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The Faber polynomial expansion method and the Taylor-Maclaurin coefficient estimates of Bi-Close-to-Convex functions connected with the q-convolution

  • In this paper, we introduce a new class of analytic and bi-close-to-convex functions connected with q-convolution, which are defined in the open unit disk. We find estimates for the general Taylor-Maclaurin coefficients of the functions in this subclass by using the Faber polynomial expansion method. Several corollaries and consequences of our main results are also briefly indicated.

    Citation: H. M. Srivastava, Sheza M. El-Deeb. The Faber polynomial expansion method and the Taylor-Maclaurin coefficient estimates of Bi-Close-to-Convex functions connected with the q-convolution[J]. AIMS Mathematics, 2020, 5(6): 7087-7106. doi: 10.3934/math.2020454

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  • In this paper, we introduce a new class of analytic and bi-close-to-convex functions connected with q-convolution, which are defined in the open unit disk. We find estimates for the general Taylor-Maclaurin coefficients of the functions in this subclass by using the Faber polynomial expansion method. Several corollaries and consequences of our main results are also briefly indicated.


    The limit theorem plays a pivotal role in the study of probability theory. Furthermore, the almost sure convergence is integral to the development of the limit theorem, a subject many scholars have studied. So far, a lot of excellent results have been obtained under the condition that the model holds with certainty. However, many uncertain phenomena of quantum mechanics and risk management cannot be explained by additive probability or expectation. To deal with this issue, many scholars have made great attempts and efforts. In particular, Peng [1,2] proposed the theory frame of the sub-linear expectations under a generic function space to solve this distributional uncertainty. In recent years, based on Peng, more and more scholars in the industry have done extensive research and obtained many related results; the study of the almost sure convergence has remained a hot-button issue. For example, Chen [3], Cheng [4], and Feng and Lan [5] obtained the SLLN (strong law of large numbers) of i.i.d.r.v. (independent identically distributed random variables), and Cheng [6] studied the SLLN of independent r.v. with supi1ˆE[|Xi|l(|Xi|)]<. Through further research, Wu and Jiang [7] obtained the SLLN of the extended independent and identically distributed r.v.; Chen and Liu [8], Gao et al. [9], and Liang and Wu [10] proved the SLLN of ND (negatively dependent) r.v.; Zhang [11] built the exponential inequality and the law of logarithm of independent and ND r.v.; Wang and Wu [12] and Feng [13] offered the almost sure covergence for weighted sums of ND r.v.; Zhang [14] derived the SLLN of the extended independent and END (extended negatively dependent) r.v.; Wang and Wu [15] obtained the almost sure convergence of END r.v.; Lin [16] achieved the SLLN of WND (widely negative dependence) r.v.; and Hwang [17] investigated the almost sure convergence of WND r.v..

    Anna [18] proposed the definition of WNOD r.v. for the first time and obtained the limiting conclusions for WNOD r.v. in Peng's theory frame. Based on Yan's results [19], this paper promotes them to the sub-linear expectation space. Compared to the previously mentioned ND and END r.v., dominating coefficients g(n) have been added to the definition of WNOD r.v., leading to a broader range. Besides, the sub-additivity property of the sub-linear expectation and capacity is added, making the research more meaningful and complex. Finally, the conclusions of almost sure convergence for WNOD r.v. are achieved. This paper contributes to the relevant research results of limiting behavior of WNOD r.v. in Peng's theory frame.

    Our essay is arranged as follows: Section 2 recommends interrelated definitions and properties as well as some important lemmas in the frame. Section 3 gives the conclusions including two theorems and two corollaries. Section 4 shows that the process of proving the conclusions is given in detail.

    Running through this essay, we point out that c will be a positive constant, its value is not important, and it may take different values according to the situation. axbx means limxaxbx=1. anbn means there must be a positive number c, satifying ancbn when n is large enough. Denote log(y)=ln(max{e,y}).

