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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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 supi≥1ˆE[|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. ax∼bx means limx→∞axbx=1. an≪bn means there must be a positive number c, satifying an≤cbn 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,…,Xn∈H, then φ(X1,X2,…,Xn)∈H for every φ∈Cl, Lip (Rn), where φ∈Cl, Lip (Rn) shows the linear space of (local Lipschitz) functions φ satisfying
|φ(x)−φ(y)|≤c(1+|x|m+|y|m)|x−y|,∀x,y∈Rn, |
for some c>0,m∈N depending on φ. H is considered as a space of random variable. In this circumstance, we denote X∈H.
Definition 2.1. (Peng [1]). A sub-linear expectation ˆE on H is a function ˆE:H→[−∞,∞] satisfying the following properties: for all X,Y∈H, we have
(a) Monotonicity: if X≥Y, 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),λ≥0.
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),∀X∈H. |
By the above definitions of ˆE and ˆE, the following inequality is feasible for all X,Y∈H,
ˆE(X)≤ˆE(X),ˆE(X−Y)≥ˆE(X)−ˆE(Y),ˆE(X+c)=ˆE(X)+c,|ˆE(X−Y)|≤ˆE|X−Y|. |
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 G⊂F, a function V:G→[0,1] is described to be a capacity, when
V(∅)=0,V(Ω)=1 and V(A)≤V(B) for A⊂B,A,B∈G. |
Similar to sub-linear expectations, it is known as sub-additive when V(A∪B)≤V(A)+V(B) for every A,B∈G. Now, represent V and V, respectively corresponing to ˆE and ˆE, using
V(A):=inf{ˆE[ξ],IA≤ξ,ξ∈H},V(A):=1−V(Ac),A∈F, |
where Ac denotes the complement set of A.
From the definion and sub-additivity property of (V, V), the following formulas are true
ˆEζ≤V(C)≤ˆEη,ˆEζ≤V(C)≤ˆEη,ifζ≤I(C)≤η,ζ,η∈H. |
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(X≥x)dx+∫0−∞[V(X≥x)−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 X≤∞∑n=1Xn,X,Xn∈H,X≥0,Xn≥0,n≥1. |
(ⅱ)V is referred to be countably sub-additive when
V(∞⋃n=1An)≤∞∑n=1V(An),∀An∈F. |
Definition 2.5. {Xn,n≥1} 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|)],forn≥1,0≤f∈Cl,Lip(R). |
Definition 2.6. (Anna [18]. Widely negative orthant dependent (WNOD)) {Xn,n≥1} is called to be widely negative orthant dependent if there is a finite positive array {g(n),n≥1} satisfying for every n≥1,
ˆE(n∏i=1φi(Xi))≤g(n)n∏i=1ˆE (φi(Xi)), |
where φi∈Cb,Lip(R),φi≥0,1≤i≤n and all functions φi are uniformly monotonous. Where the coefficients g(n) (n≥1) are known as dominating coefficients.
It is visible that, when {Xn,n≥1} is widely negative orthant dependent and all functions fk(x)∈Cl,Lip(R) (where k=1,2,⋯,n) are uniformly monotonous, then {fn(Xn),n≥1} 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
limx→∞l(tx)l(x)=1,foreacht≥0. |
(ⅱ) Each slowly varying function l(x) can be expressed as
l(x)=C(x)exp{∫x1f(y)ydy}, |
whenever limx→∞C(x)=c>0, as well as limy→∞f(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),x−αl(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, limx→∞f(x)=0.
Let
τn=nl(n)−1,n≥1. | (2.1) |
X′n=−τ1/pnI(Xn<−τ1/pn)+XnI(|Xn|≤τ1/pn)+τ1/pnI(Xn>τ1/pn). | (2.2) |
X″n=Xn−X′n=(Xn+τ1/pn)I(Xn<−τ1/pn)+(Xn−τ1/pn)I(Xn>τ1/pn). | (2.3) |
Lemma 2.2. Assume X∈H,0<p<2,τn defined by Eq (2.1).
