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Stereotyped, automatized and habitual behaviours: are they similar constructs under the control of the same cerebral areas?

  • Received: 31 January 2020 Accepted: 18 May 2020 Published: 27 May 2020
  • Comprehensive knowledge about higher executive functions of motor control has been covered in the last decades. Critical goals have been targeted through many different technological approaches. An abundant flow of new results greatly progressed our ability to respond at better-posited answers to look more than ever at the challenging neural system functioning. Behaviour is the observable result of the invisible, as complex cerebral functioning. Many pathological states are approached after symptomatology categorisation of behavioural impairments is achieved. Motor, non-motor and psychiatric signs are greatly shared by many neurological/psychiatric disorders. Together with the cerebral cortex, the basal ganglia contribute to the expression of behaviour promoting the correct action schemas and the selection of appropriate sub-goals based on the evaluation of action outcomes. The present review focus on the basic classification of higher motor control functioning, taking into account the recent advances in basal ganglia structural knowledge and the computational model of basal ganglia functioning. We discuss about the basal ganglia capability in executing ordered motor patterns in which any single movement is linked to each other into an action, and many actions are ordered into each other, giving them a syntactic value to the final behaviour. The stereotypic, automatized and habitual behaviour's constructs and controls are the expression of successive stages of rule internalization and categorisation aimed in producing the perfect spatial-temporal control of motor command.

    Citation: Tiziana M. Florio. Stereotyped, automatized and habitual behaviours: are they similar constructs under the control of the same cerebral areas?[J]. AIMS Neuroscience, 2020, 7(2): 136-152. doi: 10.3934/Neuroscience.2020010

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  • Comprehensive knowledge about higher executive functions of motor control has been covered in the last decades. Critical goals have been targeted through many different technological approaches. An abundant flow of new results greatly progressed our ability to respond at better-posited answers to look more than ever at the challenging neural system functioning. Behaviour is the observable result of the invisible, as complex cerebral functioning. Many pathological states are approached after symptomatology categorisation of behavioural impairments is achieved. Motor, non-motor and psychiatric signs are greatly shared by many neurological/psychiatric disorders. Together with the cerebral cortex, the basal ganglia contribute to the expression of behaviour promoting the correct action schemas and the selection of appropriate sub-goals based on the evaluation of action outcomes. The present review focus on the basic classification of higher motor control functioning, taking into account the recent advances in basal ganglia structural knowledge and the computational model of basal ganglia functioning. We discuss about the basal ganglia capability in executing ordered motor patterns in which any single movement is linked to each other into an action, and many actions are ordered into each other, giving them a syntactic value to the final behaviour. The stereotypic, automatized and habitual behaviour's constructs and controls are the expression of successive stages of rule internalization and categorisation aimed in producing the perfect spatial-temporal control of motor command.


    Since the 20th century, the probability theory has gained profound and extraordinary applications in the fields of mathematical statistics, information science, finance, and economics. The probability limit theory is an important branch of the probability theory. The probability limit theory has a broad range of applications. In the course of development, many important theorems and concepts have been proposed, such as the central limit theorem and the law of large numbers. These theorems are not only important in theory, but are also widely used in practical applications. Under the classical probability space, the mathematical expectation is additive, where one can solve many deterministic problems in real life. However, with the development of the society, many uncertainty phenomena have appeared in many new industries, such as insurance, finance, risk management, and other industries. In order to solve these uncertainty phenomena, Peng[1,2,3,4] broke away from the theoretical constraints of the classical probability space, constructed a sublinear expectation theoretical framework, and created a complete axiomatic system, which provides a new direction for solving these uncertainty problems.

