constants ck :=ck,n can be found, then the random variable Xn(0) :=

∑nk=1 ckX(−k) is called the best linear one-

step ahead predictor of X(0) based on the observed finite past: X(−n), . . . ,X(−1). The minimummean-squared error:

σ2n(f) := IE

∣

∣

∣X(0) − Xn(0)

∣

∣

∣

2≥ 0

is called the best linear one-step ahead prediction error of X(t) based on the past of length n.

∗Russian-Armenian University, Yerevan, Armenia, e-mail: nmbabayan@gmail.com†Boston University, Boston, USA, e-mail: ginovyan@math.bu.edu. Corresponding author.

1

One of the main problem in prediction theory of second-order stationary processes, called the’direct’ prediction problem is to describe the asymptotic behavior of the prediction error σ2

n(f) asn → ∞. This behavior depends on the regularity nature (deterministic or nondeterministic) of theobserved process X(t).

Observe that σ2n+1(f) ≤ σ2

n(f) (n ∈ N), and hence, the limit of σ2n(f) as n → ∞ exists. Denote

by σ2(f) := σ2∞(f) the prediction error of X(0) by the entire infinite past: X(t), t ≤ −1.

From the prediction point of view it is natural to distinguish the class of processes for which wehave error-free prediction by the entire infinite past, that is, σ2(f) = 0. Such processes are calleddeterministic or singular. Processes for which σ2(f) > 0 are called nondeterministic (for more aboutthese terms see, e.g., Babayan et al. [4], and Grenander and Szego [10], p.176).

Define the relative prediction error δn(f) := σ2n(f)−σ2(f), and observe that δn(f) is non-negative

and tends to zero as n → ∞. But what about the speed of convergence of δn(f) to zero as n → ∞?The paper deals with this question. Specifically, the prediction problem we are interested in isto describe the rate of decrease of δn(f) to zero as n → ∞, depending on the regularity nature(deterministic or nondeterministic) of the observed process X(t).

The prediction problem stated above goes back to classical works of A. Kolmogorov, G. Szegoand N. Wiener. It was then considered by many authors for different classes of nondeterministicprocesses (see, e.g., the survey papers Bingham [5] and Ginovyan [9], and references therein).

We focus in this paper on deterministic processes, that is, when σ2(f) = 0. This case is not onlyof theoretical interest, but is also important from the point of view of applications. For example,as pointed out by Rosenblatt [14] (see also Pierson [11]), situations of this type arise in Neumann’stheoretical model of storm-generated ocean waves. Such models are also of interest in meteorology(see Fortus [8]).

Only few works are devoted to the study of the speed of convergence of δn(f) = σ2n(f) to zero

as n → ∞, that is, the asymptotic behavior of the prediction error for deterministic processes. Oneneeds to go back to the classical work of Rosenblatt [14], where the asymptotic behavior of theprediction error σ2

n(f) was investigated in the following two cases:(a) the spectral density f(λ) is continuous and positive on an interval of the segment [−π, π] and

zero elsewhere,(b) the spectral density f(λ) has a very high order of contact with zero at points λ = 0,±π, and

is strictly positive otherwise.For the case (a) above, Rosenblatt [14] proved that the prediction error σ2

n(f) decreases to zeroexponentially as n → ∞. Later the problem (a) was studied by Babayan [1, 2] and Babayan et al.[4] (see also Davisson [7] and Fortus [8]), where some extensions of Rosenblatt’s result have beenobtained.

Concerning the case (b) above, for a specific deterministic process X(t), Rosenblatt proved in[14] that the prediction error σ2

n(f) decreases to zero like a power as n → ∞. More precisely, thedeterministic process X(t) considered in Rosenblatt [14] has the spectral density

fa(λ) :=e(2λ−π)ϕ(λ)

cosh (πϕ(λ)), fa(−λ) = fa(λ), 0 ≤ λ ≤ π, (1.1)

where ϕ(λ) = (a/2) cot λ and a is a positive parameter.Using the technique of orthogonal polynomials on the unit circle and Szego’s results, Rosenblatt

[14] proved the following theorem.

Theorem A (Rosenblatt [14]). Suppose that the process X(t) has spectral density fa given by (1.1).Then the following asymptotic relation for the prediction error σ2

n(fa) holds:

σ2n(fa) ∼

Γ2 ((a+ 1)/2)

π22−an−a as n → ∞. (1.2)

2

Note that the function in (1.1) was first considered by Pollaczek [12], and then by Szego [16], as aweight-function of a class of orthogonal polynomials that serve as illustrations for certain ’irregular’phenomena in the theory of orthogonal polynomials. For the function fa in (1.1), we have thefollowing asymptotic relation (for details see Szego [16] and Example 6.3 below):

fa(λ) ∼

2ea exp −aπ/|λ| as λ → 0,2 exp −aπ/(π − |λ|) as λ → ±π.

(1.3)

Thus, the function fa in (1.1) has a very high order of contact with zero at points λ = 0,±π, due towhich the process with spectral density fa is deterministic and the prediction error σ2

n(fa) in (1.2)decreases to zero like a power as n → ∞.

In Babayan et al. [4] was proved that if the spectral density f is such that the sequence σn(f)is weakly varying (a term defined in Section 2.3) and if, in addition, g is a nonnegative functionthat can have polynomial type singularities, then the sequences σn(fg) and σn(f) have thesame asymptotic behavior as n → ∞ (see Theorem B in Section 3). Using this result, Rosenblatt’sTheorems A was extended in Babayan et al. [4] to a class of spectral densities of the form f = fag,where fa is as in (1.1) and g is a nonnegative function that can have polynomial type singularities(see Theorem C in Section 3).

In this paper, we extend the above quoted results to a broader class of spectral densities, forwhich the function g can have arbitrary power type singularities.

Throughout the paper we will use the following notation.The standard symbols N, Z, R and C denote the sets of natural, integer, real and complex numbers,respectively. Also, we denote Λ := [−π, π], T := z ∈ C : |z| = 1. For a point λ0 ∈ Λ and a numberδ > 0 by Oδ(λ0) we denote a δ-neighborhood of λ0, that is, Oδ(λ0) := λ ∈ Λ : |λ − λ0| < δ.By Lp(µ) := Lp(T, µ) (p ≥1) we denote the weighted Lebesgue space with respect to the measureµ, and by (·, ·)p,µ and || · ||p,µ we denote the inner product and the norm in Lp(µ), respectively. Inthe special case where µ is the Lebesgue measure, we will use the notation Lp, (·, ·)p and || · ||p,respectively. For a function f ≥ 0 by G(f) we denote the geometric mean of f . For two functionsf(λ) ≥ 0 and g(λ) ≥ 0, λ ∈ Λ, we will write f(λ)∼g(λ) as λ → λ0 if limλ→λ0 f(λ/g(λ) = 1, andf(λ)≃g(λ) as λ → λ0 if limλ→λ0 f(λ/g(λ) = c > 0. We will use similar notation for sequences: fortwo sequences an ≥ 0, n ∈ N and bn > 0, n ∈ N, we will write an ∼ bn if limn→∞ an/bn = 1,an≃bn if limn→∞ an/bn = c > 0; an = O(bn) if an/bn is bounded, and an = o(bn) if an/bn → 0 asn → ∞. The letters C, c, M and m with or without indices are used to denote positive constants,the values of which can vary from line to line.

The paper is organized as follows. In Section 2 we present some necessary notions and preliminaryresults that are used throughout the paper. In Section 3 we state some auxiliary results. In Section4 we state the main results of the paper. Section 5 contains the proofs of the main results. In Section6 we discuss some examples illustrating the obtained results.