(ⅰ) For every c>0,
CV(|X|p)<∞⟺∞∑n=1l−1(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)dx∼∫∞1l(y)−yl(y)⋅f(y)yl2(y)V(|X|p>c⋅yl(y))dy(makex=yl(y))∼∫∞11l(y)V(|X|p>cτy)dy. |
So
CV(|X|p)<∞⟺∞∑n=1l−1(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=1l−1(n)V(|X|p>cτn)=∞∑k=1∑2k−1≤n<2kl−1(n)V(|X|p>cτn)≥∞∑k=1∑2k−1≤n<2kl−1(2k)V(|X|p>cτ2k)=2−1∞∑k=12kl−1(2k)V(|X|p>cτ2k). |
As such, we have completed the proof of (ⅱ).
Lemma 2.3. {Xn,n≥1} is a sequence of random variables, as well as stochastically dominated by a r.v. X and CV(|X|p)<∞,1≤p<2, ˆE has countable sub-additivity, then
∞∑n=1τ−2/pnl−1(n)ˆE(X′n)2<∞, | (2.6) |
moreover, when 1<p<2,
∞∑n=1τ−1/pnl−1(n)ˆE|X′′n|<∞. | (2.7) |
Where X′n,X′′n 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 ∀x∈R 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)≤1−h(x)≤I(|x|>μ). | (2.8) |
For α=1,2,
|X′k|α=|Xk|αI(|Xk|≤τ1/pk)+τα/pkI(|Xk|>τ1/pk)≤|X′k|αh(μ|Xk|τ1/pk)+τα/pk(1−h(|Xk|τ1/pk)). | (2.9) |
|X′′k|α=|Xk+τ1/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|−τ1/pk|αI(|Xk|>τ1/pk)≤|Xk|αI(|Xk|>τ1/pk)≤|Xk|α(1−h(|Xk|τ1/pk)). | (2.10) |
So, by (2.8) and Definition 2.7,
ˆE|X′k|α≤ˆE|Xk|αh(μ|Xk|τ1/pk)+τα/pkˆE(1−h(|Xk|τ1/pk))≤ˆE|X|αh(μ|X|τ1/pk)+τα/pkˆE(1−h(|X|τ1/pk))≤ˆE|X|αh(μ|X|τ1/pk)+τα/pkV(|X|>μτ1/pk). | (2.11) |
ˆE|X′′k|α≤ˆE|X|α(1−h(|X|τ1/pk)). | (2.12) |
Assume that hj(x)∈Cl,Lip(R),j≥1, consider that the value of hj(x) is [0,1] for ∀x∈R. hj(x)≡1 when τ1/p2j−1<|x|≤τ1/p2j; hj(x)≡0 when |x|≤μτ1/p2j−1 or |x|>(1+μ)τ1/p2j. The following formulas can be derived,
I(τ1/p2j−1<|x|≤τ1/p2j)≤hj(|x|)≤I(μτ1/p2j−1<|x|≤(1+μ)τ1/p2j). | (2.13) |
|X|rh(|X|τ1/p2k)≤1+k∑j=1|X|rhj(|X|),r>0. | (2.14) |
|X|r(1−h(|X|τ1/p2k))≤∞∑j=k|X|rhj(|X|μ),r>0. | (2.15) |
First, prove (2.6). For 1≤p<2, by (2.11) and (2.4),
H1:=∞∑n=1τ−2/pnl−1(n)ˆE(X′n)2≤∞∑n=1τ−2/pnl−1(n)[ˆE(X2h(μ|X|τ1/pn))+τ2/pnV(|X|>μτ1/pn)]=∞∑n=1τ−2/pnl−1(n)ˆE[X2h(μ|X|τ1/pn)]+∞∑n=1l−1(n)V(|X|>μτ1/pn)≪∞∑n=1τ−2/pnl−1(n)ˆE[X2h(μ|X|τ1/pn)]. |
Then, because h(x) is decreasing in (0,∞), according to Lemma 2.1, τ−2/pnl−1(n) is decreasing in (0,∞). So,
H1≪∞∑k=1∑2k−1≤n<2kτ−2/pnl−1(n)ˆE[X2h(μ|X|τ1/pn)]≤∞∑k=1∑2k−1≤n<2kτ−2/p2k−1l−1(2k−1)ˆE[X2h(μ|X|τ1/p2k)]≪∞∑k=1∑2k−1≤n<2kτ−2/p2kl−1(2k)ˆE[X2h(μ|X|τ1/p2k)]≪∞∑k=12kτ−2/p2kl−1(2k)ˆE[X2h(μ|X|τ1/p2k)]. |
Last by (2.14), (2.13), and (2.5),
H1≪∞∑k=12kτ−2/p2kl−1(2k)+∞∑k=12kτ−2/p2kl−1(2k)k∑j=1ˆE(X2hj(μ|X|))≪∞∑j=1∞∑k=j2kτ−2/p2kl−1(2k)ˆE(X2hj(μ|X|))≪∞∑j=12jτ−2/p2jl−1(2j)τ2/p2jV(|X|>τ1/p2j−1)≪∞∑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, τ−1/pnl−1(n) is decreasing in (0,∞), we have,