    Many important results and theorems in classical probability spaces can be proven and applied to the sublinear expectation spaces. Therefore, some important research directions in the classical probability space can also be extrapolated to the sublinear expectation space. More and more scholars have begun to study the related theoretical achievements under sublinear expectations. For example, Xu and Kong [5] proved the complete integral convergence and complete convergence of negatively dependent (ND) random variables under sublinear expectations. Hu and Wu [6] proved the complete convergence theorems for an array of row-wise extended negatively dependent (END) random variables utilizing truncated methods under sublinear expectations. Wang and Wu [7] used truncated methods to derive the complete convergence and complete integral convergence of the weighted sums of END random variables under sublinear expectations. In addition, many scholars have received numerous theoretical results about the law of large numbers and the law of iterated logarithms from their investigations, and have obtained many theoretical achievements under sublinear expectations. Chen [8], Hu[9,10], Zhang[11], and Song[12] studied the strong law of large numbers for independent identically distributed (IID) random variables under different conditions. Wu et al.[13] established inequalities such as the exponential inequality, the Rosenthal inequality, and obtained the Marcinkiewicz-Zygmund type strong law of large numbers for weighted sums of m-widely acceptable random variables under sublinear expectations. Chen and Wu[14] established the weak and strong law of large numbers for Pareto-type random variables, so that the relevant conclusions in the traditional probability space were extended to the sublinear expectation space. Chen et al.[15] studied the properties associated with weakly negatively dependent (WND) random variables and established the strong law of large numbers for WND random variables under sublinear expectations. Zhang[16] studied the limit behavior of linear processes under sublinear expectations and obtained a strong law of large numbers for linear processes generated by independent random variables. Zhang[17] provided the sufficient and necessary conditions of the strong law of large numbers for IID random variables under the sub-linear expectation. Guo[18] introduced the concept of pseudo-independence under sublinear expectations and derived the weak and strong laws of large numbers. Zhang [19] established some general forms of the law of the iterated logarithms for independent random variables in a sublinear expectation space. Wu and Liu [20] studied the Chover-type law of iterated logarithms for IID random variables. Zhang [21] studied the law of iterated logarithms for sequences of END random variables with different conditions. Guo et al.[22] studied two types of Hartman-Wintner iterated logarithmic laws for pseudo-independent random variables with a finite quadratic Choquet expectation and extended the existed achievements.

    The goal of this article is to prove the Marcinkiewicz-Zygmund type weak law of large numbers for an array of row-wise WND random variables, and the strong law of large numbers for linear processes generated by WND random variables under sublinear expectations. The rest of the paper is as follows: in Section 2, we recall some basic definitions, notations, and lemmas needed to prove the main theorems under sublinear expectations; in Section 3, we state our main results; in Section 4, the proofs of these theorems are given; in Section 5, we conclude the paper.

    We use the framework and notation of Peng [1,2,3,4]. Considering the following sublinear expectation space(Ω,H,ˆE), if X1,X2,,XnH, then ψ(X1,X2,,Xn)H for each ψCb,Lip(Rn), where Cb,Lip(Rn) denotes the linear space of functions ψ satisfying the following bounded Lipschitz condition:

    |ψ(x)|C,|ψ(x)ψ(y)|C|xy|,x,yRn,

    where the constant C>0 depending on ψ.

    Definition 2.1. [4] A sublinear expectation ˆE is a functional ˆE: HR satisfying the following:

    (a) Monotonicity: ˆE(X)ˆE(Y) if XY;

    (b) Constant preserving: ˆE(c)=c for cR;

    (c) Sub-additivity: For each X,YH, ˆE(X+Y)ˆE(X)+ˆE(Y);

    (d) Positive homogeneity: ˆE(λX)=λˆE(X), for λ0.

    The triple (Ω,H,ˆE) is called a sublinear expectation space.

    Through a sublinear expectation ˆE, we can use ˆεX=ˆE(X),XH to define the conjugate expectation of ˆE.

    From the above definition, for any X,YH we obtain the following:

    ˆε(X)ˆE(X),ˆE(X+c)=ˆE(X)+c,|ˆE(XY)|ˆE|XY|,ˆE(X)ˆE(Y)ˆE(XY).

    Definition 2.2. [23] A function V : F[0,1] is said a capacity satisfying the following:

    (a) V()=0, V(Ω)=1;

    (b) V(A)V(B), AB, A, BF.

    It is called to be sub-additive if V(AB)V(A)+V(B) for any A, BF with ABF. Let (Ω,H,ˆE) be a sub-linear expectation space; we define capacities of a pair (V,V) by the following:

    V(A):=inf{ˆE(ξ):IAξ,ξH},V(A)=1V(Ac),AF.

    From the above definition, we have the following:

    ˆE(f1)V(A)ˆE(f2),iff1I(A)f2,f1,f2H. (2.1)

    Because V may be not countably sub-additive in general, we define another capacity V.

    Definition 2.3.[19] A countably sub-additive extension V of V is defined by the following:

    V(A)=inf{n=1V(An):An=1An},V(A)=1V(Ac),AF.

    Then, V is a countably sub-additive capacity with V(A)V(A) and the following properties:

    (a) If V is countably sub-additive, then VV;

    (b) If I(A)g,gH, then V(A)ˆE(g). Furthermore, if ˆE is countably sub-additive, then

    ˆE(f)V(A)V(A)ˆE(g),fI(A)g,f,gH;

    (c) V is the largest countably sub-additive capacity satisfying the property that V(A)ˆE(g)whenever I(A)gH (i.e., if V is also a countably sub-additive capacity satisfying V(A)ˆE(g) whenever I(A)gH, then V(A)V(A)).

