Mingshang Hu, Xiaojuan Li, Xinpeng Li
This short note provides a new and simple proof of the convergence rate for Peng's law of large numbers under sublinear expectations, which improves the corresponding results in Song [15] and Fang et al. [3].
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In this paper, we firstly establish the weak laws of large numbers on the canonical space $(\br^\bn,\cb(\br^\bn))$ by traditional truncation method and Chebyshev's inequality as in the classical probability theory. Then we extend them from the canonical space to the general sublinear expectation space. The necessary and sufficient conditions for Peng's law of large numbers are obtained.
This paper introduces the notion of pseudo-independence on the sublinear expectation space $(Ω,\mathcal{F},\mathcal{P})$ via the classical conditional expectation, and the relations between pseudo-independence and Peng's independence are detailed discussed. Law of large numbers and central limit theorem for pseudo-independent random variables are obtained, and a purely probabilistic proof of Peng's law of large numbers is also given. In the end, some relevant counterexamples are indicated that Peng's law of large numbers and central limit theorem are invalid only with the first and second mome…
In this paper, with the notion of independent identically distributed (IID) random variables under sublinear expectations introduced by Peng [7-9], we investigate moment bounds for IID sequences under sublinear expectations. We can obtain a moment inequality for a sequence of IID random variables under sublinear expectations. As an application of this inequality, we get the following result: For any continuous function $ϕ$ satisfying the growth condition $|ϕ(x)|\leq C(1+|x|^p)$ for some $C>0$, $p\geq1$ depending on $ϕ$, central limit theorem under sublinear expectations obtained by Peng [8] st…
Peng, S. (\cite{P08b}) proved the law of large numbers under a sublinear expectation. In this paper, we give its error estimates by Stein's method.
In this paper, we investigate a central limit theorem for weighted sums of independent random variables under sublinear expectations. It is turned out that our results are natural extensions of the results obtained by Peng and Li and Shi.
Marcinkiewicz-type laws of large numbers for pseudo-independent random variables under sublinear expectations
本文在次线性期望框架下，针对弱于Peng独立性的伪独立随机变量，证明了Marcinkiewicz型弱大数定律与强大数定律。
In this paper, motivated by the notion of independent identically distributed (IID) random variables under sub-linear expectations initiated by Peng, we investigate a law of the iterated logarithm for capacities. It turns out that our theorem is a natural extension of the Kolmogorov and the Hartman-Wintner laws of the iterated logarithm.
In this paper, under some weaker conditions, we give three laws of large numbers under sublinear expectations (capacities), which extend Peng's law of large numbers under sublinear expectations in [8] and Chen's strong law of large numbers for capacities in [1]. It turns out that these theorems are natural extensions of the classical strong (weak) laws of large numbers to the case where probability measures are no longer additive.
In this paper we consider two ways to generalize the mathematical expectation of a random variable, the Choquet expectation and Peng's g-expectation. An open question has been, after making suitable restrictions to the class of random variables acted on by the Choquet expectation, for what class of expectation do these two definitions coincide? In this paper we provide a necessary and sufficient condition which proves that the only expectation which lies in both classes is the traditional linear expectation. This settles another open question about whether Choquet expectation may be used to ob…
In this paper we consider a sequence of random variables with mean uncertainty in a sublinear expectation space. Without the hypothesis of identical distributions, we show a new central limit theorem under the sublinear expectations.
Inequalities for independent random vectors under sublinear expectations
本文利用次线性期望的表示定理，为次线性期望下独立随机向量的样本均值提供了两个不等式的简单证明。
Limit theorems under nonlinear expectations dominated by sublinear expectations
本文在次线性期望控制下，利用紧性方法建立非线性期望的极限定理，允许极限分布完全非线性，并探讨了正齐次性特例。
In this paper, we define the generalized Wasserstein distance for sets of Borel probability measures and demonstrate that the weak convergence of sublinear expectations can be characterized by means of this distance.
We show that, for two non-trivial random variables X and Y under a sublinear expectation space, if X is independent from Y and Y is independent from X, then X and Y must be maximally distributed.
