# On asymptotic behavior of the prediction error for a class of deterministic
stationary sequences

N. M. Babayan and M. S. Ginovyan  Russian-Armenian University, Yerevan,
Armenia, e-mail<EMAIL_ADDRESS>University, Boston, USA, e-mail:
<EMAIL_ADDRESS>Corresponding author.

###### Abstract

One of the main problem in prediction theory of stationary processes $X(t)$ is
to describe the asymptotic behavior of the best linear mean squared prediction
error in predicting $X(0)$ given $X(t),$ $-n\leq t\leq-1$, as $n$ goes to
infinity. This behavior depends on the regularity (deterministic or non-
deterministic) of the process $X(t)$. 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 high order contact with zero M.
Rosenblatt showed that the prediction error behaves like a power as
$n\to\infty$. In the paper Babayan et al. ’Extensions of Rosenblatt’s results
on the asymptotic behavior 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 form $f=f_{d}g$,
where $f_{d}$ is the spectral density of a deterministic process that has a
very high order contact 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 have arbitrary
power type singularities. Examples illustrate the obtained results.

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

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

##  1 Introduction

Let $X(t),$ $t\in\mathbb{Z}:=\\{0,\pm 1,\ldots\\}$, be a centered discrete-
time second-order stationary process. The process is assumed to have an
absolutely continuous spectrum with spectral density function $f(\lambda),$
$\lambda\in[-\pi,\pi].$ The ’finite’ linear prediction problem is as follows.

Suppose we observe a finite realization of the process $X(t)$:
$\\{X(t),\,\,-n\leq t\leq-1\\},$ $n\in\mathbb{N}:=\\{1,2,\ldots\\}.$ We want
to make an one-step ahead prediction, that is, to predict the unobserved
random variable $X(0)$, using the linear predictor
$Y=\sum_{k=1}^{n}c_{k}X(-k)$.

The coefficients $c_{k}$, $k=1,2,\ldots,n$, are chosen so as to minimize the
mean-squared error: $\rm I\\!E\left|X(0)-Y\right|^{2},$ where $\rm
I\\!E[\cdot]$ stands for the expectation operator. If such minimizing
constants $\hat{c}_{k}:=\hat{c}_{k,n}$ can be found, then the random variable
$\hat{X}_{n}(0):=\sum_{k=1}^{n}\hat{c}_{k}X(-k)$ is called the best linear
one-step ahead predictor of $X(0)$ based on the observed finite past:
$X(-n),\ldots,X(-1)$. The minimum mean-squared error:

$\sigma_{n}^{2}(f):=\rm I\\!E\left|X(0)-\hat{X}_{n}(0)\right|^{2}\geq 0$

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

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 $\sigma_{n}^{2}(f)$ as
$n\to\infty$. This behavior depends on the regularity nature (deterministic or
nondeterministic) of the observed process $X(t)$.

Observe that $\sigma_{n+1}^{2}(f)\leq\sigma_{n}^{2}(f)$ ($n\in\mathbb{N}$),
and hence, the limit of $\sigma_{n}^{2}(f)$ as $n\to\infty$ exists. Denote by
$\sigma^{2}(f):=\sigma_{\infty}^{2}(f)$ the prediction error of $X(0)$ by the
entire infinite past: $\\{X(t)$, $t\leq-1\\}$.

From the prediction point of view it is natural to distinguish the class of
processes for which we have error-free prediction by the entire infinite past,
that is, $\sigma^{2}(f)=0$. Such processes are called deterministic or
singular. Processes for which $\sigma^{2}(f)>0$ are called nondeterministic
(for more about these terms see, e.g., Babayan et al. [5], and Grenander and
Szegő [11], p.176).

Define the relative prediction error
$\delta_{n}(f):=\sigma^{2}_{n}(f)-\sigma^{2}(f),$ and observe that
$\delta_{n}(f)$ is non-negative and tends to zero as $n\to\infty$. But what
about the speed of convergence of $\delta_{n}(f)$ to zero as $n\to\infty$? The
paper deals with this question. Specifically, the prediction problem we are
interested in is to describe the rate of decrease of $\delta_{n}(f)$ to zero
as $n\to\infty,$ 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. Szegő and N. Wiener. It was then considered by many authors for
different classes of nondeterministic processes (see, e.g., the survey papers
Bingham [6] and Ginovyan [10], and references therein).

We focus in this paper on deterministic processes, that is, when
$\sigma^{2}(f)=0$. This case is not only of theoretical interest, but is also
important from the point of view of applications. For example, as pointed out
by Rosenblatt [15] (see also Pierson [12]), situations of this type arise in
Neumann’s theoretical model of storm-generated ocean waves. Such models are
also of interest in meteorology (see Fortus [9]).

Only few works are devoted to the study of the speed of convergence of
$\delta_{n}(f)=\sigma^{2}_{n}(f)$ to zero as $n\to\infty$, that is, the
asymptotic behavior of the prediction error for deterministic processes. One
needs to go back to the classical work of Rosenblatt [15], where the
asymptotic behavior of the prediction error $\sigma^{2}_{n}(f)$ was
investigated in the following two cases:

(a) the spectral density $f(\lambda)$ is continuous and positive on an
interval of the segment $[-\pi,\pi]$ and zero elsewhere,

(b) the spectral density $f(\lambda)$ has a very high order of contact with
zero at points $\lambda=0,\pm\pi$, and is strictly positive otherwise.

For the case (a) above, Rosenblatt [15] proved that the prediction error
$\sigma^{2}_{n}(f)$ decreases to zero exponentially as $n\to\infty$. Later the
problem (a) was studied by Babayan [2, 3] and Babayan et al. [5] (see also
Davisson [8] and Fortus [9]), where some extensions of Rosenblatt’s result
have been obtained.

Concerning the case (b) above, for a specific deterministic process $X(t)$,
Rosenblatt proved in [15] that the prediction error $\sigma^{2}_{n}(f)$
decreases to zero like a power as $n\to\infty$. More precisely, the
deterministic process $X(t)$ considered in Rosenblatt [15] has the spectral
density

$f_{a}(\lambda):=\frac{e^{(2\lambda-\pi)\varphi(\lambda)}}{\cosh\left(\pi\varphi(\lambda)\right)},\quad
f_{a}(-\lambda)=f_{a}(\lambda),\quad 0\leq\lambda\leq\pi,$ (1.1)

where $\varphi(\lambda)=(a/2)\cot\lambda$ and $a$ is a positive parameter.

Using the technique of orthogonal polynomials on the unit circle and Szegő’s
results, Rosenblatt [15] proved the following theorem.

######  Theorem A (Rosenblatt [15]).

Suppose that the process $X(t)$ has spectral density $f_{a}$ given by (1.1).
Then the following asymptotic relation for the prediction error
$\sigma^{2}_{n}(f_{a})$ holds:

$\sigma^{2}_{n}(f_{a})\sim\frac{\Gamma^{2}\left(({a+1)}/2\right)}{\pi
2^{2-a}}\ n^{-a}\quad{\rm as}\quad n\to\infty.$ (1.2)

Note that the function in (1.1) was first considered by Pollaczek [13], and
then by Szegő [17], as a weight-function of a class of orthogonal polynomials
that serve as illustrations for certain ’irregular’ phenomena in the theory of
orthogonal polynomials. For the function $f_{a}$ in (1.1), we have the
following asymptotic relation (for details see Szegő [17] and Example 6.3
below):

$f_{a}(\lambda)\sim\left\\{\begin{array}[]{ll}2e^{a}\exp\left\\{-{a\pi}/{|\lambda|}\right\\}&\mbox{as
$\lambda\to 0$},\\\ 2\exp\left\\{-{a\pi}/{(\pi-|\lambda|)}\right\\}&\mbox{as
$\lambda\to\pm\pi$}.\end{array}\right.$ (1.3)

Thus, the function $f_{a}$ in (1.1) has a very high order of contact with zero
at points $\lambda=0,\pm\pi$, due to which the process with spectral density
$f_{a}$ is deterministic and the prediction error $\sigma^{2}_{n}(f_{a})$ in
(1.2) decreases to zero like a power as $n\to\infty$.

In Babayan et al. [5] was proved that if the spectral density $f$ is such that
the sequence $\\{\sigma_{n}(f)\\}$ is weakly varying (a term defined in
Section 2.3) and if, in addition, $g$ is a nonnegative function that can have
polynomial type singularities, then the sequences $\\{\sigma_{n}(fg)\\}$ and
$\\{\sigma_{n}(f)\\}$ have the same asymptotic behavior as $n\to\infty$ (see
Theorem B in Section 3). Using this result, Rosenblatt’s Theorems A was
extended in Babayan et al. [5] to a class of spectral densities of the form
$f=f_{a}g$, where $f_{a}$ 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, for which the function $g$ can have arbitrary power type
singularities.

Throughout the paper we will use the following notation.
The standard symbols $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{R}$ and $\mathbb{C}$
denote the sets of natural, integer, real and complex numbers, respectively.
Also, we denote $\Lambda:=[-\pi,\pi],$
$\mathbb{T}:=\\{z\in\mathbb{C}:\,|z|=1\\}$. For a point
$\lambda_{0}\in\Lambda$ and a number $\delta>0$ by $O_{\delta}(\lambda_{0})$
we denote a $\delta$-neighborhood of $\lambda_{0}$, that is,
$O_{\delta}(\lambda_{0}):=\\{\lambda\in\Lambda:\
|\lambda-\lambda_{0}|<\delta\\}$. By $L^{p}(\mu):=L^{p}(\mathbb{T},\mu)$
($p\geq$1) we denote the weighted Lebesgue space with respect to the measure
$\mu$, and by $(\cdot,\cdot)_{p,\mu}$ and $||\cdot||_{p,\mu}$ we denote the
inner product and the norm in $L^{p}(\mu)$, respectively. In the special case
where $\mu$ is the Lebesgue measure, we will use the notation $L^{p}$,
$(\cdot,\cdot)_{p}$ and $||\cdot||_{p}$, respectively. For a function $f\geq
0$ by $G(f)$ we denote the geometric mean of $f$. For two functions
$f(\lambda)\geq 0$ and $g(\lambda)\geq 0$, $\lambda\in\Lambda$, we will write
$f(\lambda){\sim}g(\lambda)$ as ${\lambda\to\lambda_{0}}$ if
$\lim_{\lambda\to\lambda_{0}}{f(\lambda}/{g(\lambda)}=1$, and
$f(\lambda){\simeq}g(\lambda)$ as ${\lambda\to\lambda_{0}}$ if
$\lim_{\lambda\to\lambda_{0}}{f(\lambda}/{g(\lambda)}=c>0$. We will use
similar notation for sequences: for two sequences $\\{a_{n}\geq
0,n\in\mathbb{N}\\}$ and $\\{b_{n}>0,n\in\mathbb{N}\\}$, we will write
$a_{n}\sim b_{n}$ if $\lim_{n\to\infty}{a_{n}}/{b_{n}}=1$,
$a_{n}{\simeq}b_{n}$ if $\lim_{n\to\infty}{a_{n}}/{b_{n}}=c>0$;
$a_{n}=O(b_{n})$ if ${a_{n}}/{b_{n}}$ is bounded, and $a_{n}=o(b_{n})$ if
${a_{n}}/{b_{n}}\to 0$ as $n\to\infty$. 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 preliminary results that are used throughout the paper. In Section
3 we state some auxiliary results. In Section 4 we state the main results of
the paper. Section 5 contains the proofs of the main results. In Section 6 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: the Kolmogorov-Szegő Theorem, formulas and some properties
of the finite prediction error, properties of the geometric mean of a
function, and definition and properties of weakly varying sequences.

###  2.1 Kolmogorov-Szegő’s Theorem

Let $X(t)$ be a centered discrete-time stationary process defined on a
probability space $(\Omega,\mathcal{F},P)$ with covariance function $r(t)$,
$t\in\mathbb{Z}$. By the Herglotz theorem (see, e.g., Brockwell and Davis [7],
p. 117-18), there is a finite measure $\mu$ on $\Lambda$ such that the
covariance function $r(t)$ admits the following spectral representation:

$r(t)=\int_{-\pi}^{\pi}e^{-it\lambda}d\mu(\lambda),\quad t\in\mathbb{Z}.$
(2.1)

The measure $\mu$ in (2.1) is called the spectral measure of the process
$X(t)$. If $\mu$ is absolutely continuous (with respect to the Lebesgue
measure), then the function $f(\lambda):=d\mu(\lambda)/d\lambda$ is called the
spectral density of $X(t)$. We assume that $X(t)$ is a non-degenerate process,
that is, ${\rm Var}[X(0)]:=\rm I\\!E|X(0)|^{2}=r(0)>0$ and, without loss of
generality, we may take $r(0)=1$. Also, to avoid the trivial cases, we assume
that the spectral measure $\mu$ is non-trivial, that is, the support of $\mu$
has positive Lebesgue measure.