    Definition 2.4. [24] In a sublinear expectation space (Ω,H,ˆE), let φ be a monotonically bounded function if for any X,YH that satisfies

    ˆE[φ(X+Y)]ˆE[ˆE[φ(x+Y)]x=X], (2.2)

    then the random variable Y is said to be WND on X under sublinear expectations. {Xi,iZ} is said to be a sequence of WND random variables if Xm is WND on (Xmn,Xmn+1,,Xm1) for any mZ,nN+.

    Remark 2.1. By Chen [15], if {Xn,n1} is a sequence of WND random variables under sublinear expectations, then for any XkH,1kn, we have the following:

    ˆE[exp(nk=1cXk)]nk=1ˆE[exp(cXk)],cR. (2.3)

    Definition 2.5.[3] The Choquet integral of X with respect to V is defined as following:

    CV(X)=0V(Xt)dt+0[V(Xt)1]dt.

    Usually, we denote the Choquet integral of V and V by CV and CV, respectively.

    Definition 2.6.[25] If a sublinear expectation ˆE satisfies ˆE[X]n=1ˆE[Xn]<, then ˆE is said to be countably sub-additive, where Xn=1Xn<, X,XnH, and X,Xn0,n1.

    Next, we need the following notations and lemmas. Let C be a positive constant that takes on different values in different places as needed. I(A) stands for the indicator function of A. Given a capacity V, a set A is said to be a polar set if V(A)=0. Additionally, we say a property holds "quasi-surely" (q.s.) if it holds outside a polar set. In this paper, the capacity V is countably sub-additive and lower continuous. Similar to Hu [10], we let Φc denote the set of nonnegative functions ϕ(x) defined on [0,), and ϕ(x) satisfies the following:

    (1) Function ϕ(x) is positive and nondecreasing on (0,), and the series n=11nϕ(n)<;

    (2) For any x>0 and fixed a>0, there exists C>0 such that ϕ(x+a)Cϕ(x).

    For example, functions (ln(1+x))1+α and xα(α>0) belong to the Φc.

    Lemma 2.1. [8] (Borel-Canteli's Lemma) Let {An,n1} be a sequence of events in F. Suppose that V is a countably sub-additive capacity. If n=1V(An)<, then

    V(An,i.o.)=0,

    where {An,i.o.}=n=1i=nAi.

    Lemma 2.2. Let {X,Xm,m1} be a sequence of random variables under the sublinear expectations space.

    (1) Chebyshev inequality[8]: Function f(x) is positive and nondecreasing on R; then

    V(Xx)ˆE[f(X)]f(x),V(Xx)ˆε[f(X)]f(x).

    (2) Cr inequality [3]: Let X1, X2, , XmH for m1; then

    ˆE|X1+X2++Xm|rCr[ˆE|X1|r+ˆE|X2|r++ˆE|Xm|r],

    where

    Cr={1,0<r1,mr-1,r>1.

    (3) Markov inequality [8]: For any XH, we have

    V(|X|x)ˆE(|X|p)xp,x>0,p>0.

    Lemma 2.3. [26] Let {xm,m1} and {bm,m1} be sequences of real numbers with 0<bm. If the series m=1xmbm<, then limm1bmmi=1xi=0.

    Lemma 2.4. [21] Suppose that ˆE is countably sub-additive; then, for any XH, we have ˆE(|X|)CV(|X|).

    Lemma 2.5. Let {Xni,1ikn,n1} be an array of row-wise random variables under sublinear expectation (Ω,H,ˆE) and supi1CV((|Xni|pc)+)0,c,p(0,2); if ˆE is countably sub-additive for any XniH, then we have supi1ˆE[(|Xni|pc)+]0,c.

    Proof. From Lemma 2.4, we have ˆE(|X|)CV(|X|). Let X=(|Xni|pc)+; then, we have

    supi1ˆE[(|Xni|pc)+]supi1CV((|Xni|pc)+).

    Thus, we get supi1ˆE[(|Xni|pc)+]0,c.

    Lemma 2.6. If {Xni,1ikn,n1} is an array of row-wise random variables under sublinear expectations, and supi1CV((|Xni|pc)+)0,c,p(0,2), then we have the following:

    limnkni=1V(|Xni|pakn)=0,a>0.

    Proof. From the condition supi1CV((|Xni|pc)+)0,c and the definition of a Choquet integral, it follows that for any a>0, we have the following:

    kni=1V(|Xni|pakn)2knkni=1knkn2V(|Xni|pat)dt2supi1knkn2V(|Xni|pat)dt2supi1kn2V(|Xni|pat)dt=2supi10V(1a|Xni|pkn2t)dt=2asupi1CV[(|Xni|pakn2)+].

    When kn, we obtain the following:

    kni=1V(|Xni|pakn)2asupi1CV[(|Xni|pakn2)+]0.