In this note, we will survey the existing convergence results for random variables under sublinear expectations, and prove some new results. Concretely, under the assumption that the sublinear expectation has the monotone continuity property, we will prove that $L^p$ convergence is stronger than convergence in capacity, convergence in capacity is stronger than convergence in distribution, and give some equivalent characterizations of convergence in distribution. In addition, we give a dominated convergence theorem under sublinear expectations, which may have its own interest.
Sublinear expectation structure under countable state space
本文在可数状态空间下建立次线性期望结构，提出重复求和公式进行显式计算，证明三大收敛定理，并基于各概率测度下的独立性建立次线性大数定律与最大分布。
We give a theory of sublinear expectations and martingales in discrete time. Without assuming the existence of a dominating probability measure, we derive the extensions of classical results on uniform integrability, optional stopping of martingales, and martingale convergence. We also give a theory of BSDEs in the context of sublinear expectations and a finite-state space, including general existence and comparison results.
We introduce G-Lévy processes which develop the theory of processes with independent and stationary increments under the framework of sublinear expectations. We then obtain the Lévy-Khintchine formula and the existence for G-Lévy processes. We also introduce G-Poisson processes.
In this paper, we present some multi-dimensional central limit theorems and laws of large numbers under sublinear expectations, which extend some previous results.
Prokhorov distances under sublinear expectations are presented in CLT and functional CLT, and the convergence rates for them are obtained by Lindeberg method. In particular, the obtained estimate in functional CLT yields known Borovkov's estimate in classical functional CLT with explicit constant.
This paper introduces a backward stochastic differential equation driven by both Brownian motion and a Markov chain (BSDEBM). Regime-switching is also incorporated through its driver. The existence and uniqueness of the solution of the BSDEBM are proved. A comparison theorem is also derived. Filtration consistent sublinear expectations are defined and characterized as solutions to the BSDEBM. The bid and ask prices are then represented using sublinear expectations.
Two laws of large numbers for sublinear expectations
本文在有界随机变量上定义了次线性期望，并利用不相关性和独立性概念分别证明了弱和强大数定律，为工程和统计中的质量评估与分布关系解释提供了理论支持。
In this note, we study convergence rates in the law of large numbers for independent and identically distributed random variables under sublinear expectations. We obtain a strong $L^p$-convergence version and a strongly quasi sure convergence version of the law of large numbers.
Strong laws of large numbers for sequences of blockwise $m$-dependent and orthogonal random variables under sublinear expectations
本文在次线性期望框架下，针对块wise $m$-相依和正交这两类非独立随机变量，建立了强大数定律，推广了该空间中独立随机变量的强大数定律。
In this paper, the Neyman-Pearson lemma for general sublinear expectations is studied. We weaken the assumptions for sublinear expectations in [1] and give a completely new method to study this problem. Applying Mazur-Orlicz Theorem and the decomposition theorem of finitely additive set functions, we prove that the optimal test still has the reminiscent form as in the classical Neyman-Pearson lemma. Finally, for the special sublinear expectation which can be represented by a family of probability measures, we give a sufficient condition for the existence of the optimal test and show the form o…
The complete convergence for weighted sums of sequences of independent, identically distributed random variables under sublinear expectations space was studied. By moment inequality and truncation methods, we establish the equivalent conditions of complete convergence for weighted sums of sequences of independent, identically distributed random variables under sublinear expectations space. The results extend the corresponding results obtained by Guo (2012) to those for sequences of independent, identically distributed random variables under sublinear expectations space.
In this paper, on the sublinear expectation space, we establish a comparison theorem between independent and convolutionary random vectors, which states that the partial sums of those two sequences of random vectors are identically distributed. Under the sublinear framework, through the comparison theorem, several fundamental limit theorems for convolutionary random vectors are obtained, including the law of large numbers, the central limit theorem and the law of iterated logarithm.
In this paper, we are concerned with the representation of an important sublinear expectation under which framework a new stochastic process G-Lévy process has been introduced. We show the existence of a weakly compact family of probability measures to give the representation by using two different methods.