    This article uses the theory frame and concepts proposed by Peng [1,2]. Suppose (Ω,F) is a given measurable space and H is a linear space of real functions defined on (Ω,F) so that if X1,X2,,XnH, then φ(X1,X2,,Xn)H for every φCl, Lip (Rn), where φCl, Lip (Rn) shows the linear space of (local Lipschitz) functions φ satisfying

    |φ(x)φ(y)|c(1+|x|m+|y|m)|xy|,x,yRn,

    for some c>0,mN depending on φ. H is considered as a space of random variable. In this circumstance, we denote XH.

    Definition 2.1. (Peng [1]). A sub-linear expectation ˆE on H is a function ˆE:H[,] satisfying the following properties: for all X,YH, we have

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

    (b) Constant preserving: ˆE(c)=c;

    (c) Sub-additivity: ˆE(X+Y)ˆE(X)+ˆE(Y);

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

    The triple (Ω,H,ˆE) is known as a sub-linear expectation space.

    Next, give the definition of the conjugate expectation ˆE of ˆE by

    ˆE(X):=ˆE(X),XH.

    By the above definitions of ˆE and ˆE, the following inequality is feasible for all X,YH,

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

    When we are talking about ˆE and ˆE in the course of the proof, we often use the above formula.

    Definition 2.2. (Peng [1]). Make GF, a function V:G[0,1] is described to be a capacity, when

    V()=0,V(Ω)=1 and V(A)V(B) for AB,A,BG.

    Similar to sub-linear expectations, it is known as sub-additive when V(AB)V(A)+V(B) for every A,BG. Now, represent V and V, respectively corresponing to ˆE and ˆE, using

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

    where Ac denotes the complement set of A.

    From the definion and sub-additivity property of (V, V), the following formulas are true

    ˆEζV(C)ˆEη,ˆEζV(C)ˆEη,ifζI(C)η,ζ,ηH.

    And now we have Markov inequality:

    V(|Y|y)ˆE|Y|P/yp,y>0,p>0.

    Definition 2.3. (Peng [1]). The Choquet integrals (CV) is defined as follows

    CV(X)=0V(Xx)dx+0[V(Xx)1]dx,

    where V and V can replace V when required.

    Definition 2.4. (Zhang [11]). (ⅰ) ˆE is referred to be countably sub-additive, when

    ˆE(X)n=1ˆE(Xn), whenever Xn=1Xn,X,XnH,X0,Xn0,n1.

    (ⅱ)V is referred to be countably sub-additive when

    V(n=1An)n=1V(An),AnF.

    Definition 2.5. {Xn,n1} is a sequence of r.v. and it is known to be stochastically dominated by a random variable X if for a positive number c, there has

    ˆE[f(|Xn|)]cˆE[f(|X|)],forn1,0fCl,Lip(R).

    Definition 2.6. (Anna [18]. Widely negative orthant dependent (WNOD)) {Xn,n1} is called to be widely negative orthant dependent if there is a finite positive array {g(n),n1} satisfying for every n1,

    ˆE(ni=1φi(Xi))g(n)ni=1ˆE (φi(Xi)), 

    where φiCb,Lip(R),φi0,1in and all functions φi are uniformly monotonous. Where the coefficients g(n) (n1) are known as dominating coefficients.

    It is visible that, when {Xn,n1} is widely negative orthant dependent and all functions fk(x)Cl,Lip(R) (where k=1,2,,n) are uniformly monotonous, then {fn(Xn),n1} is also widely negative orthant dependent.

    Definition 2.7. (Seneta [20]). (ⅰ) A positive function l(x) defined on [a,),a>0 is known to be a slowly varying function, satisfying

    limxl(tx)l(x)=1,foreacht0.

    (ⅱ) Each slowly varying function l(x) can be expressed as

    l(x)=C(x)exp{x1f(y)ydy},

    whenever limxC(x)=c>0, as well as limyf(y)=0.

    In this article, we want to research the almost sure convergence of WNOD sequence under sub-linear expectations. Since V is only sub-addictive, the definition of almost sure convergence is a little different and is described in detail in Wu and Jiang [7].

    Next, we give some lemmas before reaching our conclusions.

