In this paper, we establish some maximal inequalities for demimartingales which generalize and improve the results of Christofides. The maximal inequalities for demimartingales are used as key inequalities to establis...In this paper, we establish some maximal inequalities for demimartingales which generalize and improve the results of Christofides. The maximal inequalities for demimartingales are used as key inequalities to establish other results including Doob’s type maximal inequality for demimartingales, strong laws of large numbers and growth rate for demimartingales and associated random variables. At last, we give an equivalent condition of uniform integrability for demisubmartingales.展开更多
In this paper,we give an overview of representation theorems for various static risk measures:coherent or convex risk measures, risk measures with comonotonic subadditivity or convexity, law-invariant coherent or conv...In this paper,we give an overview of representation theorems for various static risk measures:coherent or convex risk measures, risk measures with comonotonic subadditivity or convexity, law-invariant coherent or convex risk measures, risk measures with comonotonic subadditivity or convexity and respecting stochastic orders.展开更多
Some exponential inequalities for partial sums of associated random variables are established. These inequalities improve the corresponding results obtained by Ioannides and Roussas (1999), and Oliveira (2005). As app...Some exponential inequalities for partial sums of associated random variables are established. These inequalities improve the corresponding results obtained by Ioannides and Roussas (1999), and Oliveira (2005). As application, some strong laws of large numbers are given. For the case of geometrically decreasing covariances, we obtain the rate of convergence n-1/2(log log n)1/2(logn) which is close to the optimal achievable convergence rate for independent random variables under an iterated logarithm, while Ioannides and Roussas (1999), and Oliveira (2005) only got n-1/3(logn)2/3 and n-1/3(logn)5/3, separately.展开更多
Using the axiomatic method,abstract concepts such as abstract mean, abstract convex function and abstract majorization are proposed. They are