    Thus, the proof of limnkni=1V(|Xni|p>akn)=0 is finished.

    Lemma 2.7. [10] If ˆE|X|<, then |X|<,q.s.V.

    Lemma 2.8. [10] Suppose ϕ(x)Φc; then, n=11nϕ(nln(1+n))<.

    Proof. Since ϕ(x)Φc, we have ϕ(nln(1+n))ϕ(n); it is only necessary to show that n=11nϕ(n)<. From n=11nϕ(n)<, we obtian the following:

    n=11nϕ(n)=i=1i2n<(i+1)21nϕ(n)i=12iϕ(i)+i=11i2ϕ(i)<.

    Then, the Lemma 2.8 is proven.

    Lemma 2.9. [10] If {εi,iZ} is a sequence of random variables, and there exists a constant c>0 such that |εn|2cnln(1+n),n1, supiZˆE[|εi|ϕ(|εi|)]<, ϕ(x)ΦC, and {αi,i0} is a sequence of real numbers, ani=nir=0αr, T=supk0|ak|<, then for any t>1,

    sup1intln(1+n)|ani|ˆE[|εi|ln(1+tln(1+n)n|ani||εi|)]0,n. (2.4)

    Proof. Becase |εn|2cnln(1+n),n1, then

    |εi|ln(1+tln(1+n)n|ani||εi|)=|εi|ln(1+tln(1+n)n|ani||εi|)I(|εi|n13)+|εi|ln(1+tln(1+n)n|ani||εi|)I(n13<|εi|2cnln(1+n)).

    Let I1=|εi|ln(1+tln(1+n)n|ani||εi|)I(|εi|n13), since T=supk0|ak|<, when n, we have

    I1n13ln(1+tTln(1+n)n23)tTln(1+n)n13. (2.5)

    Let I2=|εi|ln(1+tln(1+n)n|ani||εi|)I(n13<|εi|2cnln(1+n)), and l(x)=ϕ(x)ln(1+x); thus, we obtian the following:

    I2|εi|ϕ(|εi|)ln(1+tTln(1+n)n2cnln(1+n))ϕ(n13)|εi|ϕ(|εi|)ln(1+2ctT)ϕ(n13)|εi|ϕ(|εi|)ln(1+2ctT)ln(1+n13)l(n13). (2.6)

    Since ϕ(x)Φc, the function l(x)=ϕ(x)ln(1+x),x; then, combining (2.5) and (2.6), when n, we have the following:

    sup1intln(1+n)|ani|ˆE[|εi|ln(1+tln(1+n)n|ani||εi|)](tT)2(ln(1+n))2n13+sup1inˆE[|εi|ϕ(|εi|)]tTln(1+n)ln(1+2ctT)ln(1+n13)l(n13)(tT)2(ln(1+n))2n13+supiZˆE[|εi|ϕ(|εi|)]tTln(1+n)ln(1+2ctT)ln(1+n13)l(n13)0.

    Thus, the proof is finished.

    Lemma 2.10.[16] Suppose that {αi,i0} is a sequence of real numbers, ani=nir=0αr, T=supk0|ak|<. {εi,iZ} is a sequence of WND random variables under the sublinear expectation space (Ω,H,ˆE), ˆE[εi]=ˉμ, supiZˆE[|εi|ϕ(|εi|)]<, ϕ(x)ΦC, and there exists a constant c>0 such that |εiˉμ|2ciln(1+i), i1; then, for any t1,

    supn1ˆE[exp(tln(1+n)nni=1ani(εiˉμ))]<. (2.7)

    Proof. For any xR, we have the inequality ex1+x+|x|ln(1+|x|)e2|x|. Let x=tln(1+n)nani(εiˉμ); then,

    exp(tln(1+n)nani(εiˉμ))1+tln(1+n)nani(εiˉμ)+|tln(1+n)nani(εiˉμ)|ln(1+|tln(1+n)nani(εiˉμ)|)exp(2tln(1+n)nani(εiˉμ)). (2.8)

    Since T=supk0|ak|<, for any in, we have the following:

    |tln(1+n)nani(εiˉμ)|tln(1+n)nT2ciln(1+i)2ctT. (2.9)

    By supiZˆE[|εi|ϕ(|εi|)]< and ϕ(x+a)Cϕ(x), we have the following:

    supiZˆE[|εiˉμ|ϕ(|εiˉμ|)]supiZˆE[(|εi|+|ˉμ|)ϕ(|εi|+|ˉμ|)]CsupiZˆE[(|εi|+|ˉμ|)ϕ(|εi|)]<.

