Average case tractability of additive random fields with Korobov kernels

TL;DR

This paper studies the average case complexity of approximating additive random fields with Korobov kernels, establishing polynomial tractability and conditions for strong polynomial tractability under different error criteria.

Contribution

It provides the first comprehensive analysis of tractability for these fields, including necessary and sufficient conditions for strong polynomial tractability.

Findings

Problem is always polynomially tractable under ABS and NOR

Derived necessary and sufficient conditions for strong polynomial tractability

Analyzed non-homogeneous cases with Korobov kernels

Abstract

We investigate average case tractability of approximation of additive random fields with marginal random processes corresponding to the Korobov kernels for the non-homogeneous case. We use the absolute error criterion (ABS) or the normalized error criterion (NOR). We show that the problem is always polynomially tractable for ABS or NOR, and give sufficient and necessary conditions for strong polynomial tractability for ABS or NOR.

Equations

$$K^{X_j}f(t)={\int }_{[0,1]}{\mathcal{K}}^{X_j}(t,s)f(s)ds.$$

$$Y_d(\mathbf{t}):=\sum _{j=1}^d{X}_j(t_j),\mathbf{t}=(t_1,\cdots ,t_d)\in [0,1{]}^d,$$

$${\mathcal{K}}^{Y_d}(\mathbf{t},\mathbf{s})=\sum _{j=1}^d{\mathcal{K}}^{X_j}(t_j,s_j),$$

$$K^{Y_d}f(\mathbf{t})={\int }_{[0,1{]}^d}{\mathcal{K}}^{Y_d}(\mathbf{t},\mathbf{s})f(\mathbf{s})d\mathbf{s},$$

$$e^{Y_{d}}(n):=\inf\big{\{}\big{(}\mathbb{E}\|Y_{d}-Y_{d}^{(n)}\|^{2}_{2,d}\big{)}^{1/2}\,:\,Y_{d}^{(n)}\in\mathcal{A}^{Y_{d}}_{n}\big{\}}.$$

$$\mathcal{A}^{Y_{d}}_{n}:=\big{\{}\sum_{m=1}^{n}(Y_{d},\varphi_{m})_{2,d}\,\psi_{m}:\varphi_{m},\psi_{m}\in L_{2}([0,1]^{d})\big{\}}.$$

$$e^{Y_{d}}(0):=\big{(}\mathbb{E}\|Y_{d}\|^{2}_{2,d}\big{)}^{1/2}.$$

$$n^{Y_d,Z}(\epsilon ):=\min \{n\in \mathbb{N}:e^{Y_d}(n)\le \epsilon {\mathrm{CRI}}_d\},$$

$${\mathrm{CRI}}_d:=\{\begin{array}{cc}& 1,\text{ for}Z=\mathrm{ABS},\\ & e^{Y_d}(0),\text{for}Z=\mathrm{NOR}.\end{array}$$

$$n^{Y_d,Z}(\epsilon )\le C({\epsilon }^{-1}{)}^p;$$

$$n^{Y_d,Z}(\epsilon )\le C{d}^q({\epsilon }^{-1}{)}^p.$$

$$n^{Y_d,Z}(\epsilon )\le C\exp (t(1+\ln {\epsilon }^{-1})(1+\ln d));$$

$$\underset{{\epsilon }^{-1}+d\to \infty }{\lim }\frac{\ln n^{Y_d,Z}(\epsilon )}{({\epsilon }^{-1}{)}^s+d^t}=0;$$

$$\underset{{\epsilon }^{-1}+d\to \infty }{\lim }\frac{\ln n^{Y_d,Z}(\epsilon )}{{\epsilon }^{-1}+d}=0.$$

$$1\cup \{{\bar{\psi }}_k^{X_j}(t_j):k\in \mathbb{N},j=1,\cdots ,d\},\mathbf{t}=(t_1,\cdots ,t_d)\in [0,1{]}^d,$$