######  Remark 2.1.

The parametrization of the unit circle $\mathbb{T}$ by the formula
$z=e^{i\lambda}$ establishes a bijection between $\mathbb{T}$ and the interval
$[-\pi,\pi)$. By means of this bijection the measure $\mu$ on $\Lambda$
generates the corresponding measure on the unit circle $\mathbb{T}$, which we
also denote by $\mu$. Thus, depending on the context, the measure $\mu$ will
be supported either on $\Lambda$ or on $\mathbb{T}$. We use the standard
Lebesgue decomposition of the measure $\mu$:

$d\mu(\lambda)=d\mu_{a}(\lambda)+d\mu_{s}(\lambda)=f(\lambda)d\lambda+d\mu_{s}(\lambda),$
(2.2)

where $\mu_{a}$ is the absolutely continuous part of $\mu$ (with respect to
the Lebesgue measure) and $\mu_{s}$ is the singular part of $\mu$, which is
the sum of the discrete and continuous singular components of $\mu$.

The next result describes the asymptotic behavior of the prediction error
$\sigma_{n}^{2}(\mu)$ for a stationary process $X(t)$ with spectral measure
$\mu$ of the form (2.2) and gives a spectral characterization of deterministic
and nondeterministic processes (see, e.g., Grenander and Szegő [11], p. 44).

Kolmogorov-Szegő’s Theorem. Let $X(t)$ be a non-degenerate stationary process
with spectral measure $\mu$ of the form (2.2). The following relations hold.

$\displaystyle\lim_{n\to\infty}\sigma_{n}^{2}(\mu)=\lim_{n\to\infty}\sigma_{n}^{2}(f)=\sigma^{2}(f)=2\pi
G(f),$ (2.3)

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

$G(f):=\left\\{\begin{array}[]{ll}\exp\left\\{\frac{1}{2\pi}\int_{-\pi}^{\pi}\ln
f(\lambda)\,d\lambda\right\\}&\mbox{if \, $\ln f\in{L}^{1}(\Lambda)$}\\\
0,&\mbox{ otherwise.}\end{array}\right.$ (2.4)

It is remarkable that the limit in (2.3) is independent of the singular part
$\mu_{s}$.

The condition $\ln f\in{L}^{1}(\Lambda)$ in (2.4) is equivalent to the Szegő
condition:

$\int_{-\pi}^{\pi}\ln f(\lambda)\,d\lambda>-\infty$ (2.5)

(this equivalence follows because $\ln f(\lambda)\leq f(\lambda)$). The Szegő
condition (2.5) is also called the non-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 $[-\pi,\pi]$ (or on the unit circle $\mathbb{T}$), are
extended to $\mathbb{R}$ with period $2\pi$.

Following Rosenblatt [15], we will say that the spectral density $f(\lambda)$
has a very high order of contact with zero at a point $\lambda_{0}$ if
$f(\lambda)$ is positive everywhere except for the point $\lambda_{0}$, due to
which the Szegő condition (2.5) is violated.

###  2.2 Properties of the finite prediction error and geometric mean

Suppose we have observed the values $X(-n),\ldots,X(-1)$ of a centered, real-
valued stationary process $X(t)$ with spectral measure $\mu$ of the form
(2.2). The one-step ahead linear prediction problem in predicting a random
variable $X(0)$ based on the observed values $X(-n),\ldots,X(-1)$ involves
finding constants $\hat{c}_{k}:=\hat{c}_{k,n}$, $k=1,2,\ldots,n$, that
minimize the one-step ahead prediction error:

$\displaystyle\sigma_{n}^{2}(\mu):=\min_{\\{c_{k}\\}}{\rm
I\\!E}\left|X(0)-\sum_{k=1}^{n}c_{k}X(-k)\right|^{2}={\rm
I\\!E}\left|X(0)-\sum_{k=1}^{n}\hat{c}_{k}X(-k)\right|^{2}.$ (2.6)

Using Kolmogorov’s isometric isomorphism $V:\,X(t)\leftrightarrow
e^{it\lambda}$ (see, e.g., Babayan et al. [5]), in view of (2.6), for the
prediction error $\sigma_{n}^{2}(\mu)$ we can write

$\displaystyle\sigma_{n}^{2}(\mu)=\min_{\\{c_{k}\\}}\left\|1-\sum_{k=1}^{n}c_{k}e^{-ik\lambda}\right\|^{2}_{2,\mu}=\min_{\\{q_{n}\in\mathcal{Q}_{n}\\}}\left\|q_{n}\right\|^{2}_{2,\mu},$
(2.7)

where $||\cdot||_{2,\mu}$ is the norm in $L^{2}(\mathbb{T},\mu)$, and

$\mathcal{Q}_{n}:=\left\\{q_{n}:q_{n}(z)=z^{n}+c_{1}z^{n-1}+\cdots
c_{n}\right\\}$ (2.8)

is the class of monic polynomials (i.e. with $c_{0}=1$) of degree $n$. Thus,
the problem of finding $\sigma_{n}^{2}(\mu)$ becomes to the problem of finding
the solution of the minimum problem (2.7)-(2.8).

The polynomial $p_{n}(z):=p_{n}(z,\mu)$ which solves the minimum problem
(2.7)-(2.8) is called the optimal polynomial for $\mu$ in the class
$\mathcal{Q}_{n}$.

The next result by Szegő solves the minimum problem (2.7)-(2.8) (see, e.g.,
Grenander and Szegő [11], p. 38).

######  Proposition 2.1.

The unique solution of the minimum problem (2.7)-(2.8) is given by
$p_{n}(z)=\kappa^{-1}_{n}\varphi_{n}(z)$, and the minimum in (2.7) is equal to
$||p_{n}||_{2,\mu}=\kappa^{-2}_{n}$, where
$\varphi_{n}(z)=\kappa_{n}z^{n}+\cdots+l_{n}$ is the $n^{th}$ orthogonal
polynomial on the unit circle associated with the measure $\mu$, and
$\kappa_{n}$ is the leading coefficient of $\varphi_{n}(z)$.

Thus, for the prediction error $\sigma_{n}^{2}(\mu)$ we have the following
formula:

$\displaystyle\sigma^{2}_{n}(\mu)=\min_{\\{q_{n}\in\mathcal{Q}_{n}\\}}\left\|q_{n}\right\|^{2}_{2,\mu}=\left\|p_{n}(\mu)\right\|^{2}_{2,\mu}=\left\|\kappa^{-1}_{n}\varphi_{n}(\mu)\right\|^{2}_{2,\mu}=\kappa^{-2}_{n}.$
(2.9)

Denote by $D_{n}=D_{n}(\mu):=\det[r(t-s),\,t,s=0,1,\ldots n]$ the $n^{th}$
Toeplitz determinant generated by the measure $\mu$, where $r(t)$ is the
covariance function given by (2.1). Taking into account that
$\kappa^{2}_{n}={D_{n-1}/D_{n}}$ (see, e.g., Grenander and Szegő [11], p. 38),
in view of (2.9) we obtain the following formula for the prediction error
$\sigma_{n}^{2}(\mu)$:

$\displaystyle\sigma^{2}_{n}(\mu)=\frac{D_{n}(\mu)}{D_{n-1}(\mu)}.$ (2.10)

In what follows we assume that the spectral measure $\mu$ is absolutely
continuous with spectral density $f$, and instead of $\sigma^{2}_{n}(\mu)$,
$p_{n}(\mu)$ and $D_{n}(\mu)$ we use the notation $\sigma^{2}_{n}(f)$,
$p_{n}(f)$ and $D_{n}(f)$, respectively.

In the next proposition we list a number of properties of the prediction error
$\sigma_{n}^{2}(f)$.

######  Proposition 2.2.

The prediction error $\sigma_{n}^{2}(f)$ possesses the following properties.

  * (a)

$\sigma_{n}^{2}(f)$ is a non-decreasing functional of $f$:
$\sigma_{n}^{2}(f_{1})\leq\sigma_{n}^{2}(f_{2})$ when $f_{1}(\lambda)\leq
f_{2}(\lambda)$, $\lambda\in[-\pi,\pi]$.

  * (b)

If $f(\lambda)=g(\lambda)$ almost everywhere on $[-\pi,\pi]$, then
$\sigma_{n}^{2}(f)=\sigma_{n}^{2}(g).$

  * (c)

For any positive constant $c$ we have $\sigma_{n}^{2}(cf)=c\sigma_{n}^{2}(f)$.

  * (d)

If $\bar{f}(\lambda)=f(\lambda-\lambda_{0})$, $\lambda_{0}\in[-\pi,\pi]$, then
$\sigma_{n}^{2}(\bar{f})=\sigma_{n}^{2}(f).$

###### Proof.

To prove assertion (a), observe that by the definition of optimal polynomials
$p_{n}(z,f_{1})$ and $p_{n}(z,f_{2})$, corresponding to spectral densities
$f_{1}$ and $f_{2}$, respectively, and formula (2.9) we have

$\displaystyle\sigma_{n}^{2}(f_{1})=\|p_{n}(f_{1})\|^{2}_{2,f_{1}}\leq\|p_{n}(f_{2})\|^{2}_{2,f_{1}}\leq\|p_{n}(f_{2})\|^{2}_{2,f_{2}}=\sigma_{n}^{2}(f_{2}).$

In the last relation the first inequality follows from the optimality of the
polynomial $p_{n}(z,f_{1})$, while the second inequality follows from
assumption that $f_{1}(\lambda)\leq f_{2}(\lambda)$, $\lambda\in\Lambda$.

We show that assertions (b)–(d) follow from formula (2.10). Indeed, observing
that the elements of the Toeplitz determinant $D_{n}(f)$ 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 determinants
generated by the functions $cf$ and $f$ we have $D_{n}(cf)=c^{n}D_{n}(f)$ (see
Grenander and Szegő [11], p. 64-65). Hence in view of (2.10) we have

$\sigma_{n}^{2}(cf)=\frac{D_{n}(cf)}{D_{n-1}(cf)}=\frac{c^{n}D_{n}(f)}{c^{n-1}D_{n-1}(f)}=c\sigma_{n}^{2}(f).$

Finally, the assertion (d) follows from (2.10) and the fact that
$D_{n}(\bar{f})=D_{n}(f)$ (see Grenander and Szegő [11], p. 64-65). ∎

Recall that for a function $h\geq 0$ 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. [5]).

######  Proposition 2.3.

The following assertions hold.

  * (a)

Let $c>0$, $\alpha\in\mathbb{R}$, $f(\lambda)\geq 0$ and $g(\lambda)\geq 0$.
Then

$G(c)=c,\quad G(fg)=G(f)G(g),\quad G(f^{\alpha})=G^{\alpha}(f).$ (2.11)

  * (b)

$G(f)$ is a non-decreasing functional of $f$: if $0\leq f(\lambda)\leq
g(\lambda)$, then $0\leq G(f)\leq G(g)$. In particular, if $0\leq
f(\lambda)\leq 1$, then $0\leq G(f)\leq 1$.

  * (c)

If $t(\lambda)$ is a nonnegative trigonometric polynomial, then
$G(t^{\alpha})>0$ for $\alpha\in\mathbb{R}$.

###  2.3 Weakly varying sequences

We recall the notion of weakly varying sequences and state some of their
properties (see Babayan et al. [5])., This notion will be used in the
specification of the class of deterministic processes to be considered in this
paper.

######  Definition 2.1.

A sequence of non-zero numbers $\\{a_{n},\,n\in\mathbb{N}\\}$ is said to be
weakly varying if $\lim_{n\to\infty}{a_{n+1}}/{a_{n}}=1.$

In the next proposition we list some simple properties of the weakly varying
sequences, which can easily be verified (see Babayan et al. [5]).

######  Proposition 2.4.

The following assertions hold.

  * (a)

If $\\{a_{n},\,n\in\mathbb{N}\\}$ is a weakly varying sequence, then
$\lim_{n\to\infty}{a_{n+\nu}}/{a_{n}}=1$ for any $\nu\in\mathbb{N}$.

  * (b)

If $\\{a_{n},\,n\in\mathbb{N}\\}$ is a sequence such that $a_{n}\to a\neq 0$
as $n\to\infty$, then $\\{a_{n}\\}$ is a weakly varying sequence.

  * (c)

If $\\{a_{n},\,n\in\mathbb{N}\\}$ and $\\{b_{n},\,n\in\mathbb{N},\\}$ are
weakly varying sequences, then $ca_{n}$ $(c\neq 0),$
$a_{n}^{\alpha}\,(\alpha\in\mathbb{R},a_{n}>0)$, $a_{n}b_{n}$ and
$a_{n}/b_{n}$ also are weakly varying sequences.

  * (d)

If $\\{a_{n},\,n\in\mathbb{N}\\}$ is a weakly varying sequence, and
$\\{b_{n},\,n\in\mathbb{N}\\}$ is a sequence of non-zero numbers such that
$\lim_{n\to\infty}{b_{n}}/{a_{n}}=c\neq 0,$ then
$\\{b_{n},\,n\in\mathbb{N}\\}$ 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. [5]).