2 Preliminaries

In this section we present some notions and auxiliary results, which will be used in the sequel: theKolmogorov-Szego Theorem, formulas and some properties of the finite prediction error, propertiesof the geometric mean of a function, and definition and properties of weakly varying sequences.

2.1 Kolmogorov-Szego’s Theorem

Let X(t) be a centered discrete-time stationary process defined on a probability space (Ω,F , P )with covariance function r(t), t ∈ Z. By the Herglotz theorem (see, e.g., Brockwell and Davis [6], p.

3

117-18), there is a finite measure µ on Λ such that the covariance function r(t) admits the followingspectral representation:

r(t) =

∫ π

−πe−itλdµ(λ), t ∈ Z. (2.1)

The measure µ in (2.1) is called the spectral measure of the processX(t). If µ is absolutely continuous(with respect to the Lebesgue measure), then the function f(λ) := dµ(λ)/dλ is called the spectraldensity of X(t). We assume that X(t) is a non-degenerate process, that is, Var[X(0)] := IE|X(0)|2 =r(0) > 0 and, without loss of generality, we may take r(0) = 1. Also, to avoid the trivial cases, weassume that the spectral measure µ is non-trivial, that is, the support of µ has positive Lebesguemeasure.

Remark 2.1. The parametrization of the unit circle T by the formula z = eiλ establishes a bijectionbetween T and the interval [−π, π). By means of this bijection the measure µ on Λ generates thecorresponding measure on the unit circle T, which we also denote by µ. Thus, depending on thecontext, the measure µ will be supported either on Λ or on T. We use the standard Lebesguedecomposition of the measure µ:

dµ(λ) = dµa(λ) + dµs(λ) = f(λ)dλ+ dµs(λ), (2.2)

where µa is the absolutely continuous part of µ (with respect to the Lebesgue measure) and µs isthe singular part of µ, which is the sum of the discrete and continuous singular components of µ.

The next result describes the asymptotic behavior of the prediction error σ2n(µ) for a stationary

process X(t) with spectral measure µ of the form (2.2) and gives a spectral characterization ofdeterministic and nondeterministic processes (see, e.g., Grenander and Szego [10], p. 44).

Kolmogorov-Szego’s Theorem. Let X(t) be a non-degenerate stationary process with spectral measureµ of the form (2.2). The following relations hold.

limn→∞

σ2n(µ) = lim

n→∞σ2n(f) = σ2(f) = 2πG(f), (2.3)

where G(f) is the geometric mean of f , namely

G(f) :=

exp

12π

∫ π−π ln f(λ) dλ

if ln f ∈ L1(Λ)

0, otherwise.(2.4)

It is remarkable that the limit in (2.3) is independent of the singular part µs.The condition ln f ∈ L1(Λ) in (2.4) is equivalent to the Szego condition:

∫ π

−πln f(λ) dλ > −∞ (2.5)

(this equivalence follows because ln f(λ) ≤ f(λ)). The Szego condition (2.5) is also called thenon-determinism condition.

In this paper we consider the class of deterministic processes with absolutely continuous spectra.We will assume that the corresponding spectral densities, defined on the segment [−π, π] (or on theunit circle T), are extended to R with period 2π.

Following Rosenblatt [14], we will say that the spectral density f(λ) has a very high order ofcontact with zero at a point λ0 if f(λ) is positive everywhere except for the point λ0, due to whichthe Szego condition (2.5) is violated.

4

2.2 Properties of the finite prediction error and geometric mean

Suppose we have observed the values X(−n), . . . ,X(−1) of a centered, real-valued stationary processX(t) with spectral measure µ of the form (2.2). The one-step ahead linear prediction problem inpredicting a random variable X(0) based on the observed values X(−n), . . . ,X(−1) involves findingconstants ck := ck,n, k = 1, 2, . . . , n, that minimize the one-step ahead prediction error:

σ2n(µ) := min

ckIE

∣

∣

∣

∣

∣

X(0) −

n∑

k=1

ckX(−k)

∣

∣

∣

∣

∣

2

= IE

∣

∣

∣

∣

∣

X(0)−

n∑

k=1

ckX(−k)

∣

∣

∣

∣

∣

2

. (2.6)

Using Kolmogorov’s isometric isomorphism V : X(t) ↔ eitλ (see, e.g., Babayan et al. [4]), in viewof (2.6), for the prediction error σ2

n(µ) we can write

σ2n(µ) = min

ck

∥

∥

∥

∥

∥

1−

n∑

k=1

cke−ikλ

∥

∥

∥

∥

∥

2

2,µ

= minqn∈Qn

‖qn‖22,µ , (2.7)

where || · ||2,µ is the norm in L2(T, µ), and

Qn :=

qn : qn(z) = zn + c1zn−1 + · · · cn

(2.8)

is the class of monic polynomials (i.e. with c0 = 1) of degree n. Thus, the problem of finding σ2n(µ)

becomes to the problem of finding the solution of the minimum problem (2.7)-(2.8).The polynomial pn(z) := pn(z, µ) which solves the minimum problem (2.7)-(2.8) is called the

optimal polynomial for µ in the class Qn.The next result by Szego solves the minimum problem (2.7)-(2.8) (see, e.g., Grenander and Szego

[10], p. 38).

Proposition 2.1. The unique solution of the minimum problem (2.7)-(2.8) is given by pn(z) =κ−1n ϕn(z), and the minimum in (2.7) is equal to ||pn||2,µ = κ−2

n , where ϕn(z) = κnzn + · · · + ln is

the nth orthogonal polynomial on the unit circle associated with the measure µ, and κn is the leadingcoefficient of ϕn(z).

Thus, for the prediction error σ2n(µ) we have the following formula:

σ2n(µ) = min

qn∈Qn‖qn‖

22,µ = ‖pn(µ)‖

22,µ =

∥

∥κ−1n ϕn(µ)

∥

∥

2

2,µ= κ−2

n . (2.9)

Denote by Dn = Dn(µ) := det[r(t − s), t, s = 0, 1, . . . n] the nth Toeplitz determinant generatedby the measure µ, where r(t) is the covariance function given by (2.1). Taking into account thatκ2n = Dn−1/Dn (see, e.g., Grenander and Szego [10], p. 38), in view of (2.9) we obtain the followingformula for the prediction error σ2

n(µ):

σ2n(µ) =

Dn(µ)

Dn−1(µ). (2.10)

In what follows we assume that the spectral measure µ is absolutely continuous with spectraldensity f , and instead of σ2

n(µ), pn(µ) and Dn(µ) we use the notation σ2n(f), pn(f) and Dn(f),

respectively.In the next proposition we list a number of properties of the prediction error σ2

n(f).

Proposition 2.2. The prediction error σ2n(f) possesses the following properties.

(a) σ2n(f) is a non-decreasing functional of f : σ2

n(f1) ≤ σ2n(f2) when f1(λ) ≤ f2(λ), λ ∈ [−π, π].

5

(b) If f(λ) = g(λ) almost everywhere on [−π, π], then σ2n(f) = σ2

n(g).

(c) For any positive constant c we have σ2n(cf) = cσ2

n(f).

(d) If f(λ) = f(λ− λ0), λ0 ∈ [−π, π], then σ2n(f) = σ2

n(f).

Proof. To prove assertion (a), observe that by the definition of optimal polynomials pn(z, f1) andpn(z, f2), corresponding to spectral densities f1 and f2, respectively, and formula (2.9) we have

σ2n(f1) = ‖pn(f1)‖

22,f1 ≤ ‖pn(f2)‖

22,f1 ≤ ‖pn(f2)‖

22,f2 = σ2

n(f2).