H2:=∞∑n=1τ−1/pnl−1(n)ˆE|X′′n|≤∞∑n=1τ−1/pnl−1(n)ˆE[|X|(1−h(|X|τ1/pn))]=∞∑k=1∑2k−1≤n<2kτ−1/pnl−1(n)ˆE[|X|(1−h(|X|τ1/pn))]≤∞∑k=12kτ−1/p2k−1l−1(2k−1)ˆE[|X|(1−h(|X|τ1/p2k−1))]≪∞∑k=12kτ−1/p2kl−1(2k)ˆE[|X|(1−h(|X|τ1/p2k))]. |
Then, from (2.15), (2.13), and (2.5), countable sub-additivity of ˆE,
H2≪∞∑k=12kτ−1/p2kl−1(2k)∞∑j=kˆE(|X|hj(|X|μ))=∞∑j=1j∑k=12kτ−1/p2kl−1(2k)ˆE(|X|hj(|X|μ))≪∞∑j=12jτ−1/p2jl−1(2j)τ1/p2jV(|X|>μ2τ1/p2j−1)≪∞∑j=12jl(2j)V(|X|>μ2τ1/p2j)<∞. |
Therefore, (2.7) holds.
Lemma 2.4. (Zhang [11] Borel-Cantelli Lemma) Suppose {Bn;n≥1} 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=1⋃∞m=nBm.
Theorem 3.1. Suppose {Xn,n≥1} 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,1≤k≤n,n≥1} be a positive sequence according to
max1≤k≤nank=O(τ−1/pnl−1(n)),n→∞, | (3.2) |
where τn is defined by (2.1).
If for some 0<δ<1,
∞∑n=1e(δ−2)l(n)g(n)<∞, | (3.3) |
then,
lim supn→∞n∑k=1ank(Xk−ˆEXk)≤0a.s.V, | (3.4) |
lim infn→∞n∑k=1ank(Xk−ˆEXk)≥0a.s.V, | (3.5) |
in particular, when ˆEXk=ˆEXk, then
limn→∞n∑k=1ank(Xk−ˆEXk)=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 n≥1, then the sequence is simplified to END. When let l(n)=logn,n≥1, for 0<δ<1,
∞∑n=1e(δ−2)l(n)g(n)=M∞∑n=1n−(2−δ)<∞, |
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,n≥1} 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,1≤k≤n,n≥1} is a positive sequence according to
max1≤k≤nank=O(1n1/plog1−1/pn),n→∞. | (3.7) |
For some 0<b<1−δ,
g(n)n−b≤c, | (3.8) |
then (3.4)–(3.6) hold.
Corollary 3.2. Suppose {Xn,n≥1} 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,1≤k≤n,n≥1} is a positive sequence according to
max1≤k≤nank=O(n−1/pe(−1+1/p)(logn)ν),n→∞, | (3.9) |
where 0<ν<1.
For some m>0,
g(n)n−m≤c, | (3.10) |
then (3.4)–(3.6) hold.
Then, we will think about the situation of p=1.
Theorem 3.2. Suppose {Xn,n≥1} 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,1≤k≤n,n≥1} is a positive sequence according to
max1≤k≤nank=O(n−1),n→∞, | (3.12) |
then (3.4)–(3.6) hold.
Because the sequence {−Xk,k≥1} fulfills the criterion of Theorem 3.1, making {−Xk,k≥1} as a substitute for {Xk,k≥1} in formula (3.4), by ˆEX=−ˆE(−X), there is
0≥lim supn→∞n∑k=1ank((−Xk)−ˆE(−Xk))=lim supn→∞n∑k=1ank((−Xk)+ˆEXk)=lim supn→∞n∑k=1ank(−(Xk−ˆEXk)). |
⇒lim infn→∞n∑k=1ank(Xk−ˆEXk)≥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 X′n, X′′n respectively by equations (2.2) and (2.3). By Definition 2.6, {X′k−ˆEX′k,k≥1} is also WNOD. Denote ˜X′k:=X′k−ˆEX′k.