$$n^{Y_d,Z}(\epsilon ):=\min \{n\in \mathbb{N}:\sum _{j=n+1}^{\infty }{\lambda }_{d,j}\le {\epsilon }^2{\mathrm{CRI}}_d^2\},$$

$${\rm CRI}_{d}=\left\{\begin{matrix}&\quad 1,\,\,\qquad\quad\text{ for}\ Z={\rm ABS},\\ &\big{(}\sum_{j=1}^{\infty}\lambda_{d,j}\big{)}^{1/2},\quad\text{for}\ Z={\rm NOR},\end{matrix}\right.$$

$${\mathcal{K}}_{\alpha ,\beta ,\sigma }(x,y)=\alpha +2\beta \sum _{k=1}^{\infty }{k}^{-\sigma }\cos (2\pi k(x-y)),x,y\in [0,1].$$

$$K^{B_{\alpha ,\beta ,\sigma }}f(x):={\int }_{[0,1]}{\mathcal{K}}_{\alpha ,\beta ,\sigma }(x,y)f(y)dy.$$

$${\bar{\lambda }}_{2k-1}^{B_{\alpha ,\beta ,\sigma }}={\bar{\lambda }}_{2k}^{B_{\alpha ,\beta ,\sigma }}=\frac{\beta }{k^{\sigma }},{\bar{\psi }}_{2k-1}^{B_{\alpha ,\beta ,\sigma }}(x)=e^{-i2\pi kx},{\bar{\psi }}_{2k}^{B_{\alpha ,\beta ,\sigma }}(x)=e^{i2\pi kx},$$

$${\mathcal{K}}^{B_d}(\mathbf{x},\mathbf{y})=\sum _{j=1}^d{\mathcal{K}}_{{\alpha }_j,{\beta }_j,{\sigma }_j}(x_j,y_j),$$

$$K^{B_d}f(\mathbf{x})={\int }_{[0,1{]}^d}\sum _{j=1}^d{\mathcal{K}}_{{\alpha }_j,{\beta }_j,{\sigma }_j}(x_j,y_j)f(y_j)d\mathbf{y},$$

$${\bar{\lambda }}_0^{B_d}=\sum _{j=1}^d{\alpha }_j$$

$${\bar{\lambda }}_{2k-1}^{B_d}={\bar{\lambda }}_{2k}^{B_d}=\frac{{\beta }_j}{k^{{\sigma }_j}},k\in N,j=1,\dots ,d,$$

$${\bar{\psi }}_{2k-1}^{B_d}(x_j)=e^{-i2\pi k{x}_j},{\bar{\psi }}_{2k}^{B_d}(x_j)=e^{i2\pi k{x}_j},\mathbf{x}\in [0,1{]}^d,k\in N,j=1,\dots ,d,$$

$$\sum _{j=1}^{\infty }{\lambda }_{d,j}=\sum _{j=1}^d{\alpha }_j+2\sum _{j=1}^d\sum _{k=1}^{\infty }\frac{{\beta }_j}{k^{{\sigma }_j}}=\sum _{j=1}^d{\alpha }_j+2\sum _{j=1}^d{\beta }_j\zeta ({\sigma }_j),$$

$$\sum_{j=1}^{\infty}\lambda_{d,j}^{\tau}=\big{(}\sum_{j=1}^{d}\alpha_{j}\big{)}^{\tau}+2\sum_{j=1}^{d}\sum_{k=1}^{\infty}\frac{\beta_{j}^{\tau}}{k^{\tau\sigma_{j}}}=\big{(}\sum_{j=1}^{d}\alpha_{j}\big{)}^{\tau}+2\sum_{j=1}^{d}\beta_{j}^{\tau}\zeta(\tau\sigma_{j}),$$

$${\alpha }_j\ge 0,1\ge {\beta }_1\ge {\beta }_2\ge \cdots >0,1<{\sigma }_1\le {\sigma }_2\le \cdots .$$

$$B=\{B_d:L_2([0,1{]}^d)\to L_2([0,1{]}^d)\}$$

$$A_{\ast }:=\underset{d\to \infty }{\underline{\lim }}\frac{\ln \frac{1}{{\beta }_d}}{\ln d}>1.$$