    Lemma 2.1. (Seneta [20]). For α>0, there is a non-decreasing function φ(x) and a non-increasing function ξ(x) such that

    xαl(x)φ(x),xαl(x)ξ(x),x,

    where l(x) is a slowly varying function.

    In the following section, we assume l(x),x>0 is a non-decreasing slowly varying function that can be expressed as l(x)=cexp{x1f(y)ydy}, where c>0, limxf(x)=0.

    Let

    τn=nl(n)1,n1. (2.1)
    Xn=τ1/pnI(Xn<τ1/pn)+XnI(|Xn|τ1/pn)+τ1/pnI(Xn>τ1/pn). (2.2)
    Xn=XnXn=(Xn+τ1/pn)I(Xn<τ1/pn)+(Xnτ1/pn)I(Xn>τ1/pn). (2.3)

    Lemma 2.2. Assume XH,0<p<2,τn defined by Eq (2.1).

    (ⅰ) For every c>0,

    CV(|X|p)<n=1l1(n)V(|X|p>cτn)<. (2.4)

    (ⅱ) When CV(|X|p)<, and now for every c>0,

    k=12kl(2k)V(|X|p>cτ2k)<. (2.5)

    Proof. (ⅰ) Obviously,

    CV(|X|p)<CV(|X|p/c)<.
    CV(|X|p/c)1V(|X|p>cx)dx1l(y)yl(y)f(y)yl2(y)V(|X|p>cyl(y))dy(makex=yl(y))11l(y)V(|X|p>cτy)dy.

    So

    CV(|X|p)<n=1l1(n)V(|X|p>cτn)<.

    (ⅱ) For every positive c, using the conclusion of (ⅰ), because of the monotonically increasing property of l(x),

    >n=1l1(n)V(|X|p>cτn)=k=12k1n<2kl1(n)V(|X|p>cτn)k=12k1n<2kl1(2k)V(|X|p>cτ2k)=21k=12kl1(2k)V(|X|p>cτ2k).

    As such, we have completed the proof of (ⅱ).

    Lemma 2.3. {Xn,n1} is a sequence of random variables, as well as stochastically dominated by a r.v. X and CV(|X|p)<,1p<2, ˆE has countable sub-additivity, then

    n=1τ2/pnl1(n)ˆE(Xn)2<, (2.6)

    moreover, when 1<p<2,

    n=1τ1/pnl1(n)ˆE|Xn|<. (2.7)

    Where Xn,Xn are respectively defined by Eqs (2.2) and (2.3).

    Proof. For 0<μ<1, assume an even function h(x)Cl,Lip(R) and h(x) when x>0, so that the value of h(x) is [0,1], for xR and h(x)1 when |x|μ, h(x)0 when |x|>1. We have

    I(|x|μ)h(|x|)I(|x|1),I(|x|>1)1h(x)I(|x|>μ). (2.8)

    For α=1,2,

    |Xk|α=|Xk|αI(|Xk|τ1/pk)+τα/pkI(|Xk|>τ1/pk)|Xk|αh(μ|Xk|τ1/pk)+τα/pk(1h(|Xk|τ1/pk)). (2.9)
    |Xk|α=|Xk+τ1/pk|αI(Xk<τ1/pk)+|Xkτ1/pk|αI(Xk>τ1/pk)=||Xk|+τ1/pk|αI(Xk<τ1/pk)+||Xk|τ1/pk|αI(Xk>τ1/pk)=||Xk|τ1/pk|αI(|Xk|>τ1/pk)|Xk|αI(|Xk|>τ1/pk)|Xk|α(1h(|Xk|τ1/pk)). (2.10)

    So, by (2.8) and Definition 2.7,

    ˆE|Xk|αˆE|Xk|αh(μ|Xk|τ1/pk)+τα/pkˆE(1h(|Xk|τ1/pk))ˆE|X|αh(μ|X|τ1/pk)+τα/pkˆE(1h(|X|τ1/pk))ˆE|X|αh(μ|X|τ1/pk)+τα/pkV(|X|>μτ1/pk). (2.11)
    ˆE|Xk|αˆE|X|α(1h(|X|τ1/pk)). (2.12)