Therefore,
n∑k=1ank(Xk−ˆEXk)=n∑k=1ank˜X′k+n∑k=1ankX′′k+n∑k=1ank(ˆEX′k−ˆEXk):=I1+I2+I3. |
So, if we want to prove (3.4), just prove
lim supn→∞Ii≤0a.s.V,i=1,2,andlimn→∞I3=0. | (4.1) |
By (3.2) and the formula ex≤1+x+x22e|x|,x∈[−∞,∞], for all t>0,1≤k≤n as well as large enough n,
exp{tank˜X′k}≤1+tank˜X′k+t2a2nk(˜X′k)22exp{tank|˜X′k|}≤1+tank˜X′k+cτ−2/pnl−2(n)t2(˜X′k)2exp{ctl−1(n)}. | (4.2) |
By Definition 2.6, let φi(x)=etXi,i≥1, we can get for WNOD r.v.,
ˆEexp{tn∑i=1Xi}≤g(n)n∏i=1ˆEexp{tXi}. | (4.3) |
By (4.2), (4.3), and the inequality 1+x≤ex,∀x∈R, for all t>0 as well as large enough n,
ˆEexp{tn∑k=1ank˜X′k}≤g(n)n∏k=1ˆEexp{tank˜X′k}≤g(n)n∏k=1ˆE[1+tank˜X′k+cτ−2/pnl−2(n)t2(˜X′k)2exp{ctl−1(n)}]≤g(n)n∏k=1[1+cτ−2/pnl−2(n)t2exp{ctl−1(n)}ˆE(˜X′k)2]≤g(n)exp{cτ−2/pnl−2(n)t2exp{ctl−1(n)}n∑k=1ˆE(˜X′k)2]. |
For ε>0, let t=2ε−1l(n). According to Markov inequality, we can get
V{n∑k=1ank˜X′k>ε}≤e−εtˆEexp{tn∑k=1ank˜X′k}≤e−εtg(n)exp{cτ−2/pnl−2(n)t2exp{ctl−1(n)}n∑k=1ˆE(˜X′k)2]≤e−2l(n)g(n)exp{cε−2exp{cε−1}l(n)τ−2/pnl−1(n)n∑k=1ˆE(˜X′k)2}. |
Combining ˆE(˜X′k)2≤4ˆE(X′k)2, (2.6), and Kronecker's Lemma,
τ−2/pnl−1(n)n∑k=1ˆE(˜X′k)2→0,n→∞. |
So, for ∀0<δ<1, and large enough n, l(n) is non-decreasing in (0,∞), we can get
cε−2exp{cε−1}τ−2/pnl−1(n)n∑k=1ˆE(˜X′k)2l(n)≤δl(1)≤δl(n). |
Therefore, by (3.3),
∞∑n=1V{n∑k=1ank˜X′k>ε}≤c∞∑n=1e−2l(n)g(n)eδl(n)=c∞∑n=1e(δ−2)l(n)g(n)<∞. |
Because V has countable sub-additivity, and for every ε>0, we obtain from Lemma 2.4,
lim supn→∞I1≤0,a.s.V. | (4.4) |
For each n, there must be a m such that 2m−1≤n<2m, by (2.12) and (3.2), h(x) is decreasing in (0,∞), according to Lemma 2.1, τ−1/pnl−1(n) is decreasing in (0,∞),
H3:=n∑k=1ank|ˆEXk−ˆEX′k|≤n∑k=1ankˆE|X′′k|≤n∑k=1ankˆE[|X|(1−h(|X|τ1/pk))]≪τ−1/pnl−1(n)nˆE[|X|(1−h(|X|τ1/pn))]≤2mτ1/p2m−1l(2m−1)ˆE[|X|(1−h(|X|τ1/p2m−1))]≪2mτ1/p2ml(2m)ˆE[|X|(1−h(|X|τ1/p2m))]. |
Then, by (2.15) and (2.13), ˆE is countably sub-additive,