Full text

1. Introduction

Let $X_{j}(t),\,t\in[0,1],\,j\in\mathbb{N}$ be a sequence of independent random processes, where $\mathbb{N}=\{1,2,\cdots\}$. Suppose that every random element $X_{j}(t)$ has zero mean and a covariance function $\mathcal{K}^{X_{j}}(t,s),\,t,\,s\in[0,1]$. Let $K^{X_{j}}$ be the covariance operator with the covariance kernel $\mathcal{K}^{X_{j}}$ for every $j\in\mathbb{N}$. Then for any $f\in L_{2}([0,1])$ and $t\in[0,1]$, we have

[TABLE]

We consider the random field

[TABLE]

with the following zero-mean covariance function

[TABLE]

and covariance operator

[TABLE]

where $f\in L_{2}([0,1]^{d}),\ {\bf t}=(t_{1},\cdots,t_{d}),\,{\bf s}=(s_{1},\cdots,s_{d}),\ {\bf t},{\bf s}\in[0,1]^{d}.$ This random field is called an additive random field. There are many papers which investigated this random field, see [1, 3, 4, 5].

In this paper, we investigate the average case approximation of $Y_{d}({\bf t}),\,{\bf t}\in[0,1]^{d}$, as a random element of the space $L_{2}([0,1]^{d})$ equipped with inner product $(\cdot,\cdot)_{2,d}$ and norm $\|\cdot\|_{2,d}$, by a finite rank random field.

The $n$th minimal 2-average case error, for $n\geq 1$, is defined by

[TABLE]

Here $\mathcal{A}^{Y_{d}}_{n}$ is the class of all linear algorithms with rank $\leq n$ defined by

[TABLE]

The so-called initial average case error is given by

[TABLE]

We use either the absolute error criterion (ABS) or the normalized error criterion (NOR). For any $\varepsilon\in(0,1)$, we study information complexity $n^{Y_{d},Z}(\varepsilon)$ of approximation of the random fields $Y_{d},\,d\in\mathbb{N}$ defined by

[TABLE]

where $Z\in\{{\rm ABS},{\rm NOR}\}$,

[TABLE]

Let $Y=\{Y_{d}\}_{d\in{N}}$. First we consider average case tractability of $Y$. Various notions of tractability have been discussed for multivariate problems. We recall some of the basic tractability notions (see [7, 8, 9, 10]).

For $Z\in\{{\rm ABS,\,NOR}\}$, say $Y$ is

$\bullet$ strongly polynomially tractable (SPT) iff there exist non-negative numbers $C$ and $p$ such that for all $d\in{N},\ \varepsilon\in(0,1)$,

[TABLE]

The infimum of $p$ satisfying the above inequality is called the exponent of strong polynomial tractability and is denoted by $p^{\rm str-avg}$.

$\bullet$ polynomially tractable (PT) iff there exist non-negative numbers $C,\,p$ and $q$ such that for all $d\in{N},\ \varepsilon\in(0,1)$,

[TABLE]

$\bullet$ quasi-polynomially tractable (QPT) iff there exist two constants $C,t>0$ such that for all $d\in{N},\ \varepsilon\in(0,1)$,

[TABLE]

$\bullet$ uniformly weakly tractable (UWT) iff for all $s,t>0$,

[TABLE]

$\bullet$ weakly tractable (WT) iff

[TABLE]