In what follows we consider the class of deterministic processes possessing
spectral densities for which the sequence of prediction errors
$\\{\sigma_{n}(f)\\}$ is weakly varying, and denote by $\mathcal{F}$ the class
of the corresponding spectral densities:

$\mathcal{F}:=\left\\{f\in L^{1}(\Lambda):\,\,f\geq
0,\,\,G(f)=0,\,\,\lim_{n\to\infty}\frac{\sigma_{n+1}(f)}{\sigma_{n}(f)}=1\right\\}.$
(3.1)

######  Remark 3.1.

According to Rakhmanov’s theorem (see Rakhmanov [14] and Babayan et al. [5]) a
sufficient condition for $f\in\mathcal{F}$ is that $f>0$ almost everywhere on
$\Lambda$. Thus, the considered class $\mathcal{F}$ includes all the almost
everywhere positive spectral densities. On the other hand, according to
Theorem 3.2 and Remark 3.7 of Babayan et al. [5], the class $\mathcal{F}$ does
not contain spectral densities, which vanish on an entire segment of $\Lambda$
(or on an arc of the unit circle $\mathbb{T}$).

######  Definition 3.1.

Let $\mathcal{F}$ be the class of spectral densities defined by (3.1). For
$f\in\mathcal{F}$ denote by $\mathcal{M}_{f}$ the class of nonnegative
functions $g(\lambda)$ $(\lambda\in\Lambda)$ satisfying the following three
conditions: $G(g)>0$, $fg\in L^{1}(\Lambda)$, and

$\lim_{n\to\infty}\frac{\sigma^{2}_{n}(fg)}{\sigma^{2}_{n}(f)}=G(g),$ (3.2)

that is,

$\mathcal{M}_{f}:=\left\\{g\geq 0,\,\,G(g)>0,\,\,fg\in
L^{1}(\Lambda),\,\,\lim_{n\to\infty}\frac{\sigma^{2}_{n}(fg)}{\sigma^{2}_{n}(f)}=G(g)\right\\}.$
(3.3)

The next proposition shows that the class $\mathcal{F}$ is close under
multiplication by functions from the class $\mathcal{M}_{f}$.

######  Proposition 3.1.

If $f\in\mathcal{F}$ and $g\in\mathcal{M}_{f}$, then $fg\in\mathcal{F}$.

###### Proof.

According to the definition of the class $\mathcal{F}$, we have to show that
$fg\in L^{1}(\Lambda)$, $G(fg)=0$, and

$\lim_{n\to\infty}\frac{\sigma_{n+1}(fg)}{\sigma_{n}(fg)}=1.$ (3.4)

The assertion $fg\in L^{1}(\Lambda)$ follows from the condition
$g\in\mathcal{M}_{f}$, while $G(fg)=0$ follows from the condition
$f\in\mathcal{F}$ and Proposition 2.3(a): $G(fg)=G(f)G(g)=0$. As for the
relation (3.4), we can write

$\displaystyle\lim_{n\to\infty}\frac{\sigma_{n+1}(fg)}{\sigma_{n}(fg)}=\lim_{n\to\infty}\frac{\sigma_{n+1}(fg)}{\sigma_{n+1}(f)}\cdot\frac{\sigma_{n+1}(f)}{\sigma_{n}(f)}\cdot\frac{\sigma_{n}(f)}{\sigma_{n}(fg)}$
$\displaystyle=\lim_{n\to\infty}\frac{\sigma_{n+1}(fg)}{\sigma_{n+1}(f)}\cdot\lim_{n\to\infty}\frac{\sigma_{n+1}(f)}{\sigma_{n}(f)}\cdot\lim_{n\to\infty}\frac{\sigma_{n}(f)}{\sigma_{n}(fg)}=\sqrt{G(g)}\cdot
1/\sqrt{G(g)}=1,$

and the result follows. ∎

The next result shows that the class $\mathcal{M}_{f}$ in a certain sense is
close under multiplication.

######  Proposition 3.2.

Let $f\in\mathcal{F}$. If $g_{1}\in\mathcal{M}_{f}$ and
$g_{2}\in\mathcal{M}_{fg_{1}}$, then $g:=g_{1}g_{2}\in\mathcal{M}_{f}$ and
$fg\in\mathcal{F}$. In particular, if
$g\in\mathcal{M}_{f}\cap\mathcal{M}_{fg}$, then $g^{2}\in\mathcal{M}_{f}$.

###### Proof.

By the definition of the classes $\mathcal{M}_{f}$ and $\mathcal{M}_{fg_{1}}$,
we have $G(g_{1})>0$, $G(g_{2})>0$, and hence, by Proposition 2.3(a),
$G(g)>0$. By Proposition 3.1 we have $fg_{1}\in\mathcal{F}$. Hence, taking
into account that $g_{2}\in\mathcal{M}_{fg_{1}}$ we have $fg_{1}g_{2}\in
L^{1}(\Lambda)$, and

$\lim_{n\to\infty}\frac{\sigma^{2}_{n}(fg_{1}g_{2})}{\sigma^{2}_{n}(fg_{1})}=G(g_{2}).$
(3.5)

Then, we can write

$\displaystyle\lim_{n\to\infty}\frac{\sigma_{n}^{2}(fg)}{\sigma_{n}^{2}(f)}=\lim_{n\to\infty}\frac{\sigma_{n}^{2}(fg_{1}g_{2})}{\sigma_{n}^{2}(f)}=\lim_{n\to\infty}\frac{\sigma_{n}^{2}(fg_{1}g_{2})}{\sigma_{n}^{2}(fg_{1})}\cdot\frac{\sigma_{n}^{2}(fg_{1})}{\sigma_{n}^{2}(f)}$
$\displaystyle=\lim_{n\to\infty}\frac{\sigma_{n}^{2}(fg_{1}g_{2})}{\sigma_{n}^{2}(fg_{1})}\cdot\lim_{n\to\infty}\frac{\sigma_{n}^{2}(fg_{1})}{\sigma_{n}^{2}(f)}=G(g_{1})\cdot
G(g_{2})=G(g).$

In the last relation the fourth equality follows from (3.5) and the condition
$g_{1}\in\mathcal{M}_{f}$, while the fifth equality follows from Proposition
2.3(a). Thus, we have proved that $g\in\mathcal{M}_{f}$, from this and
Proposition 3.1 it follows that $fg\in\mathcal{F}$. ∎

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
on $\Lambda=[-\pi,\pi]$ functions $h(\lambda)$. Also, we define the following
subclasses:

$B_{+}:=\\{h\in B:\,h(\lambda)\geqslant m\\},\quad B^{-}:=\\{h\in
B:\,h(\lambda)\leqslant M\\},\quad B_{+}^{-}:=B_{+}\cap B^{-},$

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 $h\in B_{+}(B^{-})$, then $1/h\in B^{-}(B_{+})$.

  * b)

If $h_{1},h_{2}\in B_{+}(B^{-})$, then $h_{1}+h_{2}\in B_{+}(B^{-})$ and
$h_{1}h_{2}\in B_{+}(B^{-})$.

  * c)

If $h_{1},h_{2}\in B^{-}$ and $h_{1}/h_{2}$ is bounded, then $h_{1}/h_{2}\in
B^{-}$.

  * d)

If $h_{1},h_{2}\in B_{+}^{-}$, then $h_{1}+h_{2}\in B_{+}^{-}$, $h_{1}h_{2}\in
B_{+}^{-}$ and $h_{1}/h_{2}\in B_{+}^{-}$.

The following theorem, proved in Babayan et al. [5], describes the asymptotic
behavior of the ratio $\sigma_{n}^{2}(fg)/\sigma_{n}^{2}(f)$ as $n\to\infty$,
and essentially states that if the spectral density $f$ is from the class
$\mathcal{F}$ (see (3.1)), and $g$ is a nonnegative function, which can have
polynomial type singularities, then the sequences $\\{\sigma_{n}(fg)\\}$ and
$\\{\sigma_{n}(f)\\}$ have the same asymptotic behavior as $n\to\infty$ up to
a positive numerical factor.

######  Theorem B (Babayan et al. [5]).

Let $f$ be an arbitrary function from the class $\mathcal{F}$, and let $g$ be
a function of the form:

$g(\lambda)=h(\lambda)\cdot\frac{t_{1}(\lambda)}{t_{2}(\lambda)},\quad\lambda\in\Lambda,$
(3.6)

where $h\in B_{+}^{-}$, $t_{1}$ and $t_{2}$ are nonnegative trigonometric
polynomials, such that $fg\in L^{1}(\Lambda)$. Then $g\in\mathcal{M}_{f}$ and
$fg\in\mathcal{F}$, that is, $fg$ is the spectral density of a deterministic
process with weakly varying prediction error, and the relation (3.2) holds.

Taking into account that the sequence
$\\{n^{-\alpha},\,\,n\in\mathbb{N},\,\alpha>0\\}$ is weakly varying, as an
immediate consequence of Theorem B, we have the following result.

######  Corollary 3.1 (Babayan et al. [5]).

Let the functions $f$ and $g$ be as in Theorem B, and let $\sigma_{n}(f)\sim
cn^{-\alpha}$ ($c>0,\alpha>0$) as $n\to\infty$. Then

$\sigma_{n}(fg)\sim cG(g)n^{-\alpha}\quad{\rm as}\quad n\to\infty,$

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

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

######  Theorem C (Babayan et al. [5]).

Let $f=f_{a}g$, where $f_{a}$ is defined by (1.1) and $g$ satisfies the
assumptions of Theorem B. Then

$\sigma^{2}_{n}(f)\sim\frac{\Gamma^{2}\left(\frac{a+1}{2}\right)G(g)}{\pi
2^{2-a}}\ n^{-a}\quad{\rm as}\quad n\to\infty,$

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

We thus have the same limiting behavior for $\sigma^{2}_{n}(f)$ 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 and C to a broader class of spectral densities, for which
the function $g$ can have arbitrary power type singularities.

######  Theorem 4.1.

Let $f$ be an arbitrary function from the class $\mathcal{F}$, and let $g$ be
a function of the form:

$g(\lambda)=h(\lambda)\cdot|t(\lambda)|^{\alpha},\quad\alpha>0,\,\,\lambda\in\Lambda,$
(4.1)

where $h\in B_{+}^{-}$ and $t$ is an arbitrary trigonometric polynomial. Then
$g\in\mathcal{M}_{f}$ and $fg\in\mathcal{F}$, that is, $fg$ is the spectral
density of a deterministic process with weakly varying prediction error, and
the relation (3.2) holds.

######  Corollary 4.1.

The conclusion of Theorem 4.1 remains valid if the function $g$ has the
following form:

$g(\lambda)=h(\lambda)\cdot|t_{1}(\lambda)|^{\alpha_{1}}\cdot|t_{2}(\lambda)|^{\alpha_{2}}\cdot\cdots\cdot|t_{m}(\lambda)|^{\alpha_{m}},\quad\lambda\in\Lambda,$

where $h\in B_{+}^{-}$, $t_{1},t_{2},\ldots,t_{m}$ are arbitrary trigonometric
polynomials, $\alpha_{1},\alpha_{2},\ldots\alpha_{m}$ are arbitrary positive
numbers, and $m\in\mathbb{N}$.

######  Theorem 4.2.

Let $f$ be an arbitrary function from the class $\mathcal{F}$, and let $g$ be
a function of the form:

$g(\lambda)=h(\lambda)\cdot
t^{-\alpha}(\lambda),\quad\alpha>0,\,\,\lambda\in\Lambda,$ (4.2)

where $h\in B_{+}^{-}$ and $t$ is a nonnegative trigonometric polynomial. Then
$g\in\mathcal{M}_{f}$ and $fg\in\mathcal{F}$ provided that $ft^{-(k+1)}\in
L^{1}(\Lambda)$, where $k:=[\alpha]$ is the integer part of $\alpha$.

To state the next result we need the following definition.

######  Definition 4.1.

Let $E_{1}$ and $E_{2}$ be two numerical sets such that for any $x\in E_{1}$
and $y\in E_{2}$ we have $x<y$. We say that the sets $E_{1}$ and $E_{2}$ are
separated from each other if $\sup E_{1}<\inf E_{2}.$ Also, we say that a
numerical set $E$ is separated from infinity if it is bounded from above.

