The Law of the Iterated Logarithm for Linear Processes Generated by a Sequence of Stationary Independent Random Variables under the Sub-Linear Expectation

As fundamental limit theorems of probability theory, the classical law of the iterated logarithm (LIL for short) plays an important role in the development of probability theory and its applications. The original statement of LIL obtained by Khinchine [1] is for a class of Bernoulli random variables. After that, a lot of literature has performed in-depth and detailed research on LIL, we can refer to Hartman and Wintner [2], Acosta [3], Shao and Su [4], and so on. Motivated by modeling uncertainty in practice, Peng [5] introduced the reasonable framework of the sub-linear expectation of random variables in a general function space. As an alternative to the traditional probability/expectation, capacity/sub-linear expectation has been studied in many fields, such as statistics, mathematical economics, measures of risk, and super-hedging in finance. In recent years, after studying the limit theorem of sub-linear expectation (e.g., see Feng [6], Deng and Wang [7], Tan and Zong [8], and Zhang [9,10], etc.), more and more research results of LIL under this framework have been obtained, the Hartman–Winter LIL were established by Chen and Hu [11] for bounded random variables, the functional central limit and Chung’s LIL were recently obtained by Zhang [12], and the LIL for independent and negatively dependent identically distributed random variables were proven by Zhang [13].

As is well known, the linear processes are especially important in time series analysis and they arise from a wide variety of contexts (cf. Hannan [14]). Applications to economics, engineering, and physical sciences are extremely broad and a vast amount of literature is devoted to the study of linear processes under a variety of circumstances. The limit theory of linear processes has been studied in detail in many papers. Philips and Solo [15] prove the strong law of numbers and the law of iterated logarithm for linear processes, Zhang [16] gives the limit law of the iterated logarithm for linear processes. Recently, Liu and Zhang [17] obtained the central limit theorem and invariance principle for linear processes generated by independent and identically distributed (IID for short) random variables under sub-linear expectation.

A natural question is: can LIL of linear processes be realized under Peng’s framework? The main purpose of this paper is to establish the law of iterated logarithm for linear processes generated by IID random variables in sub-linear expectation space. In the classical case, the LIL of partial sum is established by decomposing the linear process. We will find that this way is also valid for proving LIL for linear process in the sub-linear expectation space, though there are some differences. Intuitively, sub-linear expectation and related non-additive probabilities (Capacities) generated by them plays a decisive role in our proof. In the sequel, c denotes a positive constant, which may take different values whenever it appears in different expressions.

To state the result, we shall first recall the framework of sub-linear expectations. We use the framework and notation of Peng [5,18,19]. Let $(\mathrm{\Omega },\mathcal{F})$ be a given measurable space. Let $\mathcal{H}$ be a linear space of real functions defined on $(\mathrm{\Omega },\mathcal{F})$, such that if $X_1,X_2,...,X_n\in \mathcal{H}$ then $\phi (X_1,X_2,...,X_n)\in \mathcal{H}$ for each $\phi \in C_{l,Lip}\left({\mathbb{R}}^n\right)$ where $\phi \in C_{l,Lip}\left({\mathbb{R}}^n\right)$ denotes the linear space of local Lipschitz continuous functions $\phi$ satisfying for some $c>0,m\in \mathbb{N}$ depending on $\phi$. $\mathcal{H}$ contains all $I_A$ where $A\in \mathcal{F}$. We also denote $\phi \in C_{b,Lip}\left({\mathbb{R}}^n\right)$ as the linear space of bounded Lipschitz continuous functions $\phi$ satisfying for some $c>0$.

The triple $(\mathrm{\Omega },\mathcal{H},\widehat{\mathbb{E}})$ is called a sub-linear expectation space. Give a sub-linear expectation $\widehat{\mathbb{E}}$, let us denote the conjugate expectation $\widehat{\mathcal{E}}$ of $\widehat{\mathbb{E}}$ by $\widehat{\mathcal{E}}\left[X\right]:=-\widehat{\mathbb{E}}[-X],\forall X\in \mathcal{H}$.

It is obvious that, if $\{X_n,n\ge 1\}$ is a sequence of independent random variables and $f_1\left(x\right),f_2\left(x\right),...\in C_{l,Lip}\left(\mathbb{R}\right)$, then $\{f_n\left(X_n\right),n\ge 1\}$ is also a sequence of independent random variables.

Next, we introduce the capacities corresponding to the sub-linear expectations.

A sub-linear expectation $\widehat{\mathbb{E}}$ could generate a pair $(\mathbb{V},\mathcal{V})$ of capacity denoted by where $A^c$ is the complement set of A. Then

In addition, a pair $(C_{\mathbb{V}},C_{\mathcal{V}})$ of the Choquet integrals/expecations denoted by with V being replaced by $\mathbb{V}$ and $\mathcal{V}$, respectively.

If ${lim}_{c\to \infty }\widehat{\mathbb{E}}{\left[\right(\left|X\right|-c)}^{+}]=0$ or $\widehat{\mathbb{E}}$ is countably sub-additive, then $\widehat{\mathbb{E}}\left[\right|X\left|\right]\le C_V\left(\right|X\left|\right)$ (See Lemma 4.5 (iii) of Zhang [13]).

It is obvious that the continuity from above and sub-additivity imply the continuity from below, and the continuity from the below and sub-additivity imply the countable sub- additivity. Therefore, we call a sub-additive capacity to be continuous if it is continuous from above. Set $\mathbb{H}=\{A:I_A\in \mathcal{H}\}$, then $\mathbb{V}$ is a countably sub-additive capacity in $\mathbb{H}$ if $\widehat{\mathbb{E}}$ is countably sub-additive in $\mathcal{H}$, and $(\mathbb{V},\mathcal{V})$ is a pair of continuous capacities in $\mathbb{H}$ if $\widehat{\mathbb{E}}$ is continuous in $\mathcal{H}$.

In this section, we shall study the LIL of linear processes under association assumption in the sub-linear expectation space. For any $I\in (k,+\infty )$, $\{X_j,j\in I\}$ is a sequence of independent random variables satisfying Definition 2; For a finite index set $I\in (-\infty ,k)$, $\{X_j,j\in I\}$ is also a sequence of independent random variables satisfying Definition 2.

First, we give the definition of strictly stationary sequence under the sub-linear expectation.

Next we give the main results: the law of the iterated logarithm for linear processes in the sub-linear expectation space.

In order to prove the main results, we need the following Lemmas. The first one was the convergence part of the Borel–Cantelli Lemma.

The second Lemma on the exponential inequality is Lemma 2.1 of Zhang [9].

The following Lemma is a law of iterated logarithm under sub-linear expectation.