This paper is devoted to studying average case tractability of the additive random field $Y=\{Y_{d}\}_{d\in\mathbb{N}}$ under ABS and NOR. For additive random fields similar problems were investigated in [2, 5, 6, 11] in various settings for the homogeneous case and in [3, 4] for the non-homogeneous case. Here, the homogeneous case means that approximated additive random fields are constructed (in a special way) from copies of one marginal process, while the non-homogeneous case means that the random fields are composed of a whole sequence of marginal random processes with generally different covariance functions. Specifically, the authors in [3] obtained the growth of $n^{Y_{d},{\rm NOR}}(\varepsilon)$ for arbitrary fixed $\varepsilon\in(0,1)$ and $d\to\infty$ for the non-homogeneous case and gave application to the additive random fields with marginal random processes corresponding to the Korobov kernels.

It should be noted, however, that all these works deal only with NOR. In this paper, we consider average case tractability of the problem ${B}=\{{B}_{d}\}_{d\in\mathbb{N}}$ of the additive random fields with marginal random processes corresponding to the Korobov kernels under ABS and NOR. We shall show that the problem ${B}$ is always polynomially tractable for ABS or NOR. Obviously, PT implies all QPT, UWT, WT. We also give sufficient and necessary conditions for which ${B}$ is SPT for ABS or NOR.

The paper is organized as follows. In Section 2 we give preliminaries about the additive random fields with marginal random processes corresponding to the Korobov kernels and introduce main results, i.e., Theorems 2.1-2.3. Section 3 is devoted to proving Theorems 2.1-2.3.

2. Preliminaries and main results

Let us consider the sequence of additive random fields $Y_{d}({\bf t})$, ${\bf t}\in[0,1]^{d}$, $d\in\mathbb{N}$, defined by (1.1). Let \big{\{}(\lambda_{k}^{X_{j}},\,\psi_{k}^{X_{j}})\big{\}}_{k\in\mathbb{N}} be the sequence of eigenpairs of the covariance operator $K^{X_{j}}$ of $X_{j}$. Under the additive structure (1.2), the eigenvalues $\lambda_{d,k},\,k\in\mathbb{N}$, are generally unknown or not easily depend on $\lambda_{k}^{X_{j}},\,k\in\mathbb{N},\,j=1,2,\cdots,d$. However, under the following condition, we can explicitly describe the eigenvalues $\lambda_{d,k},\,k\in\mathbb{N}$.

For every $j\in\mathbb{N}$ there exist $\psi_{0}\in\{\psi_{1}^{X_{j}},\psi_{2}^{X_{j}},\cdots,\psi_{k}^{X_{j}},\cdots\}$ such that $\psi_{0}(t)=1$ for all $t\in[0,1]$. We denote by $\bar{\lambda}_{0}^{X_{j}}$ the eigenvalues corresponding to eigenvector

1. Let $\{\bar{\lambda}_{k}^{X_{j}}\}_{k\in\mathbb{N}}$ and $\{\bar{\psi}_{k}^{X_{j}}\}_{k\in\mathbb{N}}$ be the non-increasing sequence of the remaining eigenvalues and the corresponding sequence of eigenvectors of $K^{X_{j}}$, respectively. It is known that the family of eigenvectors

[TABLE]

is an orthogonal system in $L_{2}([0,1]^{d})$ for every $d\in\mathbb{N}$, see [3]. Hence the identical 1 is an eigenvector of $K^{Y_{d}}$ with the eigenvalue $\bar{\lambda}_{0}^{Y_{d}}:=\sum_{j=1}^{d}\bar{\lambda}_{0}^{X_{j}}$, and the pairs $(\bar{\lambda}_{k}^{X_{j}},\,\bar{\psi}_{k}^{X_{j}})$, for all $k\in\mathbb{N}$ and $j=1,\cdots,d$, are the remaining eigenpairs of $K^{Y_{d}}$.