H3≪2mτ1/p2ml(2m)∞∑j=mˆE[|X|hj(|X|μ)]≤2mτ1/p2ml(2m)∞∑j=mτ1/p2jV(|X|>μ2τ1/p2j−1)≤∞∑j=m2jτ1/p2jl(2j)τ1/p2jV(|X|>μ2τ1/p2j)=∞∑j=m2jl(2j)V(|X|>μ2τ1/p2j). |
Combining (2.5), we get
limn→∞I3=0. | (4.5) |
If we want to prove (3.4), just prove
lim supn→∞I2≤0,a.s.V. | (4.6) |
Using (3.2) as well as the Lemma 2.1,
max2m≤n<2m+1|n∑k=1ankX′′k|≤cmax2m≤n<2m+1τ−1/pnl−1(n)n∑k=1|X′′k|≤cτ−1/p2ml−1(2m)2m+1∑k=1|X′′k|, |
for ∀ε>0, by (2.7) and Markov inequality,
∞∑m=1V(max2m≤n<2m+1|n∑k=1ankX′′k|>ε)≤∞∑m=1V(cτ−1/p2ml−1(2m)2m+1∑k=1|X′′k|>ε)≤c∞∑m=1τ−1/p2ml−1(2m)2m+1∑k=1ˆE|X′′k|=c∞∑k=1ˆE|X′′k|∑m:2m+1≥kτ−1/p2ml−1(2m)≪∞∑k=1τ−1/pkl−1(k)ˆE|X′′k|<∞. |
By Lemma 2.4, for ∀ε>0,
lim supm→∞max2m≤n<2m+1|n∑k=1ankX′′k|≤ε,a.s.V. |
Combining |∑nk=1ankX′′|≤max2m≤n<2m+1|∑nk=1ankX′′k| and the arbitrariness of ε, (4.6) holds. So far, Theorem 3.1 has been proved.
Let l(n)=log(n), for 0<b<1−δ, by (3.8), we have
∞∑n=1e(δ−2)l(n)g(n)=∞∑n=1nδ−2g(n)=∞∑n=1nδ−2+bg(n)n−b≤c∞∑n=1nδ−2+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)ν≥qloglogn, |
so,
exp{(logn)ν}≥eqloglogn=logqn≥qlogn. |
By (3.10), 0<δ<1, when q>m+12−δ, we have
∞∑n=1e(δ−2)l(n)g(n)=∞∑n=1exp{(δ−2)exp{logνn}}g(n)≤∞∑n=1exp{(δ−2)qlogn}g(n)=∞∑n=1n(δ−2)q+mg(n)n−m≤c∞∑n=1n(δ−2)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|X′′n|≤∞∑n=11nˆE|X|(1−h(|X|τn))=∞∑k=1∑2k−1≤n<2k1nˆE|X|(1−h(|X|τn))≤∞∑k=12k−112k−1ˆE|X|(1−h(|X|τ2k−1))≪∞∑k=1ˆE|X|(1−h(|X|τ2k)). |
Then, by (2.15) and (4.7),
H4≪∞∑k=1∞∑j=kˆE|X|hj(|X|μ)≤∞∑j=1jτ2jV(|X|>μ2τ2j−1)≪∞∑j=12jjl(2j)V(|X|>μ2τ2j)<∞. |
For ∀ε>0, by (3.12) and Markov inequality,
∞∑m=1V(max2m≤n<2m+1|n∑k=1ankX′′k>ε|)≤c∞∑m=1max2m≤n<2m+11nn∑k=1ˆE|X′′k|≤c∞∑m=112m2m+1∑k=1ˆE|X′′k|=c∞∑k=1ˆE|X′′k|∑m:2m+1>k12m≪c∞∑k=11kˆE|X′′k|<∞. |
By Lemma 2.4, for ∀ε>0,
lim supm→∞max2m≤n<2m+1|n∑k=1ankX′′k|≤ε,a.s.V. |
Combining |∑nk=1ankX′′|≤max2m≤n<2m+1|∑nk=1ankX′′k| 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.