Sublinear expectations for countable-state uncertain processes
针对不确定过程次线性期望扩展定义域受限的问题，本文在可数状态下温和条件里将其扩展到càdlàg路径的广义实值可测函数，并研究次线性马尔可夫半群诱导满足条件的次线性期望的情形。
We provide a general approach to construct a stochastic process with a given consistent family of finite dimensional distributions under a nonlinear expectation space. We use this approach to construct a generalized Gaussian process under a sublinear expectation and a q-Brownian motion. The later one is under a complex-valued linear expectation, with which a new type of Feynman-Kac formula can be derived to represent the solution of a Schrödinger equation.
Ergodicity and Mixing of Sublinear Expectation System and Applications
该研究利用遍历理论框架探索次线性期望系统，推导其逐点Birkhoff遍历定理及更强结果，放宽大数定律条件至α-混合，并将成果应用于G-SDEs等，还获经典遍历与容量理论的应用。
Under the sublinear expectation $\mathbb{E}[\cdot]:=\sup_{θ\in Θ} E_θ[\cdot]$ for a given set of linear expectations $\{E_θ: θ\in Θ\}$, we establish a new law of large numbers and a new central limit theorem with rate of convergence. We present some interesting special cases and discuss a related statistical inference problem. We also give an approximation and a representation of the $G$-normal distribution, which was used as the limit in Peng (2007)'s central limit theorem, in a probability space.
Convergence for sums of i. i. d. random variables under sublinear expectations
本文研究了次线性期望空间下独立同分布随机变量部分加权和的收敛性问题，证明了最大值完全矩收敛的等价条件，并建立了Baum-Katz型结果，扩展了经典线性期望下的相关理论。
Stochastic heat equations driven by space-time $G$-white noise under sublinear expectation
本文在子线性期望框架下研究由G-白噪声驱动的随机热方程，证明了解的存在性、唯一性及弱解性，并推导了解的矩估计。
We investigate the complete $p$-th moment convergence for weighted sums of independent, identically distributed random variables under sublinear expectations space. Using moment inequality and truncation methods, we prove the equivalent conditions of complete $p$-th moment convergence of weighted sums of independent, identically distributed random variables under sublinear expectations space, which complement the corresponding results obtained in Guo and Shan (2020).
In this note, we study inequality and limit theory under sublinear expectations. We mainly prove Doob's inequality for submartingale and Kolmogrov's inequality. By Kolmogrov's inequality, we obtain a special version of Kolmogrov's law of large numbers. Finally, we present a strong law of large numbers for independent and identically distributed random variables under one-order type moment condition.
In this paper, we study the ergodicity of invariant sublinear expectation of sublinear Markovian semigroup. For this, we first develop an ergodic theory of an expectation-preserving map on a sublinear expectation space. Ergodicity is defined as any invariant set either has $0$ capacity itself or its complement has $0$ capacity. We prove, under a general sublinear expectation space setting, the equivalent relation between ergodicity and the corresponding transformation operator having simple eigenvalue $1$, and also with Birkhoff type strong law of large numbers if the sublinear expectation is …
In the framework of sublinear expectation, we have introduced a new type of G-Gaussian random fields, which contain a type of spatial white noise as a special case. Based on this result, we also have introduced a spatial-temporal G-white noise. Different from the case of linear expectation, in which the probability measure need to be known, under the uncertainty of the probability measure, the spatial white noises are intrinsically different from the temporal one.
In this note we discuss uniform integrability of random variables. In a probability space, we introduce two new notions on uniform integrability of random variables, and prove that they are equivalent to the classic one. In a sublinear expectation space, we give de La Vallée Poussin criterion for the uniform integrability of random variables and do some other discussions.
In this paper we give some basic and important properties of several typical Banach spaces of functions of $G$-Brownian motion pathes induced by a sublinear expectation--G-expectation. Many results can be also applied to more general situations. A generalized version of Kolmogorov's criterion for continuous modification of a stochastic process is also obtained. The results can be applied to continuous time dynamic and coherent risk measures in finance in particular for path-dependence risky positions under situations of volatility model uncertainty.