    Assume that hj(x)Cl,Lip(R),j1, consider that the value of hj(x) is [0,1] for xR. hj(x)1 when τ1/p2j1<|x|τ1/p2j; hj(x)0 when |x|μτ1/p2j1 or |x|>(1+μ)τ1/p2j. The following formulas can be derived,

    I(τ1/p2j1<|x|τ1/p2j)hj(|x|)I(μτ1/p2j1<|x|(1+μ)τ1/p2j). (2.13)
    |X|rh(|X|τ1/p2k)1+kj=1|X|rhj(|X|),r>0. (2.14)
    |X|r(1h(|X|τ1/p2k))j=k|X|rhj(|X|μ),r>0. (2.15)

    First, prove (2.6). For 1p<2, by (2.11) and (2.4),

    H1:=n=1τ2/pnl1(n)ˆE(Xn)2n=1τ2/pnl1(n)[ˆE(X2h(μ|X|τ1/pn))+τ2/pnV(|X|>μτ1/pn)]=n=1τ2/pnl1(n)ˆE[X2h(μ|X|τ1/pn)]+n=1l1(n)V(|X|>μτ1/pn)n=1τ2/pnl1(n)ˆE[X2h(μ|X|τ1/pn)].

    Then, because h(x) is decreasing in (0,), according to Lemma 2.1, τ2/pnl1(n) is decreasing in (0,). So,

    H1k=12k1n<2kτ2/pnl1(n)ˆE[X2h(μ|X|τ1/pn)]k=12k1n<2kτ2/p2k1l1(2k1)ˆE[X2h(μ|X|τ1/p2k)]k=12k1n<2kτ2/p2kl1(2k)ˆE[X2h(μ|X|τ1/p2k)]k=12kτ2/p2kl1(2k)ˆE[X2h(μ|X|τ1/p2k)].

    Last by (2.14), (2.13), and (2.5),

    H1k=12kτ2/p2kl1(2k)+k=12kτ2/p2kl1(2k)kj=1ˆE(X2hj(μ|X|))j=1k=j2kτ2/p2kl1(2k)ˆE(X2hj(μ|X|))j=12jτ2/p2jl1(2j)τ2/p2jV(|X|>τ1/p2j1)j=12jl(2j)V(|X|>τ1/p2j)<.

    Therefore, (2.6) holds.

    Next, our proof of (2.7) is similar to (2.6). For 1<p<2, by (2.12) and the monotonically decreasing propety of h(x) in (0,), according to Lemma 2.1, τ1/pnl1(n) is decreasing in (0,), we have,

    H2:=n=1τ1/pnl1(n)ˆE|Xn|n=1τ1/pnl1(n)ˆE[|X|(1h(|X|τ1/pn))]=k=12k1n<2kτ1/pnl1(n)ˆE[|X|(1h(|X|τ1/pn))]k=12kτ1/p2k1l1(2k1)ˆE[|X|(1h(|X|τ1/p2k1))]k=12kτ1/p2kl1(2k)ˆE[|X|(1h(|X|τ1/p2k))].

    Then, from (2.15), (2.13), and (2.5), countable sub-additivity of ˆE,

    H2k=12kτ1/p2kl1(2k)j=kˆE(|X|hj(|X|μ))=j=1jk=12kτ1/p2kl1(2k)ˆE(|X|hj(|X|μ))j=12jτ1/p2jl1(2j)τ1/p2jV(|X|>μ2τ1/p2j1)j=12jl(2j)V(|X|>μ2τ1/p2j)<.

    Therefore, (2.7) holds.

    Lemma 2.4. (Zhang [11] Borel-Cantelli Lemma) Suppose {Bn;n1} is an array of matters in F. Suppose V has countable sub-additivity. We can obtain V(Bn;i.o.)=0 provided that n=1V(Bn)<, where (Bn;i.o.)=n=1m=nBm.

