It has been proved that a self-mapping with exact one discontinuity may have a continuous iterate of the second order. It actually shows that iteration can change discontinuity into continuity. Further, we can also find some examples with exact one discontinuity which have C1 smooth iterate of the second order, indicating that iteration can change discontinuity into smoothness. In this paper we investigate piecewise C1 self-mappings on the open interval (0,1) having only one removable or jumping discontinuity. We give necessary and sufficient conditions for those self-mappings to have a C1 smooth iterate of the second order.
Citation: Tianqi Luo, Xiaohua Liu. Iteration changes discontinuity into smoothness (Ⅰ): Removable and jumping cases[J]. AIMS Mathematics, 2023, 8(4): 8772-8792. doi: 10.3934/math.2023440
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[6] | Mingzhou Xu . Complete convergence and complete moment convergence for maximal weighted sums of extended negatively dependent random variables under sub-linear expectations. AIMS Mathematics, 2023, 8(8): 19442-19460. doi: 10.3934/math.2023992 |
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[9] | He Dong, Xili Tan, Yong Zhang . Complete convergence and complete integration convergence for weighted sums of arrays of rowwise m-END under sub-linear expectations space. AIMS Mathematics, 2023, 8(3): 6705-6724. doi: 10.3934/math.2023340 |
[10] | Chengcheng Jia, Qunying Wu . Complete convergence and complete integral convergence for weighted sums of widely acceptable random variables under the sub-linear expectations. AIMS Mathematics, 2022, 7(5): 8430-8448. doi: 10.3934/math.2022470 |
It has been proved that a self-mapping with exact one discontinuity may have a continuous iterate of the second order. It actually shows that iteration can change discontinuity into continuity. Further, we can also find some examples with exact one discontinuity which have C1 smooth iterate of the second order, indicating that iteration can change discontinuity into smoothness. In this paper we investigate piecewise C1 self-mappings on the open interval (0,1) having only one removable or jumping discontinuity. We give necessary and sufficient conditions for those self-mappings to have a C1 smooth iterate of the second order.
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.
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