This paper focuses on the maximal distribution on sublinear expectation space and introduces a new type of random fields with the maximally distributed finite-dimensional distribution. The corresponding spatial maximally distributed white noise is constructed, which includes the temporal-spatial situation as a special case due to the symmetrical independence property of maximal distribution. In addition, the stochastic integrals with respect to the spatial or temporal-spatial maximally distributed white noises are established in a quite direct way without the usual assumption of adaptability f…
On the Strong Law of Large Numbers under Sublinear Expectations
研究次线性期望下的强大数定律，通过特定可积性条件证明，此条件比程虎所提更弱，采用N.埃泰马迪的截断方法进行证明。
On independence and large deviations for sublinear expectations
该研究通过反例指出次线性期望框架下现有大偏差原理的错误，说明速率函数无法常规通过Fenchel变换得到，同时给出修正结果并明确原结论成立的更强独立性条件。
Peng (2006) initiated a new kind of central limit theorem under sub-linear expectations. Song (2017) gave an estimate of the rate of convergence of Peng's central limit theorem. Based on these results, we establish a new kind of almost sure central limit theorem under sub-linear expectations in this paper, which is a quasi sure convergence version of Peng's central limit theorem. Moreover, this result is a natural extension of the classical almost sure central limit theorem to the case where the probability is no longer additive. Meanwhile, we prove a new kind of strong law of large numbers fo…
The existence of invariant sublinear expectations for $G$-SDEs
本文通过建立次线性期望的完备度量空间，将Harris方法推广到非线性情形，证明了非退化弱耗散条件下$G$-SDE的不变次线性期望的存在性。
In this book, we introduce a new approach of sublinear expectation to deal with the problem of probability and distribution model uncertainty. We a new type of (robust) normal distributions and the related central limit theorem under sublinear expectation. We also present a new type of Brownian motion under sublinear expectations and the related stochastic calculus of Ito's type. The results provide robust tools for the problem of probability model uncertainty arising from financial risk management, statistics and stochastic controls.
We consider a sequence of i.i.d. random variables $\{ξ_k\}$under a sublinear expectation $\mathbb{E}=\sup_{P\inΘ}E_P$. We first give a new proof to the fact that, under each $P\inΘ$, any cluster point of the empirical averages $\barξ_n=(ξ_1+\cdots+ξ_n)/n$ lies in $[\underlineμ, \barμ]$ with $\underlineμ=-\mathbb{E}[-ξ_1], \barμ=\mathbb{E}[ξ_1]$. Then, we consider sublinear expectations on a Polish space $Ω$, and show that for each constant $μ\in [\underlineμ,\barμ]$, there exists a probability $P_μ\inΘ$ such that \begin {eqnarray}\label {intro-a.s.} \lim_{n\rightarrow\infty}\barξ_n=μ, \ P_μ\te…
Stochastic wave equations driven by space-time $G$-white noise under sublinear expectation
本文在次线性期望下研究乘性时空$G$-白噪声驱动的一维半线性随机波动方程，建立了全局温和解的存在唯一性，并证明了高阶矩估计、拟必然Hölder连续性及弱形式满足性。
It has been a well-known problem in the $G$-framework that it is hard to compute the sublinear expectation of the $G$-normal distribution $\hat{\mathbb{E}}[φ(X)]$ when $φ$ is neither convex nor concave, if not involving any PDE techniques to solve the corresponding $G$-heat equation. Recently, we have established an efficient iterative method able to compute the sublinear expectation of \emph{arbitrary} functions of the $G$-normal distribution, which directly applies the \emph{Nonlinear Central Limit Theorem} in the $G$-framework to a sequence of variance-uncertain random variables following t…
Invariant Sublinear Expectations
本文研究不变次线性期望的周期分解结构，证明强遍历连续不变次线性期望具有有限周期，其周期分量为有限个遍历概率的凸包，并由此证明相应步长时间平均收敛到上期望。
We consider coherent sublinear expectations on a measurable space, without assuming the existence of a dominating probability measure. By considering a decomposition of the space in terms of the supports of the measures representing our sublinear expectation, we give a simple construction, in a quasi-sure sense, of the (linear) conditional expectations, and hence give a representation for the conditional sublinear expectation. We also show an aggregation property holds, and give an equivalence between consistency and a pasting property of measures.