    Theorem 3.1. Suppose {Xn,n1} is a sequence of WNOD r.v., and its dominating coefficients are g(n). The sequence is stochastically dominated by a r.v. X. ˆE and V both have countable sub-additivity, and satisfying

    CV(|X|p)<,1<p<2. (3.1)

    Make {ank,1kn,n1} be a positive sequence according to

    max1knank=O(τ1/pnl1(n)),n, (3.2)

    where τn is defined by (2.1).

    If for some 0<δ<1,

    n=1e(δ2)l(n)g(n)<, (3.3)

    then,

    lim supnnk=1ank(XkˆEXk)0a.s.V, (3.4)
    lim infnnk=1ank(XkˆEXk)0a.s.V, (3.5)

    in particular, when ˆEXk=ˆEXk, then

    limnnk=1ank(XkˆEXk)=0a.s.V. (3.6)

    Remark 3.1. Theorem 3.1 under sub-linear expectations space is an extension of Theorem 2.1 of Yan [19] of the classical probability space.

    Remark 3.2. If g(n)=M, for each n1, then the sequence is simplified to END. When let l(n)=logn,n1, for 0<δ<1,

    n=1e(δ2)l(n)g(n)=Mn=1n(2δ)<,

    condition (3.3) is satisfied. By Theorem 3.1, Eqs (3.4)–(3.6) hold.

    Remark 3.3. We can obtain different conclusions by taking different forms of slowly varying function l(x). By taking l(n)=logn and l(n)=exp{(logn)ν}(0<ν<1), we will get the following two corollaries.

    Corollary 3.1. Suppose {Xn,n1} is a sequence of WNOD r.v., and its dominating coefficients are g(n). The sequence is stochastically dominated by a r.v. X. Besides, the sequence is satisfied (3.1). ˆE and V both have countable sub-additivity. Make sure {ank,1kn,n1} is a positive sequence according to

    max1knank=O(1n1/plog11/pn),n. (3.7)

    For some 0<b<1δ,

    g(n)nbc, (3.8)

    then (3.4)–(3.6) hold.

    Corollary 3.2. Suppose {Xn,n1} is a sequence of WNOD random variables, and its dominating coefficients are g(n). The sequence is stochastically dominated by a r.v. X. Besides, the sequence is satisfied (3.1). ˆE and V both have countable sub-additivity. Make sure {ank,1kn,n1} is a positive sequence according to

    max1knank=O(n1/pe(1+1/p)(logn)ν),n, (3.9)

    where 0<ν<1.

    For some m>0,

    g(n)nmc, (3.10)

    then (3.4)–(3.6) hold.

    Then, we will think about the situation of p=1.

    Theorem 3.2. Suppose {Xn,n1} is a sequence of WNOD r.v., and its dominating coefficients are g(n) and are satisfied (3.3). The sequence is stochastically dominated by a r.v. X. ˆE and V both have countable sub-additivity, and satisfying

    CV(|X|log|X|)<. (3.11)

    Suppose {ank,1kn,n1} is a positive sequence according to

    max1knank=O(n1),n, (3.12)

    then (3.4)–(3.6) hold.

    Because the sequence {Xk,k1} fulfills the criterion of Theorem 3.1, making {Xk,k1} as a substitute for {Xk,k1} in formula (3.4), by ˆEX=ˆE(X), there is

    0lim supnnk=1ank((Xk)ˆE(Xk))=lim supnnk=1ank((Xk)+ˆEXk)=lim supnnk=1ank((XkˆEXk)).
    lim infnnk=1ank(XkˆEXk)0.

    Therefore, (3.5) holds. Then, by ˆEXk=ˆEXk, (3.4) and (3.5), we can get (3.6). So we just need to prove (3.4).

    We denote Xn, Xn respectively by equations (2.2) and (2.3). By Definition 2.6, {XkˆEXk,k1} is also WNOD. Denote ˜Xk:=XkˆEXk.

    Therefore,

    nk=1ank(XkˆEXk)=nk=1ank˜Xk+nk=1ankXk+nk=1ank(ˆEXkˆEXk):=I1+I2+I3.