Let $\{\lambda_{d,j}\}_{j\in{N}}$ and $\{\psi_{d,j}\}_{j\in{N}}$ be the non-increasing sequence of the eigenvalues and the corresponding sequence of eigenvectors of $K^{Y_{d}}$ defined by (1.3). Then the average case information complexity $n^{Y_{d},Z}(\varepsilon)$ can also be described in terms of eigenvalues $\lambda_{d,j},\,j\in\mathbb{N}$ of $K^{Y_{d}}$ by

[TABLE]

where

[TABLE]

see [7].

Particularly, we study additive random fields with marginal random processes corresponding to the Korobov kernels. Let $B_{\alpha,\beta,\sigma}(x)$, $x\in[0,1]$ be a zero-mean random field with the following covariance function

[TABLE]

Here $\alpha\geq 0$, $\beta>0$ and $\sigma>1$. Let $K^{B_{\alpha,\beta,\sigma}}$ be the covariance operator with kernel $\mathcal{K}_{\alpha,\beta,\sigma}$ of $B_{\alpha,\beta,\sigma}$, and for any $L_{2}([0,1])$ and $x\in[0,1]$,

[TABLE]

The eigenpairs of the covariance operator $K^{B_{\alpha,\beta,\sigma}}$ are known, see [7]. The identical 1 is an eigenvector of $K^{B_{\alpha,\beta,\sigma}}$ with the eigenvalue $\bar{\lambda}_{0}^{B_{\alpha,\beta,\sigma}}=\alpha$. The other eigenpairs of $K^{B_{\alpha,\beta,\sigma}}$ have the following form:

[TABLE]

for any $k\in\mathbb{N},\,x\in[0,1]$.

Suppose that $B_{j}(x),\,x\in[0,1],\,j\in{N}$ is a sequence of independent zero-mean random fields with covariance functions $\mathcal{K}_{\alpha_{j},\beta_{j},\sigma_{j}}$, respectively. Let ${B}_{d}({\bf x})=\sum_{j=1}^{d}B_{j}(x_{j})$, ${\bf x}\in[0,1]^{d},\,d\in\mathbb{N}$, be the sequence of zero-mean random fields with the covariance functions

[TABLE]

where ${\bf x}=(x_{1},x_{2},\cdots,x_{d})\in[0,1]^{d}$, ${\bf y}=(y_{1},y_{2},\cdots,y_{d})\in[0,1]^{d}$, and the parameters $\alpha_{j}\geq 0$, $\beta_{j}>0$ for all $j\in\mathbb{N}$, and $\inf_{j\in\mathbb{N}}\sigma_{j}>1$.

Let $K^{{B}_{d}}$ be the covariance operator of ${B}_{d}$. We have

[TABLE]

for any $f({\bf x})\in L_{2}([0,1]^{d})$ and ${\bf x}\in[0,1]^{d}$. Then the identical 1 is an eigenvector of $K^{{B}_{d}}$ with the eigenvalue

[TABLE]

and the remaining eigenvalues and eigenvectors are

[TABLE]

and

[TABLE]

respectively. Let $\{\lambda_{d,j}\}_{j=1}^{\infty}$ be the sequence of non-increasing rearrangement of the eigenvalues of $K^{{B}_{d}}$. Then we have

[TABLE]

and for any $\tau\in(0,1)$ and $\tau\sigma_{j}>1,\ j=1,\dots,d$,

[TABLE]

where $\zeta(p)=\sum_{k=1}^{\infty}k^{-p},\,p>1$, is the Riemann zeta-function.

In the sequel we always assume that the sequences $\alpha_{j},\,\beta_{j},\,\sigma_{j}$ satisfy

[TABLE]

In this paper, we consider the tractability of the problem

[TABLE]

under ABS and NOR, where the sequences $\alpha_{j},\,\beta_{j},\,\sigma_{j}$ satisfy (2.6). Our main results can be formulated as follows.

Theorem 2.1**.**

Let the sequences $\alpha_{j},\,\beta_{j},\,\sigma_{j}$ satisfy (2.6). Then the problem ${B}$

(i) is always PT for ABS or NOR;

(ii) is SPT for ABS iff

[TABLE]

The exponent of SPT is

[TABLE]

In order to investigate the strong polynomial tractability of the problem ${B}$ for NOR, we consider two cases.