In the last relation the first inequality follows from the optimality of the polynomial pn(z, f1), whilethe second inequality follows from assumption that f1(λ) ≤ f2(λ), λ ∈ Λ.

We show that assertions (b)–(d) follow from formula (2.10). Indeed, observing that the elementsof the Toeplitz determinant Dn(f) being integrals are invariant with respect to null sets, in view of(2.10) we obtain assertion (b). To prove assertion (c), observe that for the Toeplitz determinantsgenerated by the functions cf and f we have Dn(cf) = cnDn(f) (see Grenander and Szego [10], p.64-65). Hence in view of (2.10) we have

σ2n(cf) =

Dn(cf)

Dn−1(cf)=

cnDn(f)

cn−1Dn−1(f)= cσ2

n(f).

Finally, the assertion (d) follows from (2.10) and the fact that Dn(f) = Dn(f) (see Grenanderand Szego [10], p. 64-65).

Recall that for a function h ≥ 0 by G(h) we denote the geometric mean of h (see formula (2.4)).In the next proposition we list some properties of the geometric mean G(h) (see Babayan et al. [4]).

Proposition 2.3. The following assertions hold.

(a) Let c > 0, α ∈ R, f(λ) ≥ 0 and g(λ) ≥ 0. Then

G(c) = c, G(fg) = G(f)G(g), G(fα) = Gα(f). (2.11)

(b) G(f) is a non-decreasing functional of f : if 0 ≤ f(λ) ≤ g(λ), then 0 ≤ G(f) ≤ G(g). Inparticular, if 0 ≤ f(λ) ≤ 1, then 0 ≤ G(f) ≤ 1.

(c) If t(λ) is a nonnegative trigonometric polynomial, then G(tα) > 0 for α ∈ R.

2.3 Weakly varying sequences

We recall the notion of weakly varying sequences and state some of their properties (see Babayan etal. [4])., This notion will be used in the specification of the class of deterministic processes to beconsidered in this paper.

Definition 2.1. A sequence of non-zero numbers an, n ∈ N is said to be weakly varying iflimn→∞ an+1/an = 1.

In the next proposition we list some simple properties of the weakly varying sequences, whichcan easily be verified (see Babayan et al. [4]).

Proposition 2.4. The following assertions hold.

(a) If an, n ∈ N is a weakly varying sequence, then limn→∞ an+ν/an = 1 for any ν ∈ N.

6

(b) If an, n ∈ N is a sequence such that an → a 6= 0 as n → ∞, then an is a weakly varyingsequence.

(c) If an, n ∈ N and bn, n ∈ N, are weakly varying sequences, then can (c 6= 0), aαn (α ∈R, an > 0), anbn and an/bn also are weakly varying sequences.

(d) If an, n ∈ N is a weakly varying sequence, and bn, n ∈ N is a sequence of non-zero numberssuch that limn→∞ bn/an = c 6= 0, then bn, n ∈ N is also a weakly varying sequence.

3 Asymptotic behavior of the prediction error: Auxiliary results

In this section we state some auxiliary results concerning asymptotic behavior of the prediction error(cf. Babayan et al. [4]).

In what follows we consider the class of deterministic processes possessing spectral densities forwhich the sequence of prediction errors σn(f) is weakly varying, and denote by F the class of thecorresponding spectral densities:

F :=

f ∈ L1(Λ) : f ≥ 0, G(f) = 0, limn→∞

σn+1(f)

σn(f)= 1

. (3.1)

Remark 3.1. According to Rakhmanov’s theorem (see Rakhmanov [13] and Babayan et al. [4])a sufficient condition for f ∈ F is that f > 0 almost everywhere on Λ. Thus, the considered classF includes all the almost everywhere positive spectral densities. On the other hand, according toTheorem 3.2 and Remark 3.7 of Babayan et al. [4], the class F does not contain spectral densities,which vanish on an entire segment of Λ (or on an arc of the unit circle T).

Definition 3.1. Let F be the class of spectral densities defined by (3.1). For f ∈ F denote by Mf

the class of nonnegative functions g(λ) (λ ∈ Λ) satisfying the following three conditions: G(g) > 0,fg ∈ L1(Λ), and

limn→∞

σ2n(fg)

σ2n(f)

= G(g), (3.2)

that is,

Mf :=

g ≥ 0, G(g) > 0, fg ∈ L1(Λ), limn→∞

σ2n(fg)

σ2n(f)

= G(g)

. (3.3)

The next proposition shows that the class F is close under multiplication by functions from theclass Mf .

Proposition 3.1. If f ∈ F and g ∈ Mf , then fg ∈ F .

Proof. According to the definition of the class F , we have to show that fg ∈ L1(Λ), G(fg) = 0, and

limn→∞

σn+1(fg)

σn(fg)= 1. (3.4)

The assertion fg ∈ L1(Λ) follows from the condition g ∈ Mf , while G(fg) = 0 follows from thecondition f ∈ F and Proposition 2.3(a): G(fg) = G(f)G(g) = 0. As for the relation (3.4), we canwrite

limn→∞

σn+1(fg)

σn(fg)= lim

n→∞

σn+1(fg)

σn+1(f)·σn+1(f)

σn(f)·σn(f)

σn(fg)

= limn→∞

σn+1(fg)

σn+1(f)· limn→∞

σn+1(f)

σn(f)· limn→∞

σn(f)

σn(fg)=

√

G(g) · 1/√

G(g) = 1,

and the result follows.

7

The next result shows that the class Mf in a certain sense is close under multiplication.

Proposition 3.2. Let f ∈ F . If g1 ∈ Mf and g2 ∈ Mfg1 , then g := g1g2 ∈ Mf and fg ∈ F . Inparticular, if g ∈ Mf ∩Mfg, then g2 ∈ Mf .

Proof. By the definition of the classes Mf and Mfg1 , we have G(g1) > 0, G(g2) > 0, and hence, byProposition 2.3(a), G(g) > 0. By Proposition 3.1 we have fg1 ∈ F . Hence, taking into account thatg2 ∈ Mfg1 we have fg1g2 ∈ L1(Λ), and

limn→∞

σ2n(fg1g2)

σ2n(fg1)

= G(g2). (3.5)

Then, we can write

limn→∞

σ2n(fg)

σ2n(f)

= limn→∞

σ2n(fg1g2)

σ2n(f)

= limn→∞

σ2n(fg1g2)

σ2n(fg1)

·σ2n(fg1)

σ2n(f)

= limn→∞

σ2n(fg1g2)

σ2n(fg1)

· limn→∞

σ2n(fg1)

σ2n(f)

= G(g1) ·G(g2) = G(g).

In the last relation the fourth equality follows from (3.5) and the condition g1 ∈ Mf , while thefifth equality follows from Proposition 2.3(a). Thus, we have proved that g ∈ Mf , from this andProposition 3.1 it follows that fg ∈ F .

In the next definition we introduce certain classes of bounded functions.

Definition 3.2. We define the class B to be the set of all nonnegative, Riemann integrable onΛ = [−π, π] functions h(λ). Also, we define the following subclasses:

B+ := h ∈ B : h(λ) > m, B− := h ∈ B : h(λ) 6 M, B−+ := B+ ∩B−,

where m and M are some positive constants.

In the next proposition we list some obvious properties of the classes B+, B− and B−

+ .

Proposition 3.3. The following assertions hold.

a) If h ∈ B+(B−), then 1/h ∈ B−(B+).

b) If h1, h2 ∈ B+(B−), then h1 + h2 ∈ B+(B

−) and h1h2 ∈ B+(B−).

c) If h1, h2 ∈ B− and h1/h2 is bounded, then h1/h2 ∈ B−.

d) If h1, h2 ∈ B−+, then h1 + h2 ∈ B−

+, h1h2 ∈ B−+ and h1/h2 ∈ B−

+ .