[1] |
H. Aldweby, M. Darus, On a subclass of bi-univalent functions associated with the q-derivative operator, J. Math. Comput. Sci., 19 (2019), 58-64. doi: 10.22436/jmcs.019.01.08
![]() |
[2] | M. Arif, M. Ul Haq, J. L. Liu, A subfamily of univalent functions associated with q-analogue of Noor integral operator, J. Funct. Spaces, 2018 (2018), 1-5. |
[3] |
L. de Branges, A proof of the Bieberbach conjecture, Acta Math., 154 (1985), 137-152. doi: 10.1007/BF02392821
![]() |
[4] | D. A. Brannan, J. G. Clunie, Aspects of contemporary complex analysis, Proceedings of the NATO Advanced Study Institute (University of Durham, Durham; (1979), 1-20), New York and London: Academic Press, (1980), 79-95. |
[5] |
D. A. Brannan, J. Clunie, W. E. Kirwan, Coefficient estimates for a class of star-like functions, Canad. J. Math., 22 (1970), 476-485. doi: 10.4153/CJM-1970-055-8
![]() |
[6] | D. A. Brannan, T. S. Taha, On some classes of bi-unvalent functions, In: Mathematical Analysis and Its Applications (Kuwait; (1985), 18-21) (S. M. Mazhar, A. Hamoui, N. S. Faour, Eds.), KFAS Proceedings Series, 3, Oxford: Pergamon Press (Elsevier Science Limited), (1988), 53-60; see also Studia Univ. Babeş-Bolyai Math., 31 (1986), 70-77. |
[7] | T. Bulboacă, Differential subordinations and superordinations: Recent results, House of Scientific Book Publishers, Cluj-Napoca, 2005. |
[8] |
S. Bulut, Faber polynomial coefficient estimates for a subclass of analytic bi-univalent functions, Filomat, 30 (2016), 1567-1575. doi: 10.2298/FIL1606567B
![]() |
[9] | M. Çaglar, E. Deniz, Initial coefficients for a subclass of bi-univalent functions defined by Sălăgean differential operator, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 66 (2017), 85-91. |
[10] | P. L. Duren, Univalent functions, Grundlehren der Mathematischen Wissenschaften, Bd. 259, New York, Berlin, Heidelberg and Tokyo: Springer-Verlag, 1983. |
[11] | S. M. El-Deeb, Maclaurin coefficient estimates for new subclasses of bi-univalent functions connected with a q-analogue of Bessel function, Abstr. Appl. Anal., 2020 (2020), 1-7. |
[12] |
S. M. El-Deeb, T. Bulboacă, Fekete-Szegö inequalities for certain class of analytic functions connected with q-analogue of Bessel function, J. Egyptian Math. Soc., 27 (2019), 1-11. doi: 10.1186/s42787-019-0001-5
![]() |
[13] | S. M. El-Deeb, T. Bulboacă, Differential sandwich-type results for symmetric functions connected with a q-analog integral operator, Mathematics, 7 (2019), 1-17. |
[14] | S. M. El-Deeb, T. Bulboacă, B. M. El-Matary, Maclaurin coefficient estimates of bi-univalent functions connected with the q-derivative, Mathematics, 8 (2020), 1-14. |
[15] | S. M. El-Deeb, T. Bulboacă, J. Dziok, Pascal distribution series connected with certain subclasses of univalent functions, Kyungpook Math. J., 59 (2019), 301-314. |
[16] | S. M. El-Deeb, H. Orhan, Fekete-Szegö problem for certain class of analytic functions connected with q-analogue of integral operator, Ann. Univ. Oradea Fasc. Math., 28 (2021). In press. |
[17] | S. Elhaddad, M. Darus, Coefficient estimates for a subclass of bi-univalent functions defined by q-derivative operator, Mathematics, 8 (2020), 1-14. |
[18] |
G. Faber, Über polynomische Entwickelungen, Math. Ann., 57 (1903), 389-408. doi: 10.1007/BF01444293
![]() |
[19] | G. Gasper, M. Rahman, Basic hypergeometric series (with a Foreword by Richard Askey), Encyclopedia of Mathematics and Its Applications, 35, Cambridge, New York, Port Chester, Melbourne and Sydney: Cambridge University Press, 1990; 2 Eds., Encyclopedia of Mathematics and Its Applications, 96, Cambridge, London and New York: Cambridge University Press, 2004. |
[20] |