######  Theorem 4.3.

Let $f(\lambda)$ and $\hat{f}(\lambda)$ $(\lambda\in\Lambda)$ be spectral
densities of stationary processes satisfying the following conditions:

  * 1)

$f,\hat{f}\in B^{-}$;

  * 2)

the functions $f(\lambda)$ and $\hat{f}(\lambda)$ have $k$ common essential
zeros $\lambda_{1},\lambda_{2},\ldots,\lambda_{k}\in\Lambda$
$(-\pi<\lambda_{1}<\lambda_{2}<\cdots<\lambda_{k}\leq\pi,\,k\in\mathbb{N})$,
that is,

$\lim_{\lambda\to\lambda_{j}}f(\lambda)=\lim_{\lambda\to\lambda_{j}}\hat{f}(\lambda)=0,\quad
j=1,2,\ldots,k;$ (4.3)

  * 3)

the functions $f(\lambda)$ and $\hat{f}(\lambda)$ are infinitesimal of the
same order in a neighborhood of each point $\lambda_{j}$ $(j=1,2,\ldots,k)$,
that is,

$\lim_{\lambda\to\lambda_{j}}\frac{\hat{f}(\lambda)}{f(\lambda)}=c_{j}>0,\quad
j=1,2,\ldots,k;$ (4.4)

  * 4)

the functions $f(\lambda)$ and $\hat{f}(\lambda)$ are bounded away from zero
outside any neighborhood $O_{\delta}(\lambda_{j})$ $(j=1,2,\ldots,k)$, which
is separated from the neighboring zeros $\lambda_{j-1}$ and $\lambda_{j+1}$ of
$\lambda_{j}$, that is, there is a number $m:=m_{\delta}>0$ such that
$f(\lambda)\geq m$ and $\hat{f}(\lambda)\geq m$ for almost all
$\lambda\notin\cup_{j=1}^{k}O_{\delta}(\lambda_{j})$. Then the following
assertions hold:

  * a)

$h(\lambda):=\frac{\hat{f}(\lambda)}{f(\lambda)}\in B_{+}^{-};$

  * b)

the processes with spectral densities $f$ and $\hat{f}$ either both are
deterministic or both are nondeterministic;

  * c)

if one of the functions $f$ and $\hat{f}$ is from the class $\mathcal{F}$,
then so is the other, and the following relation holds:

$\lim_{n\to\infty}\frac{\sigma_{n}^{2}(\hat{f})}{\sigma_{n}^{2}(f)}=G(h)>0.$
(4.5)

######  Remark 4.1.

The conditions of Theorem 4.3 mean that the points $\lambda_{j}$
$(j=1,2,\ldots,k)$ are the only common zeros of functions $f(\lambda)$ and
$\hat{f}(\lambda)$. Besides, in the case of deterministic processes, at least
one of these zeros should be of sufficiently high order. Also, notice that the
conditions 1) and 4) of Theorem 4.3 will be satisfied if the functions
$f(\lambda)$ and $\hat{f}(\lambda)$ are continuous on $\Lambda$.

######  Theorem 4.4.

Let $f$ be an arbitrary function from the class $\mathcal{F}$, and let $g$ be
a function of the form:

$g(\lambda)=h(\lambda)\cdot|q(\lambda)|^{\alpha},\quad\alpha\in\mathbb{R},\,\,\lambda\in\Lambda,$
(4.6)

where $h\in B_{+}^{-}$, $q$ is an arbitrary algebraic polynomial with real
coefficients, and $fg\in L^{1}(\Lambda)$. Then $fg\in\mathcal{F}$ and
$g\in\mathcal{M}_{f}$.

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

######  Corollary 4.2.

Let $f=f_{a}g$, where $f_{a}$ is defined by (1.1), and let $g$ be a function
satisfying the conditions of one of Theorems 4.1, 4.2, 4.4 or Corollary 4.1.
Then

$\delta_{n}(f)=\sigma^{2}_{n}(f)\sim\frac{\Gamma^{2}\left(\frac{a+1}{2}\right)G(g)}{\pi
2^{2-a}}\ n^{-a}\quad{\rm as}\quad n\to\infty,$

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

######  Remark 4.2.

In view of Remark 3.1 it follows that all the above stated results remain true
if the condition $f\in\mathcal{F}$ is replaced by the slightly strong but more
constructive condition: ’the spectral density $f$ is positive ($f>0$) almost
everywhere on $\Lambda$’.

##  5 Proofs

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

###  5.1 Lemmas

######  Lemma 5.1.

Let $f$ be an arbitrary function from the class $\mathcal{F}$, and let $t$ be
a nonnegative trigonometric polynomial. Then $t^{1/2}\in\mathcal{M}_{f}$ and
$ft^{1/2}\in\mathcal{F}$.

###### Proof.

Observe first that, by Proposition 3.1, the second assertion of the lemma
($ft^{1/2}\in\mathcal{F}$) follows from the first assertion
($t^{1/2}\in\mathcal{M}_{f}$). So, to complete the proof of the lemma we have
to prove the relation $t^{1/2}\in\mathcal{M}_{f}$. To this end, we first
verify the first two conditions for $t^{1/2}$ to belong to the class
$\mathcal{M}_{f}$, that is, the conditions: $G(t^{1/2})>0$ and $ft^{1/2}\in
L^{1}(\Lambda)$ (see (3.3)). First, the condition $G(t^{1/2})>0$ follows from
Proposition 2.3(c). Next, observing that the function $t^{1/2}$ is continuous,
and hence is bounded, in view of the condition $f\in\mathcal{F}$, we conclude
that $ft^{1/2}\in L^{1}(\Lambda)$. Also, observe that by Proposition 2.3(a),
we have $G(ft^{1/2})=G(f)G(t^{1/2})=0$, showing that $ft^{1/2}$ is the
spectral density of a deterministic process.

Now we proceed to verify the third condition for $t^{1/2}$ to belong to the
class $\mathcal{M}_{f}$, that is, the relation:

$\displaystyle\lim_{n\to\infty}\frac{\sigma_{n}^{2}(ft^{1/2})}{\sigma_{n}^{2}(f)}=G(t^{1/2})>0.$
(5.1)

To do this, observe first that since $t^{1/2}\in B^{-}$, we can apply Lemma
4.5 of Babayan et al. [5] to get

$\limsup_{n\to\infty}\frac{\sigma_{n}^{2}(ft^{1/2})}{\sigma_{n}^{2}(f)}\leq
G(t^{1/2}).$ (5.2)

So, we have to prove the inverse inequality:

$\liminf_{n\to\infty}\frac{\sigma_{n}^{2}(ft^{1/2})}{\sigma_{n}^{2}(f)}\geq
G(t^{1/2})>0.$ (5.3)

To do this, we consider the function:

$\hat{f}(\lambda):=f(\lambda)t(\lambda),\quad\lambda\in\Lambda.$ (5.4)

Applying Theorem B with $h(\lambda)\equiv t_{2}(\lambda)\equiv 1$,
$t_{1}(\lambda)=t(\lambda)$ we conclude that $t\in\mathcal{M}_{f}$ and
$\hat{f}\in\mathcal{F}$, and, in particular,

$\lim_{n\to\infty}\frac{\sigma^{2}_{n}(\hat{f})}{\sigma^{2}_{n}(f)}=G(t)>0.$
(5.5)

Next, taking into account that $t^{-1/2}\in B_{+}$, we can apply Lemma 4.6 of
Babayan et al. [5] to obtain

$\liminf_{n\to\infty}\frac{\sigma_{n}^{2}(\hat{f}\cdot
t^{-1/2})}{\sigma_{n}^{2}(\hat{f})}\geq G(t^{-1/2})>0.$ (5.6)

Now we can write

$\displaystyle\liminf_{n\to\infty}\frac{\sigma_{n}^{2}(ft^{1/2})}{\sigma_{n}^{2}(f)}=\liminf_{n\to\infty}\frac{\sigma_{n}^{2}(ft\cdot
t^{-1/2})}{\sigma_{n}^{2}(f)}=\liminf_{n\to\infty}\frac{\sigma_{n}^{2}(ft\cdot
t^{-1/2})}{\sigma_{n}^{2}(ft)}\cdot\frac{\sigma_{n}^{2}(ft)}{\sigma_{n}^{2}(f)}$
$\displaystyle=\liminf_{n\to\infty}\frac{\sigma_{n}^{2}(\hat{f}\cdot
t^{-1/2})}{\sigma_{n}^{2}(\hat{f})}\cdot\liminf_{n\to\infty}\frac{\sigma_{n}^{2}(\hat{f})}{\sigma_{n}^{2}(f)}\geq
G(t^{-1/2})\cdot G(t)=G(t^{1/2}).$ (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 together with the above obtained facts: $G(t^{1/2})>0$ and
$ft^{1/2}\in L^{1}(\Lambda)$ means that $t^{1/2}\in\mathcal{M}_{f}$. Lemma 5.1
is proved. ∎

######  Lemma 5.2.

Let $f$ be an arbitrary function from the class $\mathcal{F}$, and let $t$ be
a nonnegative trigonometric polynomial. Then $t^{1/2^{m}}\in\mathcal{M}_{f}$
and $ft^{1/2^{m}}\in\mathcal{F}$ for any $m\in\mathbb{N}$.

###### Proof.

As in the proof of Lemma 5.1, we have only to prove the relation
$t^{1/2^{m}}\in\mathcal{M}_{f}$. To this end, similar to Lemma 5.1, we first
verify the first two conditions for $t^{1/2^{m}}$ to belong to the class
$\mathcal{M}_{f}$, that is, $G(t^{1/2^{m}})>0$ and $ft^{1/2^{m}}\in
L^{1}(\Lambda)$. Also, we observe that $G(ft^{1/2^{m}})=0$, showing that
$ft^{1/2^{m}}$ is the spectral density of a deterministic process.

Next, we use induction on $m$ to prove the third condition for $t^{1/2^{m}}$
to belong to the class $\mathcal{M}_{f}$, that is, the relation:

$\displaystyle\lim_{n\to\infty}\frac{\sigma_{n}^{2}(ft^{1/2^{m}})}{\sigma_{n}^{2}(f)}=G(t^{1/2^{m}})>0.$
(5.8)

Observe first that for $m=1$ the relation (5.8) coincides with (5.1). Now
assuming that it is satisfied for $m=\nu$, that is,

$\lim_{n\to\infty}\frac{\sigma_{n}^{2}(ft^{1/2^{\nu}})}{\sigma_{n}^{2}(f)}=G(t^{1/2^{\nu}})>0,$
(5.9)

we prove that it remains valid for $m=\nu+1$, that is,

$\lim_{n\to\infty}\frac{\sigma_{n}^{2}(ft^{1/2^{(\nu+1)}})}{\sigma_{n}^{2}(f)}=G(t^{1/2^{(\nu+1)}})>0.$
(5.10)

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

$\limsup_{n\to\infty}\frac{\sigma_{n}^{2}(ft^{1/2^{(\nu+1)}})}{\sigma_{n}^{2}(f)}\leq
G(t^{1/2^{(\nu+1)}}).$ (5.11)

The proof of the inverse inequality

$\liminf_{n\to\infty}\frac{\sigma_{n}^{2}(ft^{1/2^{(\nu+1)}})}{\sigma_{n}^{2}(f)}\geq
G(t^{1/2^{(\nu+1)}})$ (5.12)

is similar to that of the inequality (5.3). Indeed, observe that
$t^{-1/2^{(\nu+1)}}\in B_{+}$, and by the inductive assumption the function

$\hat{f}(\lambda):=f(\lambda)t^{1/2^{\nu}}(\lambda),\quad\lambda\in\Lambda$
(5.13)

belongs to the class $\mathcal{F}$. Hence, we can apply Lemma 4.6 of Babayan
et al. [5] to obtain

$\liminf_{n\to\infty}\frac{\sigma_{n}^{2}(\hat{f}t^{-1/2^{(\nu+1)}})}{\sigma_{n}^{2}(\hat{f})}\geq
G(t^{-1/2^{(\nu+1)}})>0.$ (5.14)

Next, we can write

$\displaystyle\liminf_{n\to\infty}\frac{\sigma_{n}^{2}(ft^{1/2^{(\nu+1)}})}{\sigma_{n}^{2}(f)}=\liminf_{n\to\infty}\frac{\sigma_{n}^{2}(ft^{1/2^{\nu}}\cdot
t^{-1/2^{(\nu+1)}})}{\sigma_{n}^{2}(f)}$
$\displaystyle=\liminf_{n\to\infty}\frac{\sigma_{n}^{2}(ft^{1/2^{\nu}}\cdot
t^{-1/2^{(\nu+1)}})}{\sigma_{n}^{2}(ft^{1/2^{\nu}})}\cdot\frac{\sigma_{n}^{2}(ft^{1/2^{\nu}})}{\sigma_{n}^{2}(f)}$
$\displaystyle=\liminf_{n\to\infty}\frac{\sigma_{n}^{2}(\hat{f}\cdot
t^{-1/2^{(\nu+1)}})}{\sigma_{n}^{2}(\hat{f})}\cdot\liminf_{n\to\infty}\frac{\sigma_{n}^{2}(ft^{1/2^{\nu}})}{\sigma_{n}^{2}(f)}$
$\displaystyle\geq G(t^{-1/2^{(\nu+1)}})\cdot
G(t^{1/2^{\nu}})=G(t^{1/2^{(\nu+1)}}).$ (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: $G(t^{1/2^{(\nu+1)}}>0$
and $ft^{1/2^{(\nu+1)}}\in L^{1}(\Lambda)$, means that
$t^{1/2^{(\nu+1)}}\in\mathcal{M}_{f}$. Lemma 5.2 is proved. ∎

######  Lemma 5.3.

Let $f$ be an arbitrary function from the class $\mathcal{F}$, and let $t$ be
a nonnegative trigonometric polynomial. Then $t^{k/2^{m}}\in\mathcal{M}_{f}$
and $ft^{k/2^{m}}\in\mathcal{F}$ for any $k,m\in\mathbb{N}$.