The following theorem, proved in Babayan et al. [4], describes the asymptotic behavior of theratio σ2

n(fg)/σ2n(f) as n → ∞, and essentially states that if the spectral density f is from the class

F (see (3.1)), and g is a nonnegative function, which can have polynomial type singularities, thenthe sequences σn(fg) and σn(f) have the same asymptotic behavior as n → ∞ up to a positivenumerical factor.

Theorem B (Babayan et al. [4]). Let f be an arbitrary function from the class F , and let g be afunction of the form:

g(λ) = h(λ) ·t1(λ)

t2(λ), λ ∈ Λ, (3.6)

where h ∈ B−+ , t1 and t2 are nonnegative trigonometric polynomials, such that fg ∈ L1(Λ). Then

g ∈ Mf and fg ∈ F , that is, fg is the spectral density of a deterministic process with weakly varyingprediction error, and the relation (3.2) holds.

8

Taking into account that the sequence n−α, n ∈ N, α > 0 is weakly varying, as an immediateconsequence of Theorem B, we have the following result.

Corollary 3.1 (Babayan et al. [4]). Let the functions f and g be as in Theorem B, and let σn(f) ∼cn−α (c > 0, α > 0) as n → ∞. Then

σn(fg) ∼ cG(g)n−α as n → ∞,

where G(g) is the geometric mean of g.

The next result, which immediately follows from Theorem B and Corollary 3.1, extends Rosen-blatt’s Theorem A.

Theorem C (Babayan et al. [4]). Let f = fag, where fa is defined by (1.1) and g satisfies theassumptions of Theorem B. Then

σ2n(f) ∼

Γ2(

a+12

)

G(g)

π22−an−a as n → ∞,

where G(g) is the geometric mean of g.

We thus have the same limiting behavior for σ2n(f) as in the Rosenblatt’s relation (1.2) up to an

additional positive factor G(g).

4 Asymptotic behavior of the prediction error: The main results

In this section we state the main results of this paper, extending the above stated Theorems B andC to a broader class of spectral densities, for which the function g can have arbitrary power typesingularities.

Theorem 4.1. Let f be an arbitrary function from the class F , and let g be a function of the form:

g(λ) = h(λ) · |t(λ)|α, α > 0, λ ∈ Λ, (4.1)

where h ∈ B−+ and t is an arbitrary trigonometric polynomial. Then g ∈ Mf and fg ∈ F , that is,

fg is the spectral density of a deterministic process with weakly varying prediction error, and therelation (3.2) holds.

Corollary 4.1. The conclusion of Theorem 4.1 remains valid if the function g has the followingform:

g(λ) = h(λ) · |t1(λ)|α1 · |t2(λ)|

α2 · · · · · |tm(λ)|αm , λ ∈ Λ,

where h ∈ B−+, t1, t2, . . . , tm are arbitrary trigonometric polynomials, α1, α2, . . . αm are arbitrary

positive numbers, and m ∈ N.

Theorem 4.2. Let f be an arbitrary function from the class F , and let g be a function of the form:

g(λ) = h(λ) · t−α(λ), α > 0, λ ∈ Λ, (4.2)

where h ∈ B−+ and t is a nonnegative trigonometric polynomial. Then g ∈ Mf and fg ∈ F provided

that ft−(k+1) ∈ L1(Λ), where k := [α] is the integer part of α.

To state the next result we need the following definition.

9

Definition 4.1. Let E1 and E2 be two numerical sets such that for any x ∈ E1 and y ∈ E2 we havex < y. We say that the sets E1 and E2 are separated from each other if supE1 < inf E2. Also, wesay that a numerical set E is separated from infinity if it is bounded from above.

Theorem 4.3. Let f(λ) and f(λ) (λ ∈ Λ) be spectral densities of stationary processes satisfying thefollowing conditions:

1) f, f ∈ B−;

2) the functions f(λ) and f(λ) have k common essential zeros λ1, λ2, . . . , λk ∈ Λ (−π < λ1 <λ2 < · · · < λk ≤ π, k ∈ N), that is,

limλ→λj

f(λ) = limλ→λj

f(λ) = 0, j = 1, 2, . . . , k; (4.3)

3) the functions f(λ) and f(λ) are infinitesimal of the same order in a neighborhood of each pointλj (j = 1, 2, . . . , k), that is,

limλ→λj

f(λ)

f(λ)= cj > 0, j = 1, 2, . . . , k; (4.4)

4) the functions f(λ) and f(λ) are bounded away from zero outside any neighborhood Oδ(λj) (j =1, 2, . . . , k), which is separated from the neighboring zeros λj−1 and λj+1 of λj, that is, there

is a number m := mδ > 0 such that f(λ) ≥ m and f(λ) ≥ m for almost all λ /∈ ∪kj=1Oδ(λj).

Then the following assertions hold:

a) h(λ) := f(λ)f(λ) ∈ B−

+ ;

b) the processes with spectral densities f and f either both are deterministic or both are nonde-terministic;

c) if one of the functions f and f is from the class F , then so is the other, and the followingrelation holds:

limn→∞

σ2n(f)

σ2n(f)

= G(h) > 0. (4.5)

Remark 4.1. The conditions of Theorem 4.3 mean that the points λj (j = 1, 2, . . . , k) are the only

common zeros of functions f(λ) and f(λ). Besides, in the case of deterministic processes, at leastone of these zeros should be of sufficiently high order. Also, notice that the conditions 1) and 4) ofTheorem 4.3 will be satisfied if the functions f(λ) and f(λ) are continuous on Λ.

Theorem 4.4. Let f be an arbitrary function from the class F , and let g be a function of the form:

g(λ) = h(λ) · |q(λ)|α, α ∈ R, λ ∈ Λ, (4.6)

where h ∈ B−+, q is an arbitrary algebraic polynomial with real coefficients, and fg ∈ L1(Λ). Then

fg ∈ F and g ∈ Mf .

The next result extends Theorem C to a broader class of functions g.

Corollary 4.2. Let f = fag, where fa is defined by (1.1), and let g be a function satisfying theconditions of one of Theorems 4.1, 4.2, 4.4 or Corollary 4.1. Then

δn(f) = σ2n(f) ∼

Γ2(

a+12

)

G(g)

π22−an−a as n → ∞,

where G(g) is the geometric mean of g.

10

Remark 4.2. In view of Remark 3.1 it follows that all the above stated results remain true if thecondition f ∈ F is replaced by the slightly strong but more constructive condition: ’the spectraldensity f is positive (f > 0) almost everywhere on Λ’.

5 Proofs

In this section we prove the main results of this paper stated in Section 4. We first establish anumber of lemmas.

5.1 Lemmas

Lemma 5.1. Let f be an arbitrary function from the class F , and let t be a nonnegative trigono-metric polynomial. Then t1/2 ∈ Mf and ft1/2 ∈ F .

Proof. Observe first that, by Proposition 3.1, the second assertion of the lemma (ft1/2 ∈ F) followsfrom the first assertion (t1/2 ∈ Mf ). So, to complete the proof of the lemma we have to prove therelation t1/2 ∈ Mf . To this end, we first verify the first two conditions for t1/2 to belong to theclass Mf , that is, the conditions: G(t1/2) > 0 and ft1/2 ∈ L1(Λ) (see (3.3)). First, the conditionG(t1/2) > 0 follows from Proposition 2.3(c). Next, observing that the function t1/2 is continuous,and hence is bounded, in view of the condition f ∈ F , we conclude that ft1/2 ∈ L1(Λ). Also, observethat by Proposition 2.3(a), we have G(ft1/2) = G(f)G(t1/2) = 0, showing that ft1/2 is the spectraldensity of a deterministic process.