    So, if we want to prove (3.4), just prove

    lim supnIi0a.s.V,i=1,2,andlimnI3=0. (4.1)

    By (3.2) and the formula ex1+x+x22e|x|,x[,], for all t>0,1kn as well as large enough n,

    exp{tank˜Xk}1+tank˜Xk+t2a2nk(˜Xk)22exp{tank|˜Xk|}1+tank˜Xk+cτ2/pnl2(n)t2(˜Xk)2exp{ctl1(n)}. (4.2)

    By Definition 2.6, let φi(x)=etXi,i1, we can get for WNOD r.v.,

    ˆEexp{tni=1Xi}g(n)ni=1ˆEexp{tXi}. (4.3)

    By (4.2), (4.3), and the inequality 1+xex,xR, for all t>0 as well as large enough n,

    ˆEexp{tnk=1ank˜Xk}g(n)nk=1ˆEexp{tank˜Xk}g(n)nk=1ˆE[1+tank˜Xk+cτ2/pnl2(n)t2(˜Xk)2exp{ctl1(n)}]g(n)nk=1[1+cτ2/pnl2(n)t2exp{ctl1(n)}ˆE(˜Xk)2]g(n)exp{cτ2/pnl2(n)t2exp{ctl1(n)}nk=1ˆE(˜Xk)2].

    For ε>0, let t=2ε1l(n). According to Markov inequality, we can get

    V{nk=1ank˜Xk>ε}eεtˆEexp{tnk=1ank˜Xk}eεtg(n)exp{cτ2/pnl2(n)t2exp{ctl1(n)}nk=1ˆE(˜Xk)2]e2l(n)g(n)exp{cε2exp{cε1}l(n)τ2/pnl1(n)nk=1ˆE(˜Xk)2}.

    Combining ˆE(˜Xk)24ˆE(Xk)2, (2.6), and Kronecker's Lemma,

    τ2/pnl1(n)nk=1ˆE(˜Xk)20,n.

    So, for 0<δ<1, and large enough n, l(n) is non-decreasing in (0,), we can get

    cε2exp{cε1}τ2/pnl1(n)nk=1ˆE(˜Xk)2l(n)δl(1)δl(n).

    Therefore, by (3.3),

    n=1V{nk=1ank˜Xk>ε}cn=1e2l(n)g(n)eδl(n)=cn=1e(δ2)l(n)g(n)<.

    Because V has countable sub-additivity, and for every ε>0, we obtain from Lemma 2.4,

    lim supnI10,a.s.V. (4.4)

    For each n, there must be a m such that 2m1n<2m, by (2.12) and (3.2), h(x) is decreasing in (0,), according to Lemma 2.1, τ1/pnl1(n) is decreasing in (0,),

    H3:=nk=1ank|ˆEXkˆEXk|nk=1ankˆE|Xk|nk=1ankˆE[|X|(1h(|X|τ1/pk))]τ1/pnl1(n)nˆE[|X|(1h(|X|τ1/pn))]2mτ1/p2m1l(2m1)ˆE[|X|(1h(|X|τ1/p2m1))]2mτ1/p2ml(2m)ˆE[|X|(1h(|X|τ1/p2m))].

    Then, by (2.15) and (2.13), ˆE is countably sub-additive,

    H32mτ1/p2ml(2m)j=mˆE[|X|hj(|X|μ)]2mτ1/p2ml(2m)j=mτ1/p2jV(|X|>μ2τ1/p2j1)j=m2jτ1/p2jl(2j)τ1/p2jV(|X|>μ2τ1/p2j)=j=m2jl(2j)V(|X|>μ2τ1/p2j).