###### Proof.

We use induction on $k$, and observe first that for $k=1$, the assertion of
the lemma coincides with that of Lemma 5.2. Now assuming that it is satisfied
for $k=\nu$, that is, $t^{\nu/2^{m}}\in\mathcal{M}_{f}$ and
$ft^{\nu/2^{m}}\in\mathcal{F}$, we prove it for $k=\nu+1$, that is,
$t^{(\nu+1)/2^{m}}\in\mathcal{M}_{f}$ and $ft^{(\nu+1)/2^{m}}\in\mathcal{F}$.

To this end, we set $g_{1}:=t^{\nu/2^{m}}$ and $g_{2}:=t^{1/2^{m}}$, and
observe that by inductive assumption, we have $g_{1}\in\mathcal{M}_{f}$ and
$fg_{1}\in\mathcal{F}$. Taking into account the last relation, we can apply
Lemma 5.2 with $fg_{1}$ as $f$, and conclude that
$g_{2}\in\mathcal{M}_{fg_{1}}$.

Therefore, by Proposition 3.2, we have
$g_{1}g_{2}=t^{(\nu+1)/2^{m}}\in\mathcal{M}_{f}$ and
$fg_{1}g_{2}=ft^{(\nu+1)/2^{m}}\in\mathcal{F}$. Lemma 5.3 is proved. ∎

######  Lemma 5.4.

Let $f$ be an arbitrary function from the class $\mathcal{F}$, and let $t$ be
a nonnegative trigonometric polynomial. Then $t^{-k}\in\mathcal{M}_{f}$ and
$ft^{-k}\in\mathcal{F}$ for any $k\in\mathbb{N}$, provided that $ft^{-k}\in
L^{1}(\Lambda)$. In particular, we have

$\displaystyle\lim_{n\to\infty}\frac{\sigma_{n}^{2}(ft^{-k})}{\sigma_{n}^{2}(f)}=G(t^{-k})>0.$
(5.16)

###### Proof.

We use induction on $k$, and show that the result follows from Theorem B by
using Proposition 3.2. Observe first that for $k=1$ the assertion of the lemma
coincides with Theorem B applied to $h(\lambda)\equiv t_{1}(\lambda)\equiv 1$
and $t_{2}(\lambda)=t(\lambda)$. Now assuming that it is satisfied for
$k=\nu$, that is, $t^{-\nu}\in\mathcal{M}_{f}$ and $ft^{-\nu}\in\mathcal{F}$
provided that $ft^{-\nu}\in L^{1}(\Lambda)$, we prove it for $k=\nu+1$, that
is, $t^{-(\nu+1)}\in\mathcal{M}_{f}$ and $ft^{-(\nu+1)}\in\mathcal{F}$
provided that $ft^{-(\nu+1)}\in L^{1}(\Lambda)$.

To this end, we set $g_{1}(\lambda):=t^{-\nu}(\lambda)$,
$g_{2}(\lambda)=t^{-1}(\lambda)$ and observe that by inductive assumption, we
have $g_{1}\in\mathcal{M}_{f}$ and $fg_{1}\in\mathcal{F}$. We show that
$g_{2}\in\mathcal{M}_{fg_{1}}$. Indeed, by Proposition 2.3(c) we have
$G(g_{2})=G(t^{-1})>0$. Besides, $fg_{1}g_{2}=ft^{-(\nu+1)}\in L^{1}(\Lambda)$
by assumption. The obtained relations allow to apply Theorem B with $fg_{1}$
as $f$ and $g_{2}$ as $g$, and conclude that $g_{2}\in\mathcal{M}_{fg_{1}}$.

Thus, the functions $g_{1}$ and $g_{2}$ satisfy Proposition 3.2, and hence
$g_{1}g_{2}=t^{-(\nu+1)}\in\mathcal{M}_{f}$ and
$fg_{1}g_{2}=ft^{-(\nu+1)}\in\mathcal{F}$.

Lemma 5.4 is proved. ∎

###  5.2 Proof of theorems

###### Proof of Theorem 4.1.

We first verify the first two conditions for a function $g(\lambda)$ to belong
to the class $\mathcal{M}_{f}$, that is, the conditions: $G(g)>0$ and $fg\in
L^{1}(\Lambda)$. From the condition $h\in B_{+}^{-}$ it follows that $G(h)>0$,
while by Proposition 2.3(c) we have $G(|t|^{\alpha})>0$. Therefore,
$G(g)=G(h)G(|t|^{\alpha})>0$. Since both functions $h(\lambda)$ and
$|t(\lambda)|^{\alpha}$ are bounded, the function
$g(\lambda)=h(\lambda)|t(\lambda)|^{\alpha}$ is also bounded, and $fg\in
L^{1}(\Lambda)$. Besides, we have $G(fg)=G(f)G(g)=0,$ showing that $fg$ is the
spectral density of a deterministic process.

Taking into account Proposition 3.1, to complete the proof of the theorem it
remains to verify the 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
$h(\lambda)\equiv 1$ and $t(\lambda)$ is a nonnegative trigonometric
polynomial satisfying the condition:

$t(\lambda)\leq 1,\quad\lambda\in\Lambda.$ (5.17)

For an arbitrary natural number $m$ by $k_{m}$ we denote the integer part of
the number $2^{m}\cdot\alpha$, that is, $k_{m}:=[2^{m}\cdot\alpha]$. Then we
have the following inequality:

$k_{m}\leq 2^{m}\cdot\alpha<k_{m}+1,$

or, equivalently

$\frac{k_{m}}{2^{m}}\leq\alpha<\frac{k_{m}+1}{2^{m}}.$ (5.18)

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

$t^{(k_{m}+1)/2^{m}}(\lambda)<t^{\alpha}(\lambda)\leq
t^{k_{m}/2^{m}}(\lambda),\quad\lambda\in\Lambda,$ (5.19)

implying that

$f(\lambda)t^{(k_{m}+1)/2^{m}}(\lambda)<f(\lambda)t^{\alpha}(\lambda)\leq
f(\lambda)t^{k_{m}/2^{m}}(\lambda),\quad\lambda\in\Lambda.$ (5.20)

Taking into account that by Proposition 2.2(a) the prediction error
$\sigma_{n}^{2}(f)$ is a non-decreasing functional of $f$ from (5.20), we
obtain

$\sigma_{n}^{2}\left(ft^{(k_{m}+1)/2^{m}}\right)\leq\sigma_{n}^{2}\left(ft^{\alpha}\right)\leq\sigma_{n}^{2}\left(ft^{k_{m}/2^{m}}\right).$

Dividing the last inequality by $\sigma_{n}^{2}(f)$ and passing to the limit
as $n\to\infty$, we obtain

$\lim_{n\to\infty}\frac{\sigma_{n}^{2}\left(ft^{(k_{m}+1)/2^{m}}\right)}{\sigma_{n}^{2}(f)}\leq\liminf_{n\to\infty}\frac{\sigma_{n}^{2}\left(ft^{\alpha}\right)}{\sigma_{n}^{2}(f)}\leq\limsup_{n\to\infty}\frac{\sigma_{n}^{2}\left(ft^{\alpha}\right)}{\sigma_{n}^{2}(f)}\leq\lim_{n\to\infty}\frac{\sigma_{n}^{2}\left(ft^{k_{m}/2^{m}}\right)}{\sigma_{n}^{2}(f)}.$
(5.21)

By Lemma 5.3, the first and the last limits in (5.21) are equal to
$G\left(t^{(k_{m}+1)/2^{m}}\right)$ and $G\left(t^{k_{m}/2^{m}}\right)$,
respectively. Hence (5.21) can be written as follows:

$G\left(t^{(k_{m}+1)/2^{m}}\right)\leq\liminf_{n\to\infty}\frac{\sigma_{n}^{2}\left(ft^{\alpha}\right)}{\sigma_{n}^{2}(f)}\leq\limsup_{n\to\infty}\frac{\sigma_{n}^{2}\left(ft^{\alpha}\right)}{\sigma_{n}^{2}(f)}\leq
G\left(t^{k_{m}/2^{m}}\right).$ (5.22)

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

$G\left(t^{(k_{m}+1)/2^{m}}\right)\leq G\left(t^{\alpha}\right)\leq
G\left(t^{k_{m}/2^{m}}\right).$ (5.23)

Thus, to complete the proof in the considered case, it remains to show that
for large enough $m$ the quantities $G\left(t^{(k_{m}+1)/2^{m}}\right)$ and
$G\left(t^{k_{m}/2^{m}}\right)$ 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

$1\leq\frac{G\left(t^{k_{m}/2^{m}}\right)}{G\left(t^{(k_{m}+1)/2^{m}}\right)}=G\left(t^{k_{m}/2^{m}-(k_{m}+1)/2^{m}}\right)=G\left(t^{-1/2^{m}}\right)=G^{-1/2^{m}}\left(t\right).$
(5.24)

Therefore, in view of the limiting relation $\lim_{n\to\infty}c^{1/n}=1$
($c>0$), it follows that

$\lim_{m\to\infty}G^{-1/2^{m}}\left(t\right)=1.$ (5.25)

Finally, passing to the limit in (5.22) as $m\to\infty$ 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 case where $h(\lambda)\equiv 1$ and $t(\lambda)$ is an
arbitrary nonnegative trigonometric polynomial. To this end, we denote
$c:=\max_{\lambda\in\Lambda}t(\lambda)>0$, and consider the trigonometric
polynomial $\hat{t}(\lambda):=(1/c)t(\lambda)$. Observe that the polynomial
$\hat{t}(\lambda)$ satisfies the condition (5.17), that is,
$\hat{t}(\lambda)\leq 1$ for all $\lambda\in\Lambda$. Therefore for
$\hat{t}(\lambda)$ the relation (3.2) is satisfied, that is, we have

$\lim_{n\to\infty}\frac{\sigma_{n}^{2}(f\hat{t}^{\alpha})}{\sigma_{n}^{2}(f)}=G(\hat{t}^{\alpha}).$
(5.26)

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

$\sigma_{n}^{2}(f{\hat{t}}^{\alpha})=c^{-\alpha}\sigma_{n}^{2}(ft^{\alpha})\quad{\rm
and}\quad G(f\hat{t}^{\alpha})=c^{-\alpha}G(ft^{\alpha}).$ (5.27)

From (5.26) and (5.27) we get

$\frac{1}{c^{\alpha}}\lim_{n\to\infty}\frac{\sigma_{n}^{2}(ft^{\alpha})}{\sigma_{n}^{2}(f)}=c^{-\alpha}G(t^{\alpha}),$

which is equivalent to (3.2).

Step 3. Now we prove the relation (3.2) in the case where $h(\lambda)\equiv 1$
and $t(\lambda)$ is an arbitrary trigonometric polynomial. Denoting
$\check{t}(\lambda):=t^{2}(\lambda)$, we can write

$|t(\lambda)|^{\alpha}=\left(t^{2}(\lambda)\right)^{{}^{\alpha}/2}=\left(\check{t}(\lambda)\right)^{{}^{\alpha}/2},\quad\lambda\in\Lambda.$
(5.28)

Since $\check{t}(\lambda)$ is a nonnegative trigonometric polynomial,
according to Step 2, for $\check{t}(\lambda)$ the relation (3.2) is satisfied,
that is, we have

$\displaystyle\lim_{n\to\infty}\frac{\sigma_{n}^{2}(f\check{t}^{\alpha/2})}{\sigma_{n}^{2}(f)}=G(\check{t}^{\alpha/2}),$

and, in view of (5.28), we get

$\displaystyle\lim_{n\to\infty}\frac{\sigma_{n}^{2}(f|t|^{\alpha})}{\sigma_{n}^{2}(f)}=G(|t|^{\alpha}),$
(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, when $h(\lambda)$ is an arbitrary function from the
class $B_{+}^{-}$ and $t$ is an arbitrary trigonometric polynomial. By Step 3,
we have $g_{1}:=|t|^{\alpha}\in\mathcal{M}_{f}$ and
$fg_{1}=f|t|^{\alpha}\in\mathcal{F}$. Also, according to Theorem B with
$fg_{1}$ as $f$, the function $g_{2}:=h$ belongs to the class
$\mathcal{M}_{fg_{1}}$. Thus, the functions $g_{1}$ and $g_{2}$ satisfy the
conditions of Proposition 3.2, and hence
$g:=g_{1}g_{2}=h|t|^{\alpha}\in\mathcal{M}_{f}$ and
$fg=fh|t|^{\alpha}\in\mathcal{F}$. This completes the proof of Theorem 4.1. ∎

###### Proof of Corollary 4.1.

For $m=1$ the corollary coincides with Theorem 4.1. The result then follows
from inductive arguments and Proposition 3.2.