Now we proceed to verify the third condition for t1/2 to belong to the class Mf , that is, therelation:

limn→∞

σ2n(ft

1/2)

σ2n(f)

= G(t1/2) > 0. (5.1)

To do this, observe first that since t1/2 ∈ B−, we can apply Lemma 4.5 of Babayan et al. [4] toget

lim supn→∞

σ2n(ft

1/2)

σ2n(f)

≤ G(t1/2). (5.2)

So, we have to prove the inverse inequality:

lim infn→∞

σ2n(ft

1/2)

σ2n(f)

≥ G(t1/2) > 0. (5.3)

To do this, we consider the function:

f(λ) := f(λ)t(λ), λ ∈ Λ. (5.4)

Applying Theorem B with h(λ) ≡ t2(λ) ≡ 1, t1(λ) = t(λ) we conclude that t ∈ Mf and f ∈ F , and,in particular,

limn→∞

σ2n(f)

σ2n(f)

= G(t) > 0. (5.5)

Next, taking into account that t−1/2 ∈ B+, we can apply Lemma 4.6 of Babayan et al. [4] to obtain

lim infn→∞

σ2n(f · t−1/2)

σ2n(f)

≥ G(t−1/2) > 0. (5.6)

11

Now we can write

lim infn→∞

σ2n(ft

1/2)

σ2n(f)

= lim infn→∞

σ2n(ft · t

−1/2)

σ2n(f)

= lim infn→∞

σ2n(ft · t

−1/2)

σ2n(ft)

·σ2n(ft)

σ2n(f)

= lim infn→∞

σ2n(f · t−1/2)

σ2n(f)

· lim infn→∞

σ2n(f)

σ2n(f)

≥ G(t−1/2) ·G(t) = G(t1/2). (5.7)

In the last relation, the third equality follows from (5.4), the inequality after that follows from (5.5)and (5.6), and the last equality follows from Proposition 2.3(a). Thus, the inequality (5.3) is proved.

Combining the inequalities (5.2) and (5.3) we obtain the desired equality (5.1), which togetherwith the above obtained facts: G(t1/2) > 0 and ft1/2 ∈ L1(Λ) means that t1/2 ∈ Mf . Lemma 5.1 isproved.

Lemma 5.2. Let f be an arbitrary function from the class F , and let t be a nonnegative trigono-metric polynomial. Then t1/2

m

∈ Mf and ft1/2m

∈ F for any m ∈ N.

Proof. As in the proof of Lemma 5.1, we have only to prove the relation t1/2m

∈ Mf . To this end,similar to Lemma 5.1, we first verify the first two conditions for t1/2

m

to belong to the class Mf ,that is, G(t1/2

m

) > 0 and ft1/2m

∈ L1(Λ). Also, we observe that G(ft1/2m

) = 0, showing that ft1/2m

is the spectral density of a deterministic process.Next, we use induction on m to prove the third condition for t1/2

m

to belong to the class Mf ,that is, the relation:

limn→∞

σ2n(ft

1/2m)

σ2n(f)

= G(t1/2m

) > 0. (5.8)

Observe first that for m = 1 the relation (5.8) coincides with (5.1). Now assuming that it issatisfied for m = ν, that is,

limn→∞

σ2n(ft

1/2ν )

σ2n(f)

= G(t1/2ν

) > 0, (5.9)

we prove that it remains valid for m = ν + 1, that is,

limn→∞

σ2n(ft

1/2(ν+1))

σ2n(f)

= G(t1/2(ν+1)

) > 0. (5.10)

To this end, observe first that arguments similar to those used in the proof of Lemma 5.1 can beapplied to obtain (cf. (5.2)):

lim supn→∞

σ2n(ft

1/2(ν+1))

σ2n(f)

≤ G(t1/2(ν+1)

). (5.11)

The proof of the inverse inequality

lim infn→∞

σ2n(ft

1/2(ν+1))

σ2n(f)

≥ G(t1/2(ν+1)

) (5.12)

is similar to that of the inequality (5.3). Indeed, observe that t−1/2(ν+1)∈ B+, and by the inductive

assumption the functionf(λ) := f(λ)t1/2

ν

(λ), λ ∈ Λ (5.13)

belongs to the class F . Hence, we can apply Lemma 4.6 of Babayan et al. [4] to obtain

lim infn→∞

σ2n(f t

−1/2(ν+1))

σ2n(f)

≥ G(t−1/2(ν+1)) > 0. (5.14)

12

Next, we can write

lim infn→∞

σ2n(ft

1/2(ν+1))

σ2n(f)

= lim infn→∞

σ2n(ft

1/2ν · t−1/2(ν+1))

σ2n(f)

= lim infn→∞

σ2n(ft

1/2ν · t−1/2(ν+1))

σ2n(ft

1/2ν )·σ2n(ft

1/2ν )

σ2n(f)

= lim infn→∞

σ2n(f · t−1/2(ν+1)

)

σ2n(f)

· lim infn→∞

σ2n(ft

1/2ν )

σ2n(f)

≥ G(t−1/2(ν+1)) ·G(t1/2

ν

) = G(t1/2(ν+1)

). (5.15)

In the last relation, the third equality follows from (5.13), the inequality after that follows from(5.14) and inductive assumption (5.9), and the last equality follows from Proposition 2.3(a). Thus,the inequality (5.12) is proved.

Combining the inequalities (5.11) and (5.12) we obtain the desired equality (5.10), which together

with the above obtained facts: G(t1/2(ν+1)

> 0 and ft1/2(ν+1)

∈ L1(Λ), means that t1/2(ν+1)

∈ Mf .Lemma 5.2 is proved.

Lemma 5.3. Let f be an arbitrary function from the class F , and let t be a nonnegative trigono-metric polynomial. Then tk/2

m

∈ Mf and ftk/2m

∈ F for any k,m ∈ N.

Proof. We use induction on k, and observe first that for k = 1, the assertion of the lemma coincideswith that of Lemma 5.2. Now assuming that it is satisfied for k = ν, that is, tν/2

m

∈ Mf andftν/2

m

∈ F , we prove it for k = ν + 1, that is, t(ν+1)/2m ∈ Mf and ft(ν+1)/2m ∈ F .To this end, we set g1 := tν/2

m

and g2 := t1/2m

, and observe that by inductive assumption, wehave g1 ∈ Mf and fg1 ∈ F . Taking into account the last relation, we can apply Lemma 5.2 withfg1 as f , and conclude that g2 ∈ Mfg1 .

Therefore, by Proposition 3.2, we have g1g2 = t(ν+1)/2m ∈ Mf and fg1g2 = ft(ν+1)/2m ∈ F .Lemma 5.3 is proved.

Lemma 5.4. Let f be an arbitrary function from the class F , and let t be a nonnegative trigono-metric polynomial. Then t−k ∈ Mf and ft−k ∈ F for any k ∈ N, provided that ft−k ∈ L1(Λ). Inparticular, we have

limn→∞

σ2n(ft

−k)

σ2n(f)

= G(t−k) > 0. (5.16)

Proof. We use induction on k, and show that the result follows from Theorem B by using Proposition3.2. Observe first that for k = 1 the assertion of the lemma coincides with Theorem B applied toh(λ) ≡ t1(λ) ≡ 1 and t2(λ) = t(λ). Now assuming that it is satisfied for k = ν, that is, t−ν ∈ Mf

and ft−ν ∈ F provided that ft−ν ∈ L1(Λ), we prove it for k = ν + 1, that is, t−(ν+1) ∈ Mf andft−(ν+1) ∈ F provided that ft−(ν+1) ∈ L1(Λ).