Peng (2008)(\cite{P08b}) proved the Central Limit Theorem under a sublinear expectation: \textit{Let $(X_i)_{i\ge 1}$ be a sequence of i.i.d random variables under a sublinear expectation $\hat{\mathbf{E}}$ with $\hat{\mathbf{E}}[X_1]=\hat{\mathbf{E}}[-X_1]=0$ and $\hat{\mathbf{E}}[|X_1|^3]<\infty$. Setting $W_n:=\frac{X_1+\cdots+X_n}{\sqrt{n}}$, we have, for each bounded and Lipschitz function $φ$, \[\lim_{n\rightarrow\infty}\bigg|\hat{\mathbf{E}}[φ(W_n)]-\mathcal{N}_G(φ)\bigg|=0,\] where $\mathcal{N}_G$ is the $G$-normal distribution with $G(a)=\frac{1}{2}\hat{\mathbf{E}}[aX_1^2]$, $a\in \ma…
Functional central limit theorem with mean-uncertainty under sublinear expectation
本文在次线性期望框架下研究带均值不确定性的泛函中心极限定理，通过概率核建模、鞅方法与表示定理推广了Peng的结果，给出纯概率证明并应用于两臂老虎机问题。
We show that many well-known transforms in convex geometry (in particular, centroid body, convex floating body, and Ulam floating body) are special instances of a general construction, relying on applying sublinear expectations to random vectors in Euclidean space. We identify the dual representation of such convex bodies and describe a construction that serves as a building block for all so defined convex bodies.
We provide a general construction of time-consistent sublinear expectations on the space of continuous paths. It yields the existence of the conditional G-expectation of a Borel-measurable (rather than quasi-continuous) random variable, a generalization of the random G-expectation, and an optional sampling theorem that holds without exceptional set. Our results also shed light on the inherent limitations to constructing sublinear expectations through aggregation.
We describe a new framework of a sublinear expectation space and the related notions and results of distributions, independence. A new notion of G-distributions is introduced which generalizes our G-normal-distribution in the sense that mean-uncertainty can be also described. W present our new result of central limit theorem under sublinear expectation. This theorem can be also regarded as a generalization of the law of large number in the case of mean-uncertainty.
For $α\in (1,2)$, we present a generalized central limit theorem for $α$-stable random variables under sublinear expectation. The foundation of our proof is an interior regularity estimate for partial integro-differential equations (PIDEs). A classical generalized central limit theorem is recovered as a special case, provided a mild but natural additional condition holds. Our approach contrasts with previous arguments for the result in the linear setting which have typically relied upon tools that are non-existent in the sublinear framework, for example, characteristic functions.
In this paper, we study the minimum mean square estimator for a sublinear operator. Under some mild assumptions, we prove the existence and uniqueness of the minimum mean square estimator. Several characterizations of the minimum mean square estimator are obtained. We also explore the relationship between the minimum mean square estimator and the conditional coherent risk measure and conditional g-expectation.
On the rate of convergence for an $α$-stable central limit theorem under sublinear expectation
本文通过构造单调逼近格式并建立误差界，给出了次线性期望下α-稳定中心极限定理的显式Berry-Esseen界和收敛速度。
This article establishes a universal robust limit theorem under a sublinear expectation framework. Under moment and consistency conditions, we show that, for $α\in(1,2)$, the i.i.d. sequence \[ \left \{ \left( \frac{1}{\sqrt{n}}\sum_{i=1}^{n}X_{i},\frac{1}{n}\sum _{i=1}^{n}Y_{i},\frac{1}{\sqrt[α]{n}}\sum_{i=1}^{n}Z_{i}\right) \right \} _{n=1}^{\infty} \] converges in distribution to $\tilde{L}_{1}$, where $\tilde{L}_{t}=(\tilde ξ_{t},\tildeη_{t},\tildeζ_{t})$, $t\in [0,1]$, is a multidimensional nonlinear Lévy process with an uncertainty set $Θ$ as a set of Lévy triplets. This nonlinear Lévy p…
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