∎

###### Proof of Theorem 4.2.

Observe first that by Proposition 2.3(c) we have $G(t^{-\alpha})>0$. Next,
since $k+1-\alpha>0$ (recall that $k$ is the integer part of $\alpha$), from
the equality $ft^{-\alpha}=ft^{-(k+1)}t^{(k+1-\alpha)}$, we have
$ft^{-\alpha}\in L^{1}(\Lambda)$. Denote $g_{1}:=t^{-(k+1)}$ and
$g_{2}:=t^{(k+1-\alpha)}$. It follows from Lemma 5.4 that the function $g_{1}$
satisfies the conditions: $g_{1}\in\mathcal{M}_{f}$ and
$fg_{1}\in\mathcal{F}$. Also, according to Theorem 4.1, we have
$g_{2}\in\mathcal{M}_{fg_{1}}$. Therefore, by Proposition 3.2, we have
$g_{1}g_{2}=t^{-\alpha}\in\mathcal{M}_{f}$ and
$fg_{1}g_{2}=ft^{-\alpha}\in\mathcal{F}$. Applying Proposition 3.2 now to the
functions $t^{-\alpha}$ and $h$, we conclude that
$g:=ht^{-\alpha}\in\mathcal{M}_{f}$ and $fg\in\mathcal{F}$. Theorem 4.2 is
proved. ∎

###### Proof of Theorem 4.3.

We first prove assertion a), that is, that the function $h:=\hat{f}/f$ belongs
to the class $B_{+}^{-}$. To do this we denote $c:=\min_{1\leq j\leq k}c_{j}$,
where $c_{j}$ is as in (4.4). Then, in view of (4.4), for $\varepsilon=c/2$
and each $j=1,2,\ldots,k$, there is a number
$\delta_{j}=\delta_{j}(\varepsilon)>0$ such that
$|h(\lambda)-c_{j}|<\varepsilon$ for all $\lambda\in
O_{\delta_{j}}(\lambda_{j})$, or equivalently

$c_{j}-c/2<h(\lambda)<c_{j}+c/2\quad\text{for all}\quad\lambda\in
O_{\delta_{j}}(\lambda_{j}),\quad j=1,2,\ldots,k.$ (5.30)

Denote $C:=\max_{1\leq j\leq k}c_{j}$, $\lambda_{0}=-\pi$,
$\Delta\lambda_{j}=\lambda_{j}-\lambda_{j-1}$, and $\delta:=\min_{1\leq j\leq
k}\\{\delta_{j},\Delta\lambda_{j}/3\\}$. Then, in view of the inequalities
$c\leq c_{j}\leq C$ and $\delta_{j}\leq\delta$, from (5.30) we obtain

$c/2<h(\lambda)<3C/2\quad\text{for
all}\quad\lambda\in\cup_{j=1}^{k}O_{\delta}(\lambda_{j}).$ (5.31)

Thus, the function $h(\lambda)$ is bounded away from zero and infinity in the
set $\cup_{j=1}^{k}O_{\delta}(\lambda_{j})$. On the other hand, according to
the conditions 1) and 4) of the theorem, positive constants $m$ and $M$ can be
found to satisfy

$f(\lambda)\leq M,\,\,\hat{f}(\lambda)\leq M\quad\text{for
all}\quad\lambda\in\Lambda.$ (5.32)

and

$f(\lambda)\geq m,\,\,\hat{f}(\lambda)\geq m\quad\text{for
all}\quad\lambda\in\Lambda\setminus\cup_{j=1}^{k}O_{\delta}(\lambda_{j}).$
(5.33)

Therefore

$m/M\leq h(\lambda)\leq M/m\quad\text{for
all}\quad\lambda\in\Lambda\setminus\cup_{j=1}^{k}O_{\delta}(\lambda_{j}).$
(5.34)

Hence, denoting $\hat{m}:=\min\\{c/2,m/M\\}$ and
$\hat{M}:=\max\\{3C/2,M/m\\}$, in view of (5.31) and (5.34) we obtain

$\hat{m}<h(\lambda)<\hat{M}\quad\text{for all}\quad\lambda\in\Lambda.$

Thus, the function $h(\lambda)$ ia bounded away from zero and infinity.
Therefore, $h(\lambda)$, being a ratio of two functions from the class
$B^{-}$, by Proposition 3.3 c), also belongs to the class $B^{-}$. Therefore,
$h(\lambda)\in B_{+}^{-}$, and hence $G(h)>0$.

To prove assertion b), assume that the process with spectral density
$f(\lambda)$ is nondeterministic, that is, $G(f)>0$. Then taking into account
that $\hat{f}(\lambda)=f(\lambda)h(\lambda)$, we have $G(\hat{f})=G(f)G(h)>0$,
implying that the process with spectral density $\hat{f}(\lambda)$ is also
nondeterministic. So, in view of (2.3), we have

$\displaystyle\lim_{n\to\infty}\sigma_{n}^{2}(\hat{f})=2\pi
G(\hat{f})\quad\text{and}\quad\lim_{n\to\infty}\sigma_{n}^{2}(f)=2\pi G(f).$
(5.35)

From (5.35) and Proposition 2.3(a) we easily obtain the relation (4.5).
Besides, from the first relation in (5.35) and Proposition 2.4(b), we infer
that the sequence $\\{\sigma_{n}(\hat{f})\\}$ is weakly varying. This
completes the proof of assertion b) of the theorem.

The assertion c) of the theorem immediately follows from Theorem 4.1. Indeed,
if $f\in\mathcal{F}$, then applying Theorem 4.1 with $t(\lambda)\equiv 1$
(that is, $g(\lambda)=h(\lambda)$), we conclude that $\hat{f}=fh$ is the
spectral density of a deterministic process with a weakly varying sequence
$\\{\sigma_{n}(\hat{f})\\}$ of prediction errors and the relation (4.5) holds.
This completes the proof of Theorem 4.3. ∎

###### Proof of Theorem 4.4.

Let the polynomial $q(\lambda)$ in (4.6) be of degree $m\geq 1$, and let
$\lambda_{i}\in(-\pi,\pi]$ be the zeros of $q(\lambda)$ of multiplicities
$m_{i}\in\mathbb{N}$ $(i=1,2,\ldots,k,k\in\mathbb{N})$, respectively. Then by
the Fundamental Theorem of Algebra we can write

$q(\lambda)=c_{0}(\lambda-\lambda_{1})^{m_{1}}(\lambda-\lambda_{2})^{m_{2}}\cdots(\lambda-\lambda_{k})^{m_{k}}\hat{q}(\lambda),\quad\lambda\in\Lambda,$

where $\hat{q}(\lambda)$ consists of a product of linear binomial factors with
zeros outside $(-\pi,\pi]$ and quadratic trinomial factors, which are positive
on $\mathbb{R}$, implying that $\hat{q}(\lambda)\in B_{+}^{-}$.

Consider the trigonometric polynomial:

$t(\lambda)=\sin^{m_{1}}(\lambda-\lambda_{1})\sin^{m_{2}}(\lambda-\lambda_{2})\cdots\sin^{m_{k}}(\lambda-\lambda_{k}),\quad\lambda\in\Lambda,$

and the function

$\hat{h}(\lambda):=\frac{|q(\lambda)|^{\alpha}}{|t(\lambda)|^{\alpha}},\quad\lambda\in\Lambda.$
(5.36)

From the relation $\sin(\lambda-\lambda_{i})\sim(\lambda-\lambda_{i})$ as
$\lambda\to\lambda_{i}$ it follows that with some positive constants $c_{i}$,
$|q(\lambda)|^{\alpha}\sim c_{i}|t(\lambda)|^{\alpha}$ as
$\lambda\to\lambda_{i}$ $(i=1,2,\ldots,k)$. Hence the functions
$|q(\lambda)|^{\alpha}$ and $|t(\lambda)|^{\alpha}$, with regard to their
continuity, satisfy the conditions of Theorem 4.3. Therefore,
$\hat{h}(\lambda)\in B_{+}^{-}$ and, according to Proposition 3.3 d), we have
$h(\lambda)\hat{h}(\lambda)\in B_{+}^{-}$.

Now we can apply Theorem 4.2 to conclude the function $g(\lambda)$ in (4.6) is
the spectral density of a nondeterministic process and to obtain

$\displaystyle\lim_{n\to\infty}\frac{\sigma_{n}^{2}(fg)}{\sigma_{n}^{2}(f)}$
$\displaystyle=$
$\displaystyle\lim_{n\to\infty}\frac{\sigma_{n}^{2}(fh|q|^{\alpha})}{\sigma_{n}^{2}(f)}=\lim_{n\to\infty}\frac{\sigma_{n}^{2}(fh\hat{h}|t|^{\alpha})}{\sigma_{n}^{2}(f)}$
$\displaystyle=$ $\displaystyle
G(h\hat{h}|t|^{\alpha})=G(h|q|^{\alpha})=G(g)>0.$

Here the first and the fifth relations follow from (4.6), the second and the
fourth relations follow from (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 $g(\lambda)$ ($\lambda\in\Lambda$) be as in (4.2) with
$h(\lambda)=1$ and $t(\lambda)=\sin(\lambda-\lambda_{0})$, where $\lambda_{0}$
is an arbitrary point from $[-\pi,\pi]$, that is,
$g(\lambda)=|\sin(\lambda-\lambda_{0})|^{\alpha}$, $\alpha\in\mathbb{R}$.
Then, according to Example 4.4 of Babayan et al. [5], for the geometric mean
of $\sin^{2}(\lambda-\lambda_{0})$ we have

$G(\sin^{2}(\lambda-\lambda_{0}))=\frac{1}{4}.$ (6.1)

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

$G(g)=G(|\sin(\lambda-\lambda_{0})|^{\alpha})=G\left(\left(\sin^{2}(\lambda-\lambda_{0})\right)^{\alpha/2}\right)=G^{\alpha/2}(\sin^{2}(\lambda-\lambda_{0}))=\frac{1}{2^{\alpha}},$
(6.2)

and in view of (3.2), we get

$\lim_{n\to\infty}\frac{\sigma^{2}_{n}(fg)}{\sigma^{2}_{n}(f)}=G(g)=\frac{1}{2^{\alpha}}.$

Thus, multiplying the spectral density $f(\lambda)$ by the function
$g(\lambda)=|\sin(\lambda-\lambda_{0})|^{\alpha}$ yields a $2^{\alpha}$-fold
asymptotic reduction of the prediction error.

######  Example 6.2.

Let the function $g(\lambda)$ be as in (4.6) with $h(\lambda)=1$ and
$q(\lambda)=\lambda$, that is, $g(\lambda)=|\lambda|^{\alpha}$,
$\alpha\in\mathbb{R}$. By direct calculation we obtain

$\displaystyle\ln
G(g)=\frac{1}{2\pi}\int_{-\pi}^{\pi}\ln|\lambda|^{\alpha}\,d\lambda=\frac{\alpha}{\pi}\int_{0}^{\pi}\ln\lambda\,d\lambda=\alpha\ln(\pi/e).$

Therefore

$\displaystyle G(g)=\left(\pi/e\right)^{\alpha}\approx(1.156)^{\alpha},$

and in view of (3.2), we get

$\lim_{n\to\infty}\frac{\sigma^{2}_{n}(fg)}{\sigma^{2}_{n}(f)}=G(g)=\left(\frac{\pi}{e}\right)^{\alpha}\approx(1.156)^{\alpha}.$

Thus, multiplying the spectral density $f(\lambda)$ by the function
$g(\lambda)=|\lambda|^{\alpha}$ multiplies the prediction error asymptotically
by $(\pi/e)^{\alpha}\approx(1.156)^{\alpha}$.

It follows from Proposition 2.2(d) that the same asymptotic is true for the
prediction error with spectral density
$\bar{g}(\lambda)=|\lambda-\lambda_{0}|^{\alpha}$, $\lambda_{0}\in[-\pi,\pi]$.