To this end, we set g1(λ) := t−ν(λ), g2(λ) = t−1(λ) and observe that by inductive assumption,we have g1 ∈ Mf and fg1 ∈ F . We show that g2 ∈ Mfg1 . Indeed, by Proposition 2.3(c) we haveG(g2) = G(t−1) > 0. Besides, fg1g2 = ft−(ν+1) ∈ L1(Λ) by assumption. The obtained relationsallow to apply Theorem B with fg1 as f and g2 as g, and conclude that g2 ∈ Mfg1 .

Thus, the functions g1 and g2 satisfy Proposition 3.2, and hence g1g2 = t−(ν+1) ∈ Mf andfg1g2 = ft−(ν+1) ∈ F .

Lemma 5.4 is proved.

13

5.2 Proof of theorems

Proof of Theorem 4.1. We first verify the first two conditions for a function g(λ) to belong to theclass Mf , that is, the conditions: G(g) > 0 and fg ∈ L1(Λ). From the condition h ∈ B−

+ it followsthat G(h) > 0, while by Proposition 2.3(c) we have G(|t|α) > 0. Therefore, G(g) = G(h)G(|t|α) > 0.Since both functions h(λ) and |t(λ)|α are bounded, the function g(λ) = h(λ)|t(λ)|α is also bounded,and fg ∈ L1(Λ). Besides, we have G(fg) = G(f)G(g) = 0, showing that fg is the spectral densityof a deterministic process.

Taking into account Proposition 3.1, to complete the proof of the theorem it remains to verifythe relation (3.2). The proof of relation (3.2) we split into four steps.Step 1. We prove the relation (3.2) in the special case where h(λ) ≡ 1 and t(λ) is a nonnegativetrigonometric polynomial satisfying the condition:

t(λ) ≤ 1, λ ∈ Λ. (5.17)

For an arbitrary natural number m by km we denote the integer part of the number 2m · α, that is,km := [2m · α]. Then we have the following inequality:

km ≤ 2m · α < km + 1,

or, equivalentlykm2m

≤ α <km + 1

2m. (5.18)

In view of (5.17) and (5.18) we can write

t(km+1)/2m(λ) < tα(λ) ≤ tkm/2m(λ), λ ∈ Λ, (5.19)

implying thatf(λ)t(km+1)/2m(λ) < f(λ)tα(λ) ≤ f(λ)tkm/2m(λ), λ ∈ Λ. (5.20)

Taking into account that by Proposition 2.2(a) the prediction error σ2n(f) is a non-decreasing func-

tional of f from (5.20), we obtain

σ2n

(

ft(km+1)/2m)

≤ σ2n (ft

α) ≤ σ2n

(

ftkm/2m)

.

Dividing the last inequality by σ2n(f) and passing to the limit as n → ∞, we obtain

limn→∞

σ2n

(

ft(km+1)/2m)

σ2n(f)

≤ lim infn→∞

σ2n (ft

α)

σ2n(f)

≤ lim supn→∞

σ2n (ft

α)

σ2n(f)

≤ limn→∞

σ2n

(

ftkm/2m)

σ2n(f)

. (5.21)

By Lemma 5.3, the first and the last limits in (5.21) are equal to G(

t(km+1)/2m)

and G(

tkm/2m)

,respectively. Hence (5.21) can be written as follows:

G(

t(km+1)/2m)

≤ lim infn→∞

σ2n (ft

α)

σ2n(f)

≤ lim supn→∞

σ2n (ft

α)

σ2n(f)

≤ G(

tkm/2m)

. (5.22)

On the other hand, taking into account that the geometric mean G(f) is a non-decreasing functionalof f (see Proposition 2.3(b)), from (5.19) we obtain

G(

t(km+1)/2m)

≤ G (tα) ≤ G(

tkm/2m)

. (5.23)

14

Thus, to complete the proof in the considered case, it remains to show that for large enough m thequantities G

(

t(km+1)/2m)

and G(

tkm/2m)

are arbitrarily close. To do this, observe that in view of(5.17) and the properties of the geometric mean (see Proposition 2.3(a),(b)), we can write

1 ≤G(

tkm/2m)

G(

t(km+1)/2m) = G

(

tkm/2m−(km+1)/2m)

= G(

t−1/2m)

= G−1/2m (t) . (5.24)

Therefore, in view of the limiting relation limn→∞ c1/n = 1 (c > 0), it follows that

limm→∞

G−1/2m (t) = 1. (5.25)

Finally, passing to the limit in (5.22) as m → ∞ and taking into account the relations (5.23)-(5.25),we obtain the desired relation (3.2).Step 2. Now we prove the relation (3.2) without assuming the condition (5.17), that is, in the casewhere h(λ) ≡ 1 and t(λ) is an arbitrary nonnegative trigonometric polynomial. To this end, wedenote c := maxλ∈Λ t(λ) > 0, and consider the trigonometric polynomial t(λ) := (1/c)t(λ). Observethat the polynomial t(λ) satisfies the condition (5.17), that is, t(λ) ≤ 1 for all λ ∈ Λ. Therefore fort(λ) the relation (3.2) is satisfied, that is, we have

limn→∞

σ2n(f t

α)

σ2n(f)

= G(tα). (5.26)

On the other hand, by Propositions 2.2(c) and 2.3(a), we have

σ2n(f t

α) = c−ασ2n(ft

α) and G(f tα) = c−αG(ftα). (5.27)

From (5.26) and (5.27) we get

1

cαlimn→∞

σ2n(ft

α)

σ2n(f)

= c−αG(tα),

which is equivalent to (3.2).Step 3. Now we prove the relation (3.2) in the case where h(λ) ≡ 1 and t(λ) is an arbitrarytrigonometric polynomial. Denoting t(λ) := t2(λ), we can write

|t(λ)|α =(

t2(λ))α/2

=(

t(λ))α/2

, λ ∈ Λ. (5.28)

Since t(λ) is a nonnegative trigonometric polynomial, according to Step 2, for t(λ) the relation (3.2)is satisfied, that is, we have

limn→∞

σ2n(f t

α/2)

σ2n(f)

= G(tα/2),

and, in view of (5.28), we get

limn→∞

σ2n(f |t|

α)

σ2n(f)

= G(|t|α), (5.29)

and the result follows.Step 4. Finally, we prove the relation (3.2), and thus the theorem, in the general case, that is, whenh(λ) is an arbitrary function from the class B−

+ and t is an arbitrary trigonometric polynomial. ByStep 3, we have g1 := |t|α ∈ Mf and fg1 = f |t|α ∈ F . Also, according to Theorem B with fg1 as f ,the function g2 := h belongs to the class Mfg1 . Thus, the functions g1 and g2 satisfy the conditionsof Proposition 3.2, and hence g := g1g2 = h|t|α ∈ Mf and fg = fh|t|α ∈ F . This completes theproof of Theorem 4.1.

15

Proof of Corollary 4.1. For m = 1 the corollary coincides with Theorem 4.1. The result then followsfrom inductive arguments and Proposition 3.2.

Proof of Theorem 4.2. Observe first that by Proposition 2.3(c) we have G(t−α) > 0. Next, sincek + 1 − α > 0 (recall that k is the integer part of α), from the equality ft−α = ft−(k+1)t(k+1−α),we have ft−α ∈ L1(Λ). Denote g1 := t−(k+1) and g2 := t(k+1−α). It follows from Lemma 5.4 thatthe function g1 satisfies the conditions: g1 ∈ Mf and fg1 ∈ F . Also, according to Theorem 4.1, wehave g2 ∈ Mfg1 . Therefore, by Proposition 3.2, we have g1g2 = t−α ∈ Mf and fg1g2 = ft−α ∈ F .Applying Proposition 3.2 now to the functions t−α and h, we conclude that g := ht−α ∈ Mf andfg ∈ F . Theorem 4.2 is proved.