Theorem 2.2**.**

Let $\alpha_{j},\,\beta_{j},\,\sigma_{j}$ satisfy (2.6) and

[TABLE]

where $c>0$ is a constant. Then for NOR the problem ${B}$ is SPT iff

[TABLE]

The exponent of SPT is

[TABLE]

Theorem 2.3**.**

Let $\alpha_{j},\,\beta_{j},\,\sigma_{j}$ satisfy (2.6). Further assume that

[TABLE]

and there exists a constant $c>0$ such that for every $d\in{N}$,

[TABLE]

Then the problem ${B}$ is SPT for NOR iff

[TABLE]

The exponent of SPT is

[TABLE]

In particular, if $\alpha_{j}=1$ for $j\in{N}$, then the problem ${B}$ is SPT for NOR iff

[TABLE]

Remark 2.4*.*

In [3], Khartov and Zani investigated $n^{{B}_{d},{\rm NOR}}(\varepsilon)$ for arbitrary fixed $\varepsilon\in(0,1)$ and $d\to\infty$ of the above problem ${B}$ with the parameters

[TABLE]

where $c>0$, $s>0$, and $0\leq r\leq\infty$. They obtained the following results.

(1) For $c>0$ and either $s>1$, $0\leq r\leq\infty$ or $0\leq s\leq 1$, $r=\infty$. Then

[TABLE]

(2) For $c>0$, $0\leq s<1$ and $0\leq r<\infty$,

[TABLE]

where

[TABLE]

(3) For $c>0$, $s=1$ and $0\leq r<\infty$, then

[TABLE]

We remark that in some sense the above results give the tractability results of the problem ${B}$ for NOR under the condition (2.11). Specifically, the above result (1) corresponds to SPT of ${B}$ for NOR, and results (2) and (3) relate to PT of ${B}$ for NOR, but more explicitly. Comparing with [3], we obtain the tractability of the problem ${B}$ with general parameters $\alpha_{j},\ \beta_{j},\ \sigma_{j}$ satisfying (2.6) for ABS and NOR. Also we get the exponent of SPT for ABS and NOR.

Remark 2.5*.*

For the above problem ${B}=\{{B}_{d}\}_{d\in{N}}$, let $\mathaccent 869{{B}}_{d}$ be the additive random fields with marginal random processes corresponding to the Korobov covariance functions

[TABLE]

with parameters $\mathaccent 869{\alpha}_{j},\mathaccent 869{\beta}_{j},\mathaccent 869{\sigma}_{j}$ satisfying

[TABLE]

Let $\{{\lambda}_{d,j}\}_{j=1}^{\infty}$ and $\{\mathaccent 869{\lambda}_{d,j}\}_{j=1}^{\infty}$ be the sequences of non-increasing rearrangement of the eigenvalues of the covariance operators $K^{{{B}}_{d}}$ and $K^{\mathaccent 869{{B}}_{d}}$, respectively. Then for some $j_{0}\in{N}$,

[TABLE]

and

[TABLE]

From the above equalities we obtain that

[TABLE]

It follows from (2.1) and the inequality $\lambda_{d,j}\geq\mathaccent 869{\lambda}_{d,j}$ that for $\varepsilon\in(0,1)$,

[TABLE]

On the other hand, we note that for $\varepsilon\in(0,1)$,

[TABLE]

where $n_{0}=n^{\mathaccent 869{{B}}_{d},\text{ABS}}(\varepsilon)$. This gives that

[TABLE]

which means that

[TABLE]

Hence we have for $\varepsilon\in(0,1)$,

[TABLE]

It follows that the problems

[TABLE]

have the same tractability for ABS and the same exponent of SPT for ABS if the problem $\mathaccent 869{{B}}$ is SPT for ABS.