    Thus, {εiˉμ,iZ}satisfies the conditions of Lemma 2.9; furthermore, we have

    sup1intln(1+n)n|ani|ˆE[|εiˉμ|ln(1+tln(1+n)n|ani||εiˉμ|)]Cn. (2.10)

    Taking ˆE for both sides of (2.8) and combining (2.9) and (2.10), we have the following:

    ˆE[exp(tln(1+n)nani(εiˉμ))]1+Cne4ctTeCne4ctT.

    From (2.3), we obtain the following:

    ˆE[exp(tln(1+n)nni=1ani(εiˉμ))]ni=1ˆE[exp(tln(1+n)nani(εiˉμ))](eCne4ctT)neCe4ctT<.

    Theorem 3.1. Let {kn,n1} be a sequence of positive numbers, and limnkn=. Assume that ˆE is countably sub-additive. For any i,n1, ˆE[Xni]=ˉμni, ˆE[Xni]=μ_ni.

    (1) Let {Xni,1ikn,n1} be an array of row-wise random variables under the sublinear expectation (Ω,H,ˆE). Suppose that supi1CV((|Xni|pc)+)0,c for any p(0,1); then,

    limnV(1(kn)1p|kni=1Xni|ε)=0. (3.1)

    (2) Let {Xni,1ikn,n1} be an array of row-wise WND random variables under sublinear expectation (Ω,H,ˆE). Suppose that supi1CV((|Xni|pc)+)0,c for any p[1,2); then,

    limnV({1(kn)1pkni=1Xni1(kn)1pkni=1ˉμni+ε}{1(kn)1pkni=1Xni1(kn)1pkni=1μ_niε})=0. (3.2)

    For a fixed n1 in Theorem 3.1, we obtain the Corollary 3.1.

    Corollary 3.1. Assume that ˆE is countably sub-additive.

    (1) Let {Xi,i1} be a sequence of random variables under the sublinear expectation space (Ω,H,ˆE). Suppose that supi1CV((|Xi|pc)+)0,c for any p(0,1); then,

    limnV(1n1p|ni=1Xi|ε)=0. (3.3)

    (2) Let {Xi,i1} be a sequence of WND random variables under the sublinear expectation space (Ω,H,ˆE) and for any i1,ˆE[Xi]=ˉμi,ˆE[Xi]=μ_i. Suppose that supi1CV((|Xi|pc)+)0,c for any p[1,2); then,

    limnV({1n1pni=1Xi1n1pni=1ˉμi+ε}{1n1pni=1Xi1n1pni=1μ_iε})=0. (3.4)

    Theorem 3.2. Suppose that ˆE is countably sub-additive. Let {αi,i0} be a sequence of real numbers satisfying i=0i|αi|<,i=0αi=A>0, and {εi,iZ} be a sequence of WND random variables under sublinear expectations satisfying ˆE[εi]=ˉμ,ˆE[εi]=μ_, supiZˆE[|εi|ϕ(|εi|)]<,ϕΦC. {Xt,t1} is a sequence of linear processes satisfying Xt=i=0αiεti. Note that Tn=nt=1Xt; then,

    V({lim infnTnn<Aμ_}{lim supnTnn>Aˉμ})=0. (3.5)

    Remark 3.1. Under the sub-linear expectations, the main purpose of Theorem 3.1 is to extend the range of p and improve the result of Fu [24] from the Kolmogorov type weak law of large numbers to the Marcinkiewicz-Zygmund type weak law of large numbers.

    Remark 3.2. Under the sub-linear expectations, the main purpose of Theorem 3.2 is to improve the result of Zhang [16] from IID random variables to WND random variables under a more general moment condition.

    The proof of Theorem 3.1. (1) For a fixed constant c, let Yni=((c)Xni)c and Zni=XniYni. Using the Cr inequality and the Markov inequality in Lemma 2.2, we obtain the following:

    V(1(kn)1p|kni=1Xni|>ε)V(kni=1|Yni|(kn)1pε2)+V(kni=1|Zni|(kn)1pε2)V(c(kn)1p1ε2)+2pknεpˆE[(kni=1|Zni|)p]V(c(kn)1p1ε2)+2pknεpkni=1ˆE[|Zni|p]V(c(kn)1p1ε2)+2pεpsupi1ˆE[|Zni|p].

    Thus,

    limnV(1(kn)1p|kni=1Xni|>ε)2pεpsupi1ˆE[|Zni|p]. (4.1)

    Therefore,

    |Zni|p=|Zni|pI(|Xni|c)+|Zni|pI(|Xni|c)=|Zni|pI(Xni>c)+|Zni|pI(Xni<c)=|Xnic|pI(Xni>c)+|Xni+c|pI(Xni<c)(|Xni|c)pI(|Xni|>c)C(|Xni|pc)+.