######  Example 6.3.

We first analyze the Pollaczek-Szegő function $f_{a}(\lambda)$ given by (1.1)
(cf. Pollaczek [13] and Szegő [17]). We have

$f_{a}(\lambda)=\frac{2e^{2\lambda\varphi(\lambda)}e^{-\pi\varphi(\lambda)}}{e^{\pi\varphi(\lambda)}+e^{-\pi\varphi(\lambda)}}=\frac{2e^{2\lambda\varphi(\lambda)}}{e^{2\pi\varphi(\lambda)}+1},\quad
0\leq\lambda\leq\pi,\quad\varphi(\lambda):=\varphi_{a}(\lambda)=(a/2)\cot\lambda.$
(6.3)

Observe that $\varphi(\lambda)\to+\infty$ as $\lambda\to 0^{+}$, and we have

$\varphi(\lambda)\sim a/(2\lambda),\quad e^{2\lambda\varphi(\lambda)}\sim
e^{a},\quad e^{2\pi\varphi(\lambda)}+1\sim e^{a\pi/\lambda}\quad{\rm
as}\quad\lambda\to 0^{+}.$ (6.4)

Taking into account that $f_{a}(\lambda)$ is an even function, from (6.3) and
(6.4) we obtain the following asymptotic relation for $f_{a}(\lambda)$ in a
vicinity of the point $\lambda=0$.

$f_{a}(\lambda)\sim 2e^{a}\exp\left\\{-{a\pi}/{|\lambda|}\right\\}\quad{\rm
as}\quad\lambda\to 0.$ (6.5)

Next, observe that $\varphi(\lambda)\to-\infty$ as $\lambda\to\pi$, and we
have

$\varphi(\lambda)=-\varphi(\pi-\lambda)\sim(-a/2)(\pi-\lambda),\quad
2\lambda\varphi(\lambda)\sim-a\pi/(\pi-\lambda),\quad{\rm
as}\quad\lambda\to\pi.$ (6.6)

In view of (6.3) and (6.6) we obtain the following asymptotic relation for the
function $f_{a}(\lambda)$ in a vicinity of the point $\lambda=\pi$.

$f_{a}(\lambda)\sim 2e^{2\lambda\varphi(\lambda)}\sim
2\exp\left\\{-{a\pi}/{(\pi-\lambda)}\right\\}\quad{\rm as}\quad\lambda\to\pi.$
(6.7)

Putting together (6.5) and (6.7), and taking into account evenness of
$f_{a}(\lambda)$, we conclude that

$f_{a}(\lambda)\sim\left\\{\begin{array}[]{ll}2e^{a}\exp\left\\{-{a\pi}/{|\lambda|}\right\\}&\mbox{as
$\lambda\to 0$},\\\ 2\exp\left\\{-{a\pi}/{(\pi-|\lambda|)}\right\\}&\mbox{as
$\lambda\to\pm\pi$},\end{array}\right.$ (6.8)

Thus, the function $f_{a}(\lambda)$ is positive everywhere except for points
$\lambda=0,\pm\pi,$ and has a very high order of contact with zero at these
points, so that Szegő’s condition (2.5) is violated implying that
$G(f_{a})=0$. Also, observe that $f_{a}(\lambda)$ is infinitely differentiable
at all points of the segment $[-\pi,\pi]$ including the points
$\lambda=0,\pm\pi,$ and attains it maximum value of 1 at the points
$\pm\pi/2$. For some specific values of the parameter $a$ the graph of the
function $f_{a}(\lambda)$ is represented in Figure 1a).

Figure 1: a) Graph of the function $f_{a}(\lambda)$. b) Graph of the function
$\hat{f}_{a}(\lambda)$.

For $a>0$ and $\lambda\in[-\pi,\pi]$, consider the pair of functions
$\hat{f}_{1}(\lambda)$ and $\hat{f}_{2}(\lambda)$ defined by formulas:

$\hat{f}_{1}(\lambda):=\exp\left\\{-{a\pi}/{|\lambda|}\right\\},\quad\hat{f}_{2}(\lambda):=\exp\left\\{-{a\pi}/{(\pi-|\lambda|)}\right\\}.$
(6.9)

Observe that the function $\hat{f}_{1}(\lambda)$ is positive everywhere except
for point $\lambda=0$ at which it has the same order of contact with zero as
$f_{a}(\lambda)$, and hence $G(\hat{f}_{1})=0$. Also, $\hat{f}_{1}(\lambda)$
is infinitely differentiable at all points of the segment $[-\pi,\pi]$ except
for the points $\lambda=\pm\pi,$ where it attains its maximum value equal to
$e^{-a}$. As for the function $\hat{f}_{2}(\lambda)$, it is positive
everywhere except for points $\lambda=\pm\pi,$ at which it has the same order
of contact with zero as $f_{a}(\lambda)$, and hence $G(\hat{f}_{2})=0$. Also,
$\hat{f}_{2}(\lambda)$ is infinitely differentiable at all points of the
segment $[-\pi,\pi]$ except for the point $\lambda=0$, where it attains its
maximum value equal to $e^{-a}$. For some specific values of the parameter $a$
the graphs of functions $\hat{f}_{1}(\lambda)$ and $\hat{f}_{2}(\lambda)$ are
represented in Figure 2.

Figure 2: a) Graph of the function $\hat{f}_{1}(\lambda)$. b) Graph of the
function $\hat{f}_{2}(\lambda)$.

Denote by $\hat{f}_{a}(\lambda)$ the product of functions
$\hat{f}_{1}(\lambda)$ and $\hat{f}_{2}(\lambda)$ defined in (6.9) and
normalized by the factor $e^{4a}$:

$\hat{f}_{a}(\lambda):=e^{4a}\hat{f}_{1}(\lambda)\hat{f}_{2}(\lambda)=e^{4a}\exp\left\\{-{a\pi^{2}}/{(|\lambda|(\pi-|\lambda|))}\right\\},$
(6.10)

and observe that $\hat{f}_{a}(\lambda)$ behaves similar to $f_{a}(\lambda)$.
Indeed, the function $\hat{f}_{a}(\lambda)$ also is positive everywhere except
for points $\lambda=0,\pm\pi,$ it is infinitely differentiable at all points
of the segment $[-\pi,\pi]$ including the points $\lambda=0,\pm\pi,$ and
attains it maximum value of 1 at the points $\pm\pi/2$. Also, in view of (6.8)
and (6.10), at points $\lambda=0,\pm\pi$ the function $\hat{f}_{a}(\lambda)$
has the same order of zeros as $f_{a}(\lambda)$, and hence $G(\hat{f}_{a})=0$.
Thus, the process $X(t)$ with spectral density $\hat{f}_{a}(\lambda)$ is
deterministic. For some specific values of the parameter $a$ the graph of the
function $\hat{f}_{a}(\lambda)$ is represented in Figure 1b).

The functions $f_{a}(\lambda)$ and $\hat{f}_{a}(\lambda)$ defined by (1.1) and
(6.10), respectively, satisfy the conditions of Theorem 4.3. Therefore, we
have (see (4.5))

$\lim_{n\to\infty}\frac{\sigma_{n}^{2}(\hat{f}_{a})}{\sigma_{n}^{2}(f_{a})}=G(\hat{f}_{a}/f_{a}):=\hat{C}(a)>0.$
(6.11)

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

$\sigma^{2}_{n}(\hat{f}_{a})\sim C(a)\cdot n^{-a}\quad{\rm as}\quad
n\to\infty.$ (6.12)

where

$C(a):=\frac{\Gamma^{2}\left(({a+1)}/2\right)}{\pi 2^{2-a}}\cdot\hat{C}(a).$
(6.13)

The values of the constants $\hat{C}(a)$ and $C(a)$ for some specific values
of the parameter $a$ are given in Table 1.

Table 1. The values of constants $\hat{C}(a)$ and $C(a)$

$a$ | $\frac{\Gamma^{2}\left((a+1)/2\right)}{\pi 2^{2-a}}$ | $\hat{C}(a)$ | $C(a)$
---|---|---|---
0.1 | 0.223 | 0.797 | 0.178
0.5 | 0.169 | 1.113 | 0.188
1.0 | 0.159 | 2.545 | 0.406
1.5 | 0.185 | 6.446 | 1.193
2.0 | 0.250 | 16.830 | 4.214
3.0 | 0.637 | 119.220 | 76.379
3.3 | 0.902 | 215.715 | 194.656
3.4 | 1.020 | 263.173 | 268.375
5.0 | 10.186 | 6128.990 | 62429.000
10.0 | 223256 | 1.104 $\cdot 10^{8}$ | 2.428 $\cdot 10^{13}$

Now we compare the prediction errors $\sigma^{2}_{n}(\hat{f}_{1})$ and
$\sigma^{2}_{n}(\hat{f}_{2})$ with $\sigma^{2}_{n}(f_{a})$. To this end,
observe first that the function
$g_{1}(\lambda):=f_{a}(\lambda)/\hat{f}_{1}(\lambda)$ has a very high order of
contact with zero at points $\lambda=\pm\pi,$ so that Szegő’s condition (2.5)
is violated implying that $G(g_{1})=0$. Besides, the function $g_{1}(\lambda)$
is continuous on $[-\pi,\pi]$, and hence $g_{1}\in B^{-}$. Therefore,
according to Corollary 4.5 of Babayan et al. [5], we have

$\sigma^{2}_{n}(f_{a})=o\left(\sigma^{2}_{n}(\hat{f}_{1})\right)\quad{\rm
as}\quad n\to\infty.$ (6.14)

Similar arguments applied to the function $f_{2}(\lambda)$ yield

$\sigma^{2}_{n}(f_{a})=o\left(\sigma^{2}_{n}(\hat{f}_{2})\right)\quad{\rm
as}\quad n\to\infty.$ (6.15)

The relations (6.14) and (6.15) show that the rate of convergence to zero of
the prediction errors $\sigma^{2}_{n}(\hat{f}_{1})$ and
$\sigma^{2}_{n}(\hat{f}_{2})$ is less than the one for
$\sigma^{2}_{n}(f_{a})$, 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 $\lambda=0,\pm\pi$ of the function $f_{a}(\lambda)$,
whereas each of these zeros separately does not guarantee the rate of
convergence $n^{-a}$.

## References

  * [1]
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# SuperCLUE-Math6: Graded Multi-Step Math Reasoning Benchmark for LLMs in
Chinese

Liang Xu, Hang Xue, Lei Zhu, Kangkang Zhao

###### Abstract

We introduce SuperCLUE-Math6(SC-Math6), a new benchmark dataset to evaluate
the mathematical reasoning abilities of Chinese language models. SC-Math6 is
designed as an upgraded Chinese version of the GSM8K dataset with enhanced
difficulty, diversity, and application scope. It consists of over 2000
mathematical word problems requiring multi-step reasoning and providing
natural language solutions. We propose an innovative scheme to quantify the
reasoning capability of large models based on performance over problems with
different reasoning steps. Experiments on 13 representative Chinese models
demonstrate a clear stratification of reasoning levels, with top models like
GPT-4 showing superior performance. SC-Math6 fills the gap in Chinese
mathematical reasoning benchmarks and provides a comprehensive testbed to
advance the intelligence of Chinese language models. 111Our benchmark can be
found at https://www.CLUEbenchmarks.com/superclue˙math6.html.

##  1 Introduction

Recent advances in large language models like GPT-4 [1] have sparked great
interest in evaluating their proficiency in solving reasoning problems. While
benchmarks like GSM8K [2] have been influential, they are limited to English
and do not sufficiently test multi-step inference. To overcome these
limitations and systematically assess the mathematical reasoning of Chinese
models, we introduce SC-Math6 as an upgraded Chinese version of GSM8K.

SC-Math6 has 1072 unique problems covering a diverse range of grade school
math topics. Each problem is presented in a native Chinese context and
accompanied by a detailed natural language solution walkthrough. Moreover, SC-
Math6 provides a follow-up question for each initial query to assess the
model’s continuous reasoning ability during interaction with users (As shown
in Figure 1. For more examples, please refer to the appendix A). We also
propose a novel scoring scheme that combines performance over problems with
different reasoning steps and overall accuracy to produce interpretable and
fair reasoning levels from 1 to 5. Disparities and Correlations of SC-Math6
and GSM8K is presented in Table 1

Figure 1: An example of a problem in SC-Math6

Comparison Item SC-Math6 GSM8K Mathematical Logic Reasoning YES YES Natural
Language Solutions YES YES Elementary Mathematical Knowledge YES YES Multi-
step Reasoning YES YES Native Chinese Context YES NO Multi-round In-depth
Reasoning YES NO Reasoning Steps in Problems YES NO Interpretable Reasoning
Level for LLMs YES NO Number of Test Questions 2144 (1072 Pairs) 1300

Table 1: SC-Math6 and GSM8K: Disparities and Correlations

Our experiments on 13 major Chinese models demonstrate a clear stratification
of reasoning capabilities. Advanced models like GPT-4 exhibit remarkably high
accuracy on multi-step problems, while lower-level models show large
performance gaps. The diversified grading scheme provides a reference for
model selection and evaluation. SC-Math6 thus contributes the first
comprehensive Chinese benchmark to assess and improve the mathematical
reasoning abilities of Chinese language models.

Figure 2: Trend analysis between GSM8k and SC-Math6. SC-Math6 aligns with
GSM8K yet demands in-depth reasoning, whereas some models may struggle with
SC-Math6.