Proof of Theorem 4.3. We first prove assertion a), that is, that the function h := f/f belongs to theclass B−

+ . To do this we denote c := min1≤j≤k cj , where cj is as in (4.4). Then, in view of (4.4), forε = c/2 and each j = 1, 2, . . . , k, there is a number δj = δj(ε) > 0 such that |h(λ) − cj | < ε for allλ ∈ Oδj (λj), or equivalently

cj − c/2 < h(λ) < cj + c/2 for all λ ∈ Oδj (λj), j = 1, 2, . . . , k. (5.30)

Denote C := max1≤j≤k cj , λ0 = −π, ∆λj = λj − λj−1, and δ := min1≤j≤kδj ,∆λj/3. Then, inview of the inequalities c ≤ cj ≤ C and δj ≤ δ, from (5.30) we obtain

c/2 < h(λ) < 3C/2 for all λ ∈ ∪kj=1Oδ(λj). (5.31)

Thus, the function h(λ) is bounded away from zero and infinity in the set ∪kj=1Oδ(λj). On the other

hand, according to the conditions 1) and 4) of the theorem, positive constants m and M can befound to satisfy

f(λ) ≤ M, f(λ) ≤ M for all λ ∈ Λ. (5.32)

andf(λ) ≥ m, f(λ) ≥ m for all λ ∈ Λ \ ∪k

j=1Oδ(λj). (5.33)

Thereforem/M ≤ h(λ) ≤ M/m for all λ ∈ Λ \ ∪k

j=1Oδ(λj). (5.34)

Hence, denoting m := minc/2,m/M and M := max3C/2,M/m, in view of (5.31) and (5.34)we obtain

m < h(λ) < M for all λ ∈ Λ.

Thus, the function h(λ) ia bounded away from zero and infinity. Therefore, h(λ), being a ratio oftwo functions from the class B−, by Proposition 3.3 c), also belongs to the class B−. Therefore,h(λ) ∈ B−

+ , and hence G(h) > 0.To prove assertion b), assume that the process with spectral density f(λ) is nondeterministic,

that is, G(f) > 0. Then taking into account that f(λ) = f(λ)h(λ), we have G(f) = G(f)G(h) > 0,implying that the process with spectral density f(λ) is also nondeterministic. So, in view of (2.3),we have

limn→∞

σ2n(f) = 2πG(f ) and lim

n→∞σ2n(f) = 2πG(f). (5.35)

From (5.35) and Proposition 2.3(a) we easily obtain the relation (4.5). Besides, from the firstrelation in (5.35) and Proposition 2.4(b), we infer that the sequence σn(f) is weakly varying. Thiscompletes the proof of assertion b) of the theorem.

The assertion c) of the theorem immediately follows from Theorem 4.1. Indeed, if f ∈ F , thenapplying Theorem 4.1 with t(λ) ≡ 1 (that is, g(λ) = h(λ)), we conclude that f = fh is the spectraldensity of a deterministic process with a weakly varying sequence σn(f) of prediction errors andthe relation (4.5) holds. This completes the proof of Theorem 4.3.

16

Proof of Theorem 4.4. Let the polynomial q(λ) in (4.6) be of degree m ≥ 1, and let λi ∈ (−π, π]be the zeros of q(λ) of multiplicities mi ∈ N (i = 1, 2, . . . , k, k ∈ N), respectively. Then by theFundamental Theorem of Algebra we can write

q(λ) = c0(λ− λ1)m1(λ− λ2)

m2 · · · (λ− λk)mk q(λ), λ ∈ Λ,

where q(λ) consists of a product of linear binomial factors with zeros outside (−π, π] and quadratictrinomial factors, which are positive on R, implying that q(λ) ∈ B−

+ .Consider the trigonometric polynomial:

t(λ) = sinm1(λ− λ1) sinm2(λ− λ2) · · · sin

mk(λ− λk), λ ∈ Λ,

and the function

h(λ) :=|q(λ)|α

|t(λ)|α, λ ∈ Λ. (5.36)

From the relation sin(λ − λi) ∼ (λ − λi) as λ → λi it follows that with some positive constants ci,|q(λ)|α ∼ ci|t(λ)|

α as λ → λi (i = 1, 2, . . . , k). Hence the functions |q(λ)|α and |t(λ)|α, with regardto their continuity, satisfy the conditions of Theorem 4.3. Therefore, h(λ) ∈ B−

+ and, according to

Proposition 3.3 d), we have h(λ)h(λ) ∈ B−+ .

Now we can apply Theorem 4.2 to conclude the function g(λ) in (4.6) is the spectral density ofa nondeterministic process and to obtain

limn→∞

σ2n(fg)

σ2n(f)

= limn→∞

σ2n(fh|q|

α)

σ2n(f)

= limn→∞

σ2n(fhh|t|

α)

σ2n(f)

= G(hh|t|α) = G(h|q|α) = G(g) > 0.

Here the first and the fifth relations follow from (4.6), the second and the fourth relations followfrom (5.36), and the third relation follows from Theorem 4.2.

This completes the proof of Theorem 4.4.

6 Examples

In this section we discuss examples demonstrating the obtained results.

Example 6.1. Let the function g(λ) (λ ∈ Λ) be as in (4.2) with h(λ) = 1 and t(λ) = sin(λ − λ0),where λ0 is an arbitrary point from [−π, π], that is, g(λ) = | sin(λ− λ0)|

α, α ∈ R. Then, accordingto Example 4.4 of Babayan et al. [4], for the geometric mean of sin2(λ− λ0) we have

G(sin2(λ− λ0)) =1

4. (6.1)

According to Proposition 2.3(a) and (6.1), for the geometric mean of g(λ), we obtain

G(g) = G(| sin(λ− λ0)|α) = G

(

(

sin2(λ− λ0))α/2

)

= Gα/2(sin2(λ− λ0)) =1

2α, (6.2)

and in view of (3.2), we get

limn→∞

σ2n(fg)

σ2n(f)

= G(g) =1

2α.

Thus, multiplying the spectral density f(λ) by the function g(λ) = | sin(λ − λ0)|α yields a 2α-fold

asymptotic reduction of the prediction error.

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Example 6.2. Let the function g(λ) be as in (4.6) with h(λ) = 1 and q(λ) = λ, that is, g(λ) = |λ|α,α ∈ R. By direct calculation we obtain

lnG(g) =1

2π

∫ π

−πln |λ|α dλ =

α

π

∫ π

0lnλdλ = α ln(π/e).

Therefore

G(g) = (π/e)α ≈ (1.156)α ,

and in view of (3.2), we get

limn→∞

σ2n(fg)

σ2n(f)

= G(g) =(π

e

)α≈ (1.156)α .

Thus, multiplying the spectral density f(λ) by the function g(λ) = |λ|α multiplies the predictionerror asymptotically by (π/e)α ≈ (1.156)α.

It follows from Proposition 2.2(d) that the same asymptotic is true for the prediction error withspectral density g(λ) = |λ− λ0|

α, λ0 ∈ [−π, π].