M. E. H. Ismail, E. Merkes, D. Styer, A generalization of starlike functions, Complex Variables Theory Appl., 14 (1990), 77-84. doi: 10.1080/17476939008814407
![]() |
[21] | F. H. Jackson, On q-functions and a certain difference operator, Trans. Royal Soc. Edinburgh, 46 (1908), 253-281. |
[22] | F. H. Jackson, On q-definite integrals, Quart. J. Pure Appl. Math., 41 (1910), 193-203. |
[23] | P. N. Kamble, M. G. Shrigan, Coefficient estimates for a subclass of bi-univalent functions defined by Sălăgean type q-calculus operator, Kyungpook Math. J., 58 (2018), 677-688. |
[24] | W. Kaplan, Close-to-convex schlicht functions, Michigan Math. J., 1 (1952), 169-185. |
[25] | Q. Khan, M. Arif, M. Raza, et al. Some applications of a new integral operator in q-analog for multivalent functions, Mathematics, 7 (2019), 1-13. |
[26] |
M. Lewin, On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc., 18 (1967), 63-68. doi: 10.1090/S0002-9939-1967-0206255-1
![]() |
[27] | S. Mahmood, N. Raza, E. S. A. Abujarad, et al. Geometric properties of certain classes of analytic functions associated with a q-integral operator, Symmetry, 11 (2019), 1-14. |
[28] | S. S. Miller, P. T. Mocanu, Differential subordination: Theory and applications, Series on Monographs and Textbooks in Pure and Applied Mathematics, New York and Basel: Marcel Dekker Incorporated, 225 (2000). |
[29] | E. Netanyahu, The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in |z| < 1, Arch. Rational Mech. Anal., 32 (1969), 100-112. |
[30] | S. Porwal, An application of a Poisson distribution series on certain analytic functions, J. Complex Anal., 2014 (2014), 1-3. |
[31] |
F. M. Sakar, H. Ö. Güeny, Coefficient bounds for a new subclass of analytic bi-close-to-convex functions by making use of Faber polynomial expansion, Turkish J. Math., 41 (2017), 888-895. doi: 10.3906/mat-1605-117
![]() |
[32] | L. Shi, Q. Khan, G. Srivastava, et al. A study of multivalent q-starlike functions connected with circular domain, Mathematics, 7 (2019), 1-12. |
[33] |
H. M. Srivastava, Certain q-polynomial expansions for functions of several variables, I and II, IMA J. Appl. Math., 30 (1983), 315-323; ibid. 33 (1984), 205-209. doi: 10.1093/imamat/30.3.315
![]() |
[34] | H. M. Srivastava, Univalent functions, fractional calculus, and associated generalized hypergeometric functions, In: Univalent functions, fractional calculus, and their applications (H. M. Srivastava, S. Owa, Eds.), New York, Chichester, Brisbane and Toronto: Halsted Press (Ellis Horwood Limited, Chichester) and John Wiley and Sons, (1989), 329-354. |
[35] |
H. M. Srivastava, Operators of basic (or q-) calculus and fractional q-calculus and their applications in geometric function theory of complex analysis, Iran. J. Sci. Technol. Trans. A: Sci., 44 (2020), 327-344. doi: 10.1007/s40995-019-00815-0
![]() |
[36] | H. M. Srivastava, Q. Z. Ahmad, N. Khan, et al. Some applications of higher-order derivatives involving certain subclasses of analytic and multivalent functions, J. Nonlinear Var. Anal., 2 (2018), 343-353. |
[37] |
H. M. Srivastava, Ş. Altinkaya, S. Yalçn, Hankel determinant for a subclass of bi-univalent functions defined by using a symmetric q-derivative operator, Filomat, 32 (2018), 503-516. doi: 10.2298/FIL1802503S
![]() |
[38] |
H. M. Srivastava, Ş. Altınkaya, S. Yalçin, Certain subclasses of bi-univalent functions associated with the Horadam polynomials, Iran. J. Sci. Technol. Trans. A: Sci., 43 (2019), 1873-1879. doi: 10.1007/s40995-018-0647-0
![]() |
[39] | H. M. Srivastava, S. Arjika, A. S. Kelil, Some homogeneous q-difference operators and the associated generalized Hahn polynomials, Appl. Set-Valued Anal. Optim., 1 (2019), 187-201. |