    Taking ˆE for both sides of the above inequality, when c, we have the following:

    supi1ˆE[|Zni|p]Csupi1ˆE((|Xni|pc)+)Csupi1CV((|Xni|pc)+)0. (4.2)

    Substituting (4.2) into (4.1), we get that (3.1) holds.

    (2) When 1p<2, we construct a function Ψ(y)C2b(R); for any ε>0, we have Ψ(y)=0 when y0, 0<Ψ(y)<1 when 0<y<ε, and Ψ(y)=1 when yε. It is obvious that I(yε)Ψ(y). Let Yni=Xniˉμni; then, we have the following:

    V(1(kn)1pkni=1Yniε)ˆE[Ψ(1(kn)1pkni=1Yni)]=knm=1{ˆE[Ψ(1(kn)1pmi=1Yni)]ˆE[Ψ(1(kn)1pm1i=1Yni)]}. (4.3)

    Let h(y)=ˆE[Ψ(y+Ynm(kn)1p)]; by Definition 2.4 and the sub-additivity of ˆE, then we obtain the following:

    ˆE[Ψ(1(kn)1pmi=1Yni)]ˆE[Ψ(1(kn)1pm1i=1Yni)]ˆE[ˆE[Ψ(y+Ynm(kn)1p)]y=1(kn)1pm1i=1Yni]ˆE[Ψ(1(kn)1pm1i=1Yni)]=ˆE[h(1(kn)1pm1i=1Yni)]ˆE[Ψ(1(kn)1pm1i=1Yni)]ˆE[h(1(kn)1pm1i=1Yni)Ψ(1(kn)1pm1i=1Yni)]supyR{h(y)Ψ(y)}=supyR{ˆE[Ψ(y+Ynm(kn)1p)]Ψ(y)}=supyRˆE[Ψ(y+Ynm(kn)1p)Ψ(y)]. (4.4)

    Let g(x)Cl,Lip(R); for any x, we have 0g(x)1, g(x)=1 when |x|μ, and g(x)=0 when |x|>1. Then, we have the following:

    I(|x|μ)g(x)I(|x|1),I(|x|>1)1g(x)I(|x|>μ). (4.5)

    For any 1mkn, there exist λnm,ˉλnm[0,1] such that

    Ψ(y+Ynm(kn)1p)Ψ(y)=Ψ(y)Ynm(kn)1p+(Ψ(y+λnmYnm(kn)1p)Ψ(y))Ynm(kn)1p,Ψ(y+λnmYnm(kn)1p)Ψ(y)=Ψ(y+λnmˉλnmYnm(kn)1p)λnmYnm(kn)1p. (4.6)

    Since Ψ(y)C2b(R), then we have |Ψ(y)|supyR|Ψ(y)|C |Ψ(y)|supyR|Ψ(y)|C and |Ψ(y)|supyR|Ψ(y)|C. Combining (4.5), (4.6), and the Cr-inequality in Lemma 2.2, then for any δ>0, we have the following:

    Ψ(y+Ynm(kn)1p)Ψ(y)Ψ(y)Ynm(kn)1p+|Ψ(y+λnmYnm(kn)1p)Ψ(y)||Ynm|(kn)1pCYnm(kn)1p+|Ψ(y+λnmYnm(kn)1p)Ψ(y)||Ynm|(kn)1pI(|Xnm|>δ(kn)1p)+|Ψ(y+λnmˉλnmYnm(kn)1p)||λnm||Ynm|2(kn)2pI(|Xnm|δ(kn)1p)CYnm(kn)1p+2C(kn)1p|Xnm|I(|Xnm|>δ(kn)1p)+2C(kn)1p|ˉμnm|I(|Xnm|>δ(kn)1p)+2C(kn)2p|Xnm|2I(|Xnm|δ(kn)1p)+2C(kn)2p|ˉμnm|2I(|Xnm|δ(kn)1p)CYnm(kn)1p+2Cknδp1|Xnm|pI(|Xnm|>δ(kn)1p)+2C|ˉμnm|(kn)1p+1δp|Xnm|p+2Cδ2pkn|Xnm|p+2C(kn)2p|ˉμnm|2CYnm(kn)1p+2Cknδp1[(|Xnm|pkn)++knI(|Xnm|>δ(kn)1p)]+2C|ˉμnm|(kn)1p+1δp|Xnm|p+2Cδ2pkn|Xnm|p+2C(kn)2p|ˉμnm|2CYnm(kn)1p+2Cknδp1(|Xnm|pkn)++2Cδp1I(|Xnm|>δ(kn)1p)+2C|ˉμnm|(kn)1p+1δp|Xnm|p+2Cδ2pkn|Xnm|p+2C(kn)2p|ˉμnm|2CYnm(kn)1p+2Cknδp1(|Xnm|pkn)++2Cδp1(1g(Xnmδ(kn)1p))+2C|ˉμnm|(kn)1p+1δp|Xn,m|p+2Cδ2pkn|Xnm|p+2C(kn)2p|ˉμnm|2. (4.7)

    Substituting (4.4), (4.7), into (4.3), then combining (2.1) and (4.5), we obtain the following:

    V(1(kn)1pkni=1Yniε)2Cδp1supm1ˆE(|Xnm|pkn)++2Cδp1knm=1V(|Xnm|p>μpδpkn)+2C|ˉμnm|(kn)1pδpsupm1CV(|Xnm|p)+2Cδ2psupm1CV(|Xnm|p)+2C(kn)2p1|ˉμnm|2.