    Combining (2.5), we get

    limnI3=0. (4.5)

    If we want to prove (3.4), just prove

    lim supnI20,a.s.V. (4.6)

    Using (3.2) as well as the Lemma 2.1,

    max2mn<2m+1|nk=1ankXk|cmax2mn<2m+1τ1/pnl1(n)nk=1|Xk|cτ1/p2ml1(2m)2m+1k=1|Xk|,

    for ε>0, by (2.7) and Markov inequality,

    m=1V(max2mn<2m+1|nk=1ankXk|>ε)m=1V(cτ1/p2ml1(2m)2m+1k=1|Xk|>ε)cm=1τ1/p2ml1(2m)2m+1k=1ˆE|Xk|=ck=1ˆE|Xk|m:2m+1kτ1/p2ml1(2m)k=1τ1/pkl1(k)ˆE|Xk|<.

    By Lemma 2.4, for ε>0,

    lim supmmax2mn<2m+1|nk=1ankXk|ε,a.s.V.

    Combining |nk=1ankX|max2mn<2m+1|nk=1ankXk| and the arbitrariness of ε, (4.6) holds. So far, Theorem 3.1 has been proved.

    Let l(n)=log(n), for 0<b<1δ, by (3.8), we have

    n=1e(δ2)l(n)g(n)=n=1nδ2g(n)=n=1nδ2+bg(n)nbcn=1nδ2+b<.

    Then, (3.4) holds. From Theorem 3.1, Eqs (3.4)–(3.6) hold.

    Let l(n)=exp{(logn)ν},0<ν<1. For q>0, we have

    (logn)νqloglogn,

    so,

    exp{(logn)ν}eqloglogn=logqnqlogn.

    By (3.10), 0<δ<1, when q>m+12δ, we have

    n=1e(δ2)l(n)g(n)=n=1exp{(δ2)exp{logνn}}g(n)n=1exp{(δ2)qlogn}g(n)=n=1n(δ2)q+mg(n)nmcn=1n(δ2)q+m<.

    Then, (3.4) holds. From Theorem 3.1, Eqs (3.4)–(3.6) hold.

    When p=1, CV(|X|)CV(|X|log|X|)<, thus (4.4) and (4.5) are still valid, we just need to prove (4.6). Imitating the proof of Lemma 2.2, from CV(|X|log|X|)<, we can obtain

    k=12kkl(2k)V(|X|>cτ2k)<. (4.7)

    Combining (2.12) and the monotonically decreasing property of h(x) in (0,),

    H4:=n=11nˆE|Xn|n=11nˆE|X|(1h(|X|τn))=k=12k1n<2k1nˆE|X|(1h(|X|τn))k=12k112k1ˆE|X|(1h(|X|τ2k1))k=1ˆE|X|(1h(|X|τ2k)).

    Then, by (2.15) and (4.7),

    H4k=1j=kˆE|X|hj(|X|μ)j=1jτ2jV(|X|>μ2τ2j1)j=12jjl(2j)V(|X|>μ2τ2j)<.

    For ε>0, by (3.12) and Markov inequality,

    m=1V(max2mn<2m+1|nk=1ankXk>ε|)cm=1max2mn<2m+11nnk=1ˆE|Xk|cm=112m2m+1k=1ˆE|Xk|=ck=1ˆE|Xk|m:2m+1>k12mck=11kˆE|Xk|<.

    By Lemma 2.4, for ε>0,

    lim supmmax2mn<2m+1|nk=1ankXk|ε,a.s.V.

    Combining |nk=1ankX|max2mn<2m+1|nk=1ankXk| and the arbitrariness of ε, (4.6) holds. So far, Theorem 3.2 has been proved.

    Almost sure convergence of WNOD r.v. in Peng's theory frame is built through this essay. It is based on the corresponding definition of stochastic domination in the sub-linear expectation space, as well as the properties of WNOD r.v. and the related proving methods. Compared with the previous research of ND, END, and so on, the research in this paper is suitable for a wider range of r.v.. So, broader conclusions are reached. In future research work, we will further consider investigating more meaningful conclusions.

    Baozhen Wang: Conceptualization, Formal analysis, Investigation, Methodology, Writing-original draft, Writing-review & editing; Qunying Wu: Funding acquisition, Formal analysis, Writing-review & 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 National Natural Science Foundation of China (12061028) and Guangxi Colleges and Universities Key Laboratory of Applied Statistics.

    In this article, all authors disclaim any conflict of interest.



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