[40] | H. M. Srivastava, D. Bansal, Close-to-convexity of a certain family of q-Mittag-Leffler functions, J. Nonlinear Var. Anal., 1 (2017), 61-69. |
[41] |
H. M. Srivastava, S. S. Eker, R. M. Ali, Coefficient bounds for a certain class of analytic and bi-univalent functions, Filomat, 29 (2015), 1839-1845. doi: 10.2298/FIL1508839S
![]() |
[42] |
H. M. Srivastava, S. S. Eker, S. G. Hamidi, et al. Faber polynomial coefficient estimates for biunivalent functions defined by the Tremblay fractional derivative operator, Bull. Iran. Math. Soc., 44 (2018), 149-157. doi: 10.1007/s41980-018-0011-3
![]() |
[43] |
H. M. Srivastava, S. M. El-Deeb, A certain class of analytic functions of complex order connected with a q-analogue of integral operators, Miskolc Math. Notes, 21 (2020), 417-433. doi: 10.18514/MMN.2020.3102
![]() |
[44] | H. M. Srivastava, S. Gaboury, F. Ghanim, Coefficient estimates for a general subclass of analytic and bi-univalent functions of the Ma-Minda type, Rev. Real Acad. Cienc. Exactas Fís. Natur. Ser. A Mat., (RACSAM), 112 (2018), 1157-1168. |
[45] | H. M. Srivastava, P. W. Karlsson, Multiple Gaussian Hypergeometric Series, New York, Chichester, Brisbane and Toronto: Halsted Press (Ellis Horwood Limited, Chichester) and John Wiley and Sons, 1985. |
[46] |
H. M. Srivastava, B. Khan, N. Khan, et al. Coefficient inequalities for q-starlike functions associated with the Janowski functions, Hokkaido Math. J., 48 (2019), 407-425. doi: 10.14492/hokmj/1562810517
![]() |
[47] | H. M. Srivastava, S. Khan, Q. Z. Ahmad, et al. The Faber polynomial expansion method and its application to the general coefficient problem for some subclasses of bi-univalent functions associated with a certain q-integral operator, Stud. Univ. Babeş-Bolyai Math., 63 (2018), 419- 436. |
[48] |
H. M. Srivastava, A. K. Mishra, P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett., 23 (2010), 1188-1192. doi: 10.1016/j.aml.2010.05.009
![]() |
[49] | H. M. Srivastava, A. Motamednezhad, E. A. Adegan, Faber polynomial coefficient estimates for biunivalent functions defined by using differential subordination and a certain fractional derivative operator, Mathematics, 8 (2020), 1-12. |
[50] | H. M. Srivastava, F. M. Sakar, H. Ö. Güney, Some general coefficient estimates for a new class of analytic and bi-univalent functions defined by a linear combination, Filomat, 34 (2018), 1313- 1322. |
[51] | H. M. Srivastava, N. Raza, E. S. A. AbuJarad, et al. Fekete-Szegö inequality for classes of (p, q)- starlike and (p, q)-convex functions, Rev. Real Acad. Cienc. Exactas Fís. Natur. Ser. A Mat., (RACSAM), 113 (2019), 3563-3584. |
[52] | H. M. Srivastava, M. Tahir, B. Khan, et al. Some general classes of q-starlike functions associated with the Janowski functions, Symmetry, 11 (2019), 1-14. |
[53] |
H. M. Srivastava, M. Tahir, B. Khan, et al. Some general families of q-starlike functions associated with the Janowski functions, Filomat, 33 (2019), 2613-2626. doi: 10.2298/FIL1909613S
![]() |
[54] | H. M. Srivastava, A. K. Wanas, Initial Maclaurin coefficient bounds for new subclasses of analytic and m-fold symmetric bi-univalent functions defined by a linear combination, Kyungpook Math. J., 59 (2019), 493-503. |
[55] | H. Tang, Q. Khan, M. Arif, et al. Some applications of a new integral operator in q-analog for multivalent functions, Mathematics, 7 (2019), 1-13. |
[56] | H. E. Ö. Uçar, Coefficient inequality for q-starlike functions, Appl. Math. Comput., 276 (2016), 122-126. |
[57] | B. Wongsaijai, N. Sukantamala, Certain properties of some families of generalized starlike functions with respect to q-calculus, Abstr. Appl. Anal., 2016 (2016), 1-8. |