    Taking the limit of the above inequality at both sides, then by Lemma 2.6, we obtain

    limnV(1(kn)1pkni=1Yniε)=2Cδ2psupm1CV(|Xnm|p).

    Because supm1CV((|Xnm|c)+)0,c means supm1CV(|Xnm|p)<, and from the arbitrariness of δ, we obtain the following:

    limnV(1(kn)1pkni=1Xni1(kn)1pkni=1ˉμni+ε)=0. (4.8)

    Similarly, for {Xni,1ikn,n1}, we obtain the following:

    limnV(1(kn)1pkni=1Xni1(kn)1pkni=1μ_niε)=0. (4.9)

    Using the sub-additivity of V and combining (4.8) and (4.9), we obtain the following:

    limnV({1(kn)1pkni=1Xni1(kn)1pkni=1ˉμni+ε}{1(kn)1pkni=1Xni1(kn)1pkni=1μ_niε})=0.

    The proof of Theorem 3.1 is completed.

    The proof of Theorem 3.2. To prove Theorem 3.2, we only need to show that

    V(lim supnTnn>Aˉμ)=0, (4.10)

    and

    V(lim infnTnn<Aμ_)=0. (4.11)

    First, we prove Eq (4.10); then, we need to show that

    V(lim supnTnn>Aˉμ+ϵ)=0,ϵ>0.

    It is obvious that

    Tn=nt=1Xt=nt=1i=0αiεti=nt=1i=tαiεti+ni=1εinit=0αt:=Nn+Mn.

    It is only necessary to show that

    limnNnn=0,q.s.V, (4.12)

    and

    V(lim supnMnn>Aˉμ+ϵ)=0,ϵ>0. (4.13)

    To prove(4.12), we need to prove limti=tαiεti=0,q.s.V.

    For any ϵ>0, using the Chebyshev inequality in Lemma 2.2, and the countable sub-additivity of ˆE, we obtain the following:

    t=1V(|i=tαiεti|>ϵ)=t=1ˆE[|i=tαiεti|]ϵ1ϵt=1i=t|αi|ˆE|εti|1ϵsupiZˆE|εi|t=1i=t|αi|=1ϵsupiZˆE|εi|i=1i|αi|<.

    By Lemma 2.1, it follows that

    V(lim supt|i=tαiεti|>ϵ)=0.

    Therefore, by the arbitrariness of ϵ, it follows that

    limti=tαiεti=0,q.s.V.

    Thus, (4.12) holds. Let ani=nir=0αr and T=supk0|ak|<; we prove Eq (4.13) in two steps.

    Step 1: If for any i1 we have |εiˉμ|2ciln(1+i),c>0, then we can directly utilize the conclusion of Lemma 2.10; for any t1, we have the following:

    supn1ˆE[exp(tln(1+n)nni=1ani(εiˉμ))]<.

    Since limnnk=1ankn=A, then V(lim supnni=1ani(εiˉμ)n>ϵ)=0 is equivalent to (4.13). Choosing a suitable t, such that t>1ϵ, using the Chebyshev inequality in Lemma 2.2, we have the following:

    V(ni=1ani(εiˉμ)nϵ)=V(tln(1+n)ni=1ani(εiˉμ)nϵtln(1+n))1(1+n)ϵtsupn1ˆE[exp(tln(1+n)nni=1ani(εiˉμ))].

    By Lemma 2.10 and the convergence of infinite series n=11(1+n)ϵt, we obtain the following:

    n=1V(ni=1ani(εiˉμ)nϵ)n=11(1+n)ϵtsupn1ˆE[exp(tln(1+n)nni=1ani(εiˉμ))]<.

    By Lemma 2.1, it follows that

    V(lim supnni=1ani(εiˉμ)n>ϵ)=0.

    Therefore, (4.13) is proven.