Remark 2.6*.*

Let the sequences $\alpha_{j},\,\beta_{j},\,\sigma_{j}$ satisfy (2.6). If the problem ${B}$ is SPT (or PT) for ABS, then it is also SPT (or PT) for NOR.

Indeed, it follows from (2.4) that

[TABLE]

By (2.1) we have

[TABLE]

which means

[TABLE]

Therefore if the problem ${B}$ is SPT (or PT) for ABS, then it is also SPT (or PT) for NOR.

3. Proofs of Theorems 2.1-2.3

Proof of Theorem 2.1.

(i) From Remark 2.6, it is sufficient to show that ${B}$ is PT for ABS. By Remark 2.5 we get that ${B}$ is PT for ABS iff $\mathaccent 869{{B}}$ is PT for ABS. Hence it suffices to prove that $\mathaccent 869{{B}}$ is PT for ABS.

We note

[TABLE]

and

[TABLE]

It follows that for $\tau\in(\frac{1}{\sigma_{1}},1)$,

[TABLE]

where in the last inequality we used $\beta_{j}\leq 1$ for all $j\in\mathbb{N}$. This forces

[TABLE]

Due to [7, Theorem 6.1] we obtain that $\mathaccent 869{{B}}$ is PT for ABS. Therefore the problem ${B}$ is always PT for ABS or NOR.

(ii) From Remark 2.5, it is sufficient to prove that the problem

[TABLE]

is SPT for ABS iff (2.7) holds, and that the exponent of $\mathaccent 869{{B}}$ satisfies (2.8).

Assume that (2.7) holds, i.e.,

[TABLE]

We want to prove that $\mathaccent 869{{B}}$ is SPT for ABS. Indeed, by (3.1) we have for any $\tau\in(\frac{1}{\sigma_{1}},1)$,

[TABLE]

Next, we shall prove that for any $\tau\in(\frac{1}{A_{*}},1)$,

[TABLE]

Indeed, for such $\tau\in(\frac{1}{A_{*}},1)$, there exists a $\delta\in(0,1)$ for which

[TABLE]

Since

[TABLE]

there exists a $d_{0}\in\mathbb{N}$ such that

[TABLE]

which means

[TABLE]

It follows that

[TABLE]

for any $\tau>\frac{1}{A_{*}}$. Hence we obtain

[TABLE]

for any \tau\in\big{(}\max\{\frac{1}{\sigma_{1}},\frac{1}{A_{*}}\},1\big{)}. We note that \tau\in\big{(}\max\{\frac{1}{\sigma_{1}},\frac{1}{A_{*}}\},1) is equivalent to

[TABLE]

due to the monotonicity of the function

[TABLE]

It follows from [7, Theorem 6.1] that if (2.7) holds, then $\mathaccent 869{{B}}$ is SPT for ABS, and the exponent of SPT satisfies

[TABLE]

On the other hand, assume that $\mathaccent 869{{B}}$ is SPT for ABS. Then there exist positive $C,\,C_{1},$ and $\tau\in(0,1)$ such that

[TABLE]

where we used

[TABLE]

in the above inequalities. Obviously, by (3) we have

[TABLE]

It follows from (3) that

[TABLE]

where $C_{2}:=\frac{C^{\tau}+\lceil C_{1}\rceil-1}{2}>0$, which yields that

[TABLE]

Letting $d\to\infty$ we get

[TABLE]

which means

[TABLE]

Hence if $\mathaccent 869{{B}}$ is SPT for ABS, then we have

[TABLE]

and the exponent of SPT satisfies

[TABLE]

Therefore ${{B}}$ is SPT for ABS iff

[TABLE]

and the exponent of SPT is

[TABLE]

Theorem 2.1 is proved. $\hfill\Box$

Proof of Theorem 2.2.