This work pioneers the systematic evaluation and benchmarking of mathematical
reasoning capabilities of major Chinese language models. The key contributions
are three-fold:

First, the construction of SuperCLUE-Math6, the first native Chinese multi-
turn, multi-step mathematical reasoning dataset for assessing model logical
thinking and reasoning skills.

Second, the proposal of a novel transparent and consistent framework to parse
and evaluate model reasoning levels, providing quantifiable metrics of model
intellectual capabilities.

Third, comprehensive benchmarking and analysis of leading Chinese models on
SuperCLUE-Math6, offering valuable insights into current model strengths,
weaknesses and factors impacting reasoning performance.

Overall, this research fills the gap in Chinese mathematical reasoning
evaluation and establishes an important benchmark for advancing the reasoning
abilities of Chinese language models. The benchmark and insights lay a solid
foundation for developing models with more human-like reasoning.

##  2 SuperCLUE-Math6

### Data Collection

We first curated a large pool of Chinese math problems from elementary school
exams and books and altered it manually to ensure it was unique. Two criteria
were then applied to select problems requiring: 1) At least one step of
reasoning, and 2) Error-free natural language solutions. This yielded 1072
unique arithmetic problems.

To evaluate the model’s proficiency in sustained inferential reasoning
throughout the interactive engagement, we designed multi-turn follow-up
questions for each problem, bringing the total size to 2144.

Quality Control and Inspection of Questions. In the second round process, all
questions were subjected to manual verification, which required the annotators
to solve the questions themselves and record their answers. These were then
compared with the provided reference answers and solution steps. If any
inconsistencies were discovered, the problem was pinpointed and corrected.
Corrections were made if there were issues with the answer or solution steps;
if the question itself was ambiguous, it required clarification. Once
consistency was confirmed, the process advanced to the next question. After
the manual verification, a final round of random sampling checks was
conducted. Out of 50 pairs of questions, 1 pair was found to potentially have
ambiguities and needed to be corrected, giving us a sampling accuracy of 98%.

The distribution of reasoning steps is controlled to prevent biases and test
varied capabilities: 15-20% with 1 step, 15-20% with 2 steps, 45-50% with 3
steps, and 5-10% with 4-5 steps. The textual lengths of problems and solutions
also exhibit high variability.

### Scoring Scheme and Reasoning Level

To produce interpretable and fair quantification of reasoning capabilities, we
propose a scoring scheme that combines:

  * •

Reasoning Steps Score: Higher weight assigned for more steps based on the
insight that longer reasoning chains are more difficult. Firstly, we
separately compute the average score of the model corresponding to each
inference step across the set of problems. Subsequently, we employ the number
of inference steps as weights to calculate a weighted average of the scores at
each step, thereby deriving a score that is weighted according to the number
of inference steps.

  * •

Overall Accuracy Score: The Overall Average Score is derived as the mean value
of the Mean Accuracy and the Strict Interaction Accuracy. Mean Accuracy is
computed by considering each question and its corresponding follow-up question
as two separate items, thus calculating the average accuracy across 2144
questions. Conversely, Strict Interaction Accuracy is calculated by treating
the question and its follow-up question as a unified interactive pair, with a
point awarded only if both the question and the follow-up question are
correctly answered, demonstrating proper reasoning. This Strict Interaction
Accuracy is evaluated over a cohort of 1072 test pairs to establish the
average interaction accuracy.

  * •

Comprehensive Score: Calculated as the weighted sum of the Reasoning Steps
Score and the Overall Accuracy Score, each component contributing equally to
the final score.

  * •

Reasoning Level: The Reasoning Level of a language model is based on the
Comprehensive Score, with levels ranging from 1 to 5, where level 5 is the
highest and level 1 is the lowest. A threshold of 5 points is used to
determine the levels. If the composite scores of the two models differ by less
than 5 points, they are considered to be within the same level. This provides
a transparent system to classify model capabilities.

The accuracy score is a commonly used evaluation metric to quantify the
mathematical reasoning abilities of language models, but it fails to account
for the varying difficulty levels of individual questions. Solving a more
challenging problem should be awarded more points than solving an easier one,
and the number of reasoning steps involved usually correlates with the
difficulty level. Therefore, we incorporate the Reasoning Steps Score into our
assessment framework.

The advantage of the Reasoning Step Score lies in its ability to account for
the varying difficulty levels of different questions. This score is validated
by manual problem-solving, ensuring a high level of accuracy in the reasoning
steps involved. Given that mathematical problems may have multiple solution
methods, each with a potentially different number of reasoning steps, the
reasoning step count is not necessarily unique. Therefore, the calculation of
Reasoning Step Scores cannot eliminate the discrepancies in weighted
precision.

On the other hand, the Overall Accuracy Score, while not considering the
difficulty level of each question, ensures fairness and avoids bias that might
be introduced during the weighting process. Therefore, we have not completely
discarded the Overall Accuracy Score. Instead, we employ a weighted summation
of Reasoning Steps Score and Overall Accuracy Score to calculate the unified
score. This method balances the precision given by the Reasoning Steps Score
with the fairness ensured by the Overall Accuracy Score, aiming to provide a
more comprehensive evaluation of mathematical reasoning performance.

### Experiments and Analysis

We evaluated 13 major Chinese models on SC-Math6 covering capacities from 13B
to Proprietary APIs. Table 2 presents the overall accuracy, Reasoning Steps
Score, Comprehensive Score, and resulting Reasoning Level. Information on
models is shown in Table 3.

Model Name R Level Comp. Score Reas. Steps Score OvrAcc Score
GPT-4-1106-Preview 5 90.71 91.65 89.77 GPT-4 5 88.40 89.10 87.71 Ernie-bot 4.0
5 85.60 86.82 84.38 GLM-4 5 84.24 85.72 82.77 Xinghuo 3.5 5 83.73 85.37 82.09
ChatGLM-Turbo 4 57.70 60.32 55.09 GPT-3.5-Turbo 4 57.05 59.61 54.50
Qwen-14B-Chat 4 53.12 55.99 50.26 ChatGLM3-6B 3 40.90 44.20 37.60 Xinghuo 3.0
3 40.08 45.27 34.89 Baichuan2-13B-Chat 3 39.40 42.63 36.18 Ernie-3.5-turbo 2
25.19 27.70 22.67 Chinese-Alpaca2-13B 2 20.55 22.52 18.58

Table 2: SC-Math6 Model Reasoning Level. ’R Level’ for Reasoning Level, ’Comp.
Score’ stands for Comprehensive Score, ’Reas. Steps Score’ stands for
Reasoning Steps Score, ’OvrAcc Score’ stands for Overall Accuracy Score.

Model Name Organization Access GPT-4-1106-Preview OpenAI API GPT-4 OpenAI API
Ernie-bot 4.0 Baidu API GLM-4 ZhiPu Web Page Xinghuo 3.5 Iflytek API ChatGLM-
Turbo ZhiPu API GPT-3.5-Turbo OpenAI API Qwen-14B-Chat Alibaba API ChatGLM3-6B
ZhiPu Weight Xinghuo 3.0 Iflytek API Baichuan2-13B-Chat Baichuan Weight
Ernie-3.5-turbo Baidu Weight Chinese-Alpaca2-13B Yiming Cui Weight

Table 3: Model information

Top models like GPT-4 exhibit remarkably high performance on multi-step
problems, demonstrating advanced reasoning skills. There is also a clear
stratification of capabilities, with higher-scoring models significantly
outperforming lower ones. The diverse levels allow the selection of
appropriate models based on application requirements.

We highlight several observations:

1) Comparison between GSM8k Score and Comprehensive Score in SC-Math6

The comparative analysis of performance on the GSM8k and SC-Math6 benchmark.
SC-Math6(Average model score is 55.34) aligns with GSM8K(Average model score
is 59.21) yet demands more advanced reasoning, whereas models excelling on
GSM8K may struggle on SC-Math6. It suggests that the SC-Math6 benchmark
presents a greater level of difficulty. It was observed that, across the
board, models tend to score lower on the SC-Math6 benchmark compared to GSM8k,
with this trend being particularly pronounced for the ChatGLM3-6B and
Baichuan2-13B-Chat models(As shown in Figure 2).

2) Performance Declining during Multi-turn Interaction

In all models observed, the accuracy rate of the second iteration generally
falls below that of the first, indicating a decline in model performance with
increasing task complexity from the first to the second iteration(As shown in
Table 4). This trend is ubiquitous across all models, suggesting that special
attention should be given to the stability and adaptability of models in
sustained tasks during their design and optimization processes.

Model Name Accuracy of Turn 1 Accuracy of Turn 2 Difference GPT-4-1106-Preview
95.43 89.37 -6.06 GPT-4 94.12 87.13 -6.99 Ernie-bot 4.0 91.98 83.96 -8.02
GLM-4 90.39 83.02 -7.37 Xinghuo 3.5 91.70 81.44 -10.26 ChatGLM-Turbo 73.69
52.80 -20.89 GPT-3.5-Turbo 70.99 53.59 -17.40 Qwen-14B-Chat 73.23 46.26 -26.97
ChatGLM3-6B 61.10 35.35 -25.75 Xinghuo 3.0 70.99 25.65 -45.34
Baichuan2-13B-Chat 58.86 33.77 -25.09 Chinese-Alpaca2-13B 35.63 16.62 -19.01
Ernie-3.5-turbo 43.00 20.43 -22.57

Table 4: SC-Math6 Accuracy during interaction

For instance, the GPT-4-1106-Preview model exhibited a first iteration
accuracy rate of 95.43%, which decreased to 89.37% in the second iteration,
marking an 6.06% reduction in accuracy. Similarly, the ERNIE_35_Turbo model’s
accuracy rate declined from 43.00% in the first iteration to 20.43% in the
second, constituting a 22.57% decrease.

3) Correlation between Instruction Compliance Ratio and Comprehensive Score

Models with high compliance to the instructed output formats also tend to
achieve higher Comprehensive Scores, suggesting instruction understanding as
an important indicator(As shown in Figure 3).

Figure 3: Correlation between instruction compliance ratio and Comprehensive
Score

4) The Potential Relationship between Mathematical Reasoning Proficiency and
Response Length

It has been observed that models yielding longer average response lengths tend
to receive higher evaluation scores(As shown in Figure 4). In certain models,
such as GPT-4-1106-Preview, a higher accuracy rate is accompanied by a longer
average response length, which may suggest that these models are more precise
when generating comprehensive responses. However, this trend is not
consistently observed across all models, indicating that the relationship
between response length and accuracy rate may be influenced by a multitude of
factors, including the design of the model and the training data employed.

Figure 4: Relation between response length and Comprehensive Score

The GPT-4-1106-Preview exhibits an average response length of 179.78,
correlating with a higher accuracy rate, while the ChatGLM3-6B shows a
comparatively shorter average response length of 66.66, with a corresponding
lower accuracy rate. This implies that, in certain instances, there may be a
correlation between response length and accuracy rate.

5) Performance Declining with Reasoning Steps Increasing

Figure 5: Relation between reasoning steps and Mean Accuracy

Analysis of the step-by-step scores reveals declining performance as problem
complexity increases from 1 to 5 reasoning steps(As shown in Figure 5). This
highlights the need to improve the model’s capability to tackle more
challenging problems requiring more reasoning steps. The results provide
comprehensive insights to guide further progress on mathematical and general
reasoning for Chinese language models.

##  3 Related Work

Benchmarks to evaluate the reasoning skills of language models have gained
increasing research attention. Existing datasets mostly focus on English,
including GSM8K for mathematical reasoning [2], MATH for complexity
mathematical problem [3]. For general LLMs benchmark, we can find MT-bench
[4], AlpacaEval [5]. We can find reasoning benchmarks for NLP, such as
WinoGrad Schema Challenge for commonsense reasoning [6], and ARC for
scientific question answering [7]. Our work aims to close the gap for the
Chinese through a systematically designed mathematical reasoning benchmark.
Our focus is to provide a benchmark to evaluate the general reasoning skills
of Chinese language models. The diverse problems and reasoning patterns in SC-
Math6 complement these methods to inspire new model designs and training
strategies targeting enhanced mathematical intelligence.

##  4 Conclusion

We present SC-Math6 as the first native Chinese benchmark dataset for
assessing the multi-step mathematical reasoning skills of language models in a
multi-turn interaction. Developing human-like intelligence requires rich,
diverse datasets like SC-Math6 that test sophisticated capabilities beyond
pattern recognition. Our work aims to catalyze advances in Chinese models to
better support real-world applications. The human-annotated natural language
solutions also provide valuable data for training. We hope SC-Math6 will
inspire exciting new model designs and training strategies targeting enhanced
reasoning.

## References

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##  Appendix A Examples for problem in SC-Math6

Below are examples of problems in SuperCLUE-Math6.

Figure 6: An example of a problem in SC-Math6 Figure 7: An example of a
problem in SC-Math6