    Step 2: Assume that {εi,iZ} only satisfies the conditions of Theorem 3.2. Let g(x)Cl,Lip(R); for any x, we have 0g(x)1, g(x)=1 when |x|μ, and g(x)=0 when |x|>1. Then we have the following:

    I(|x|μ)g(x)I(|x|1),I(|x|>1)1g(x)I(|x|>μ). (4.14)

    Let ˜εi=ˆE[(εiˉμ)g(μ(εiˉμ)ln(1+i)i)]+(εiˉμ)g(μ(εiˉμ)ln(1+i)i)+ˉμ; for any i1, we have ˆE(˜εi)=ˉμ and |˜εiˉμ|2ciln(1+i). Then, {˜εi,i1} satisfies the conditions of Lemma 2.10. Let ˜Mn=ni=1ani˜εi; similar to the proof of step 1, we obtain the following:

    V(lim supn˜Mnn>Aˉμ+ϵ)=0,ϵ>0. (4.15)

    By the definition of ˜εi, we have the following:

    εi=˜εi+ˆE[(εiˉμ)g(μ(εiˉμ)ln(1+i)i)]+(εiˉμ)[1g(μ(εiˉμ)ln(1+i)i)].

    Since T=supk0|ak|<, then we have the following:

    Mnn˜Mnn+Tnni=1ˆE[(εiˉμ)g(μ(εiˉμ)ln(1+i)i)]+Tnni=1(εiˉμ)[1g(μ(εiˉμ)ln(1+i)i)]. (4.16)

    Note that

    ˆE[(εiˉμ)g(μ(εiˉμ)ln(1+i)i)]ˆE[|εiˉμ|(1g(μ(εiˉμ)ln(1+i)i))]. (4.17)

    Substituting (4.17) into (4.16), we only need to prove

    limn1nni=1ˆE[|εiˉμ|(1g(μ(εiˉμ)ln(1+i)i))]=0, (4.18)

    and

    limn1nni=1|εiˉμ|[1g(μ(εiˉμ)ln(1+i)i)]=0,q.s.V. (4.19)

    By (4.14), we have the following:

    |εiˉμ|[1g(μ(εiˉμ)ln(1+i)i)]|εiˉμ|I(|εiˉμ|>iln(1+i))|εiˉμ|ϕ(|εiˉμ|)ϕ(iln(1+i)).

    Then, combining supiZˆE[|εiˉμ|ϕ(|εiˉμ|)]< and Lemma 2.8, we obtain the following:

    i=11iˆE[|εiˉμ|(1g(μ(εiˉμ)ln(1+i)i))]supiZˆE[|εiˉμ|ϕ(|εiˉμ|)]i=11iϕ(iln(1+i))<.

    By Lemma 2.3, (4.18) holds.

    Since ˆE is countably sub-additive, we have the following:

    ˆE[i=11i|εiˉμ|(1g(μ(εiˉμ)ln(1+i)i))]i=11iˆE[|εiˉμ|(1g(μ(εiˉμ)ln(1+i)i))]<.

    From Lemma 2.7, we obtain the following:

    i=11i|εiˉμ|(1g(μ(εiˉμ)ln(1+i)i))<,q.s.V.

    By Lemma 2.3, (4.19) holds. Combining (4.14), (4.18), and (4.19), it follows that (4.13) holds.

    Similarly, for {εi,iZ}, and ˆE(εi)=ˉμ, we obtain the following:

    V(lim infnTnn<Aμ_)=0.

    Using the sub-additivity of V, the proof of Theorem 3.2 is completed.

    In the framework of sublinear expectations, we established the Marcinkiewicz-Zygmund type weak law of large numbers, and the strong law of large numbers for WND random variables using the Chebyshev inequality, the Cr inequality, and so on. Theorem 3.1 extends the result of Fu[24] from the Kolmogorov type weak law of large numbers to the Marcinkiewicz-Zygmund type weak law of large numbers. Theorem 3.2 extends the result of Zhang[16] from IID random variables to WND random variables under a more general moment condition. In the future, we will try to develop broader results for other sequences of dependent random variables under sublinear expectations.

    Yuyan Wei: conceptualization, formal analysis, investigation, methodology, writing-original draft, writing-review and editing; Xili Tan: funding acquisition, project administration, supervision; Peiyu Sun: formal analysis, writing-review and editing; Shuang Guo: writing-review and editing. All authors have read and approved the final version of the manuscript for publication.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    This paper was supported by the Department of Science and Technology of Jilin Province (Grant No.YDZJ202101ZYTS156), and Graduate Innovation Project of Beihua University (2023004).

    All authors declare no conflicts of interest in this paper.


    Acknowledgments



    Author acknowledges Profs. Marcello Alecci for general supervision of the research group and proofreading, Angelo Galante for stimulating discussions and Giuseppe Di Giovanni for encouraging comments and proofreading.

    Funding



    University of L'Aquila RIA 2019 research funds.

    Conflict of interest



    Author declares no conflicts of interest in this paper.

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