Example 6.3. We first analyze the Pollaczek-Szego function fa(λ) given by (1.1) (cf. Pollaczek [12]and Szego [16]). We have

fa(λ) =2e2λϕ(λ)e−πϕ(λ)

eπϕ(λ) + e−πϕ(λ)=

2e2λϕ(λ)

e2πϕ(λ) + 1, 0 ≤ λ ≤ π, ϕ(λ) := ϕa(λ) = (a/2) cot λ. (6.3)

Observe that ϕ(λ) → +∞ as λ → 0+, and we have

ϕ(λ) ∼ a/(2λ), e2λϕ(λ) ∼ ea, e2πϕ(λ) + 1 ∼ eaπ/λ as λ → 0+. (6.4)

Taking into account that fa(λ) is an even function, from (6.3) and (6.4) we obtain the followingasymptotic relation for fa(λ) in a vicinity of the point λ = 0.

fa(λ) ∼ 2ea exp −aπ/|λ| as λ → 0. (6.5)

Next, observe that ϕ(λ) → −∞ as λ → π, and we have

ϕ(λ) = −ϕ(π − λ) ∼ (−a/2)(π − λ), 2λϕ(λ) ∼ −aπ/(π − λ), as λ → π. (6.6)

In view of (6.3) and (6.6) we obtain the following asymptotic relation for the function fa(λ) in avicinity of the point λ = π.

fa(λ) ∼ 2e2λϕ(λ) ∼ 2 exp −aπ/(π − λ) as λ → π. (6.7)

Putting together (6.5) and (6.7), and taking into account evenness of fa(λ), we conclude that

fa(λ) ∼

2ea exp −aπ/|λ| as λ → 0,2 exp −aπ/(π − |λ|) as λ → ±π,

(6.8)

Thus, the function fa(λ) is positive everywhere except for points λ = 0,±π, and has a very highorder of contact with zero at these points, so that Szego’s condition (2.5) is violated implying thatG(fa) = 0. Also, observe that fa(λ) is infinitely differentiable at all points of the segment [−π, π]including the points λ = 0,±π, and attains it maximum value of 1 at the points ±π/2. For somespecific values of the parameter a the graph of the function fa(λ) is represented in Figure 1a).

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a) b)

Figure 1: a) Graph of the function fa(λ). b) Graph of the function fa(λ).

For a > 0 and λ ∈ [−π, π], consider the pair of functions f1(λ) and f2(λ) defined by formulas:

f1(λ) := exp −aπ/|λ| , f2(λ) := exp −aπ/(π − |λ|) . (6.9)

Observe that the function f1(λ) is positive everywhere except for point λ = 0 at which it has thesame order of contact with zero as fa(λ), and hence G(f1) = 0. Also, f1(λ) is infinitely differentiableat all points of the segment [−π, π] except for the points λ = ±π, where it attains its maximumvalue equal to e−a. As for the function f2(λ), it is positive everywhere except for points λ = ±π,at which it has the same order of contact with zero as fa(λ), and hence G(f2) = 0. Also, f2(λ)is infinitely differentiable at all points of the segment [−π, π] except for the point λ = 0, where itattains its maximum value equal to e−a. For some specific values of the parameter a the graphs offunctions f1(λ) and f2(λ) are represented in Figure 2.

a) b)

Figure 2: a) Graph of the function f1(λ). b) Graph of the function f2(λ).

Denote by fa(λ) the product of functions f1(λ) and f2(λ) defined in (6.9) and normalized by thefactor e4a:

fa(λ) := e4af1(λ)f2(λ) = e4a exp

−aπ2/(|λ|(π − |λ|))

, (6.10)

and observe that fa(λ) behaves similar to fa(λ). Indeed, the function fa(λ) also is positive every-where except for points λ = 0,±π, it is infinitely differentiable at all points of the segment [−π, π]including the points λ = 0,±π, and attains it maximum value of 1 at the points ±π/2. Also, in view

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of (6.8) and (6.10), at points λ = 0,±π the function fa(λ) has the same order of zeros as fa(λ),and hence G(fa) = 0. Thus, the process X(t) with spectral density fa(λ) is deterministic. For somespecific values of the parameter a the graph of the function fa(λ) is represented in Figure 1b).

The functions fa(λ) and fa(λ) defined by (1.1) and (6.10), respectively, satisfy the conditions ofTheorem 4.3. Therefore, we have (see (4.5))

limn→∞

σ2n(fa)

σ2n(fa)

= G(fa/fa) := C(a) > 0. (6.11)

In view of (1.2) and (6.11) we have

σ2n(fa) ∼ C(a) · n−a as n → ∞. (6.12)

where

C(a) :=Γ2 ((a+ 1)/2)

π22−a· C(a). (6.13)

The values of the constants C(a) and C(a) for some specific values of the parameter a are givenin Table 1.

Table 1. The values of constants C(a) and C(a)

a Γ2((a+1)/2)π22−a C(a) C(a)

0.1 0.223 0.797 0.1780.5 0.169 1.113 0.1881.0 0.159 2.545 0.4061.5 0.185 6.446 1.1932.0 0.250 16.830 4.2143.0 0.637 119.220 76.3793.3 0.902 215.715 194.656

3.4 1.020 263.173 268.3755.0 10.186 6128.990 62429.00010.0 223256 1.104 ·108 2.428 ·1013

Now we compare the prediction errors σ2n(f1) and σ2

n(f2) with σ2n(fa). To this end, observe first

that the function g1(λ) := fa(λ)/f1(λ) has a very high order of contact with zero at points λ = ±π,so that Szego’s condition (2.5) is violated implying that G(g1) = 0. Besides, the function g1(λ) iscontinuous on [−π, π], and hence g1 ∈ B−. Therefore, according to Corollary 4.5 of Babayan et al.[4], we have

σ2n(fa) = o

(

σ2n(f1)

)

as n → ∞. (6.14)

Similar arguments applied to the function f2(λ) yield

σ2n(fa) = o

(

σ2n(f2)

)

as n → ∞. (6.15)

The relations (6.14) and (6.15) show that the rate of convergence to zero of the prediction errorsσ2n(f1) and σ2

n(f2) is less than the one for σ2n(fa), that is, the power rate of convergence n−a (see

(6.12)). Thus, the rate of convergence n−a is due to the joint contribution of all zeros λ = 0,±π ofthe function fa(λ), whereas each of these zeros separately does not guarantee the rate of convergencen−a.

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References

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[3] Babayan, N. M. and Ginovyan, M. S. On hyperbolic decay of prediction error variance for deterministicstationary sequences. J. of Contemp. Math Analysis. 55 (2), 76 – 95, 2020.

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[9] Ginovian M. S., Asymptotic behavior of the prediction error for stationary Random sequences. Journalof Contemporary Math. Anal., 34(1), 14 – 33, 1999.

[10] Grenander, U. and Szego, G.: Toeplitz Forms and Their Applications. University of California Press,Berkeley and Los Angeles, 1958.

[11] Pierson, W. J. JR., Wind Generated Gravity Waves, Advances in Geophysics, 2, 93-178, 1955.

[12] Pollaczek, F. Sur une generalisation des polynomes de Legendre, C. R. Acad. Sci. Paris, 228, 1363-1365,1949.

[13] Rakhmanov E. A., On asymptotic properties of polynomials orthogonal on the circle with weights notsatisfying Szego’s condition. Math. USSR Sb. 58 (1987) 149–167.

[14] Rosenblatt M., Some Purely Deterministic Processes. J. of Math. and Mech., 6(6), 801 – 810, 1957.

[15] Simon, B. Orthogonal Polynomials on the Unit Circle. AMS Colloqium Publications, Vol. 54, Parts 1,2,Providence, Rhode Island, 2005.

[16] Szego, G.: On certain special sets of orthogonal polynomials, Proc. Amer. Math. Soc. 1, 731–737 (1950).

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