From Remark 2.6, we note that if ${B}$ is SPT for ABS, then it is SPT for NOR. If $A_{*}>1$ holds, by Theorem 2.1 we get that ${B}$ is SPT for ABS, and hence is SPT for NOR. It suffices to prove that if ${B}$ is SPT for NOR, then $A_{*}>1.$

Assume that ${B}$ is SPT for NOR. Then by [7, Theorem 6.2] there exists a $\tau\in(0,1)$ such that

[TABLE]

It follows that

[TABLE]

Using (3.1) and $0\leq\alpha_{j}\leq c\beta_{j}$ for all $j\in\mathbb{N}$, we get

[TABLE]

Assume that $\sum_{j=1}^{\infty}\beta_{j}=+\infty$. Then for the above $\tau\in(0,1)$ we have

[TABLE]

By the Stolz theorem we get

[TABLE]

where

[TABLE]

Since

[TABLE]

and

[TABLE]

for any $\alpha>0,$ we have

[TABLE]

where $f(x)\sim g(x)$ as $g(x)\to 0$ means that $\lim\limits_{g(x)\to 0}\frac{f(x)}{g(x)}=1$. Substituting (3.8) into (3) yields

[TABLE]

contrary to (3).

Hence we have $\sum_{j=1}^{\infty}\beta_{j}<+\infty$, and thence C_{1}:=\big{(}\sum_{j=1}^{\infty}\beta_{j}^{\tau}\big{)}^{\frac{1}{\tau}}<+\infty. It follows that

[TABLE]

We obtain further

[TABLE]

This gives

[TABLE]

which implies $\tau\geq\frac{1}{A_{*}}$. Hence we conclude that if ${B}$ is SPT for NOR, then

[TABLE]

and the exponent of SPT for NOR satisfies

[TABLE]

Theorem 2.2 is proved. $\hfill\Box$

Proof of Theorem 2.3.

First we assume that SPT holds for NOR. By [7, Theorem 6.2] there exists a $\tau\in(0,1)$ such that

[TABLE]

It follows that

[TABLE]

We get for all $d\in{N}$,

[TABLE]

where we used

[TABLE]

and $1\geq\beta_{1}\geq\beta_{2}\geq\cdots\geq 0$ in the last inequality. It follows that

[TABLE]

This means

[TABLE]

Letting $d\to\infty$ we get

[TABLE]

Hence if ${B}$ is SPT for NOR, then we have $B_{*}=\underset{d\to\infty}{\underline{\lim}}\frac{\ln\frac{\alpha_{d}}{\beta_{d}}}{\ln d}>0$, and the exponent of SPT for NOR satisfies

[TABLE]

On the other hand, we assume

[TABLE]

Then for any $\tau\in(\max\{\frac{1}{\sigma_{1}},\frac{1}{1+B_{*}}\},1)$ we shall prove

[TABLE]

Indeed, we have

[TABLE]

where in the first inequality we used

[TABLE]

and (3.1). It suffices to prove that

[TABLE]

We suppose that $\tau=\frac{1}{1+B_{*}}\cdot\frac{1}{1-\delta_{0}}$ for some $\delta_{0}\in(0,1)$. Since

[TABLE]

there exists a $d_{0}\in\mathbb{N}$ such that

[TABLE]

It follows that

[TABLE]

Note that

[TABLE]

Hence we get

[TABLE]

It follows from the Hölder inequality that

[TABLE]

Hence for any $\tau\in(\max\{\frac{1}{\sigma_{1}},\frac{1}{1+B_{*}}\},1)$ we have

[TABLE]

which means that SPT holds with the exponent of SPT for NOR satisfying

[TABLE]

Therefore SPT holds for NOR iff

[TABLE]

and the exponent of SPT for NOR is

[TABLE]

The proof of Theorem 2.2 is finished. $\hfill\Box$

Acknowledgments

The authors were supported by the National Natural Science Foundation of China (Project no. 11671271), the Beijing Natural Science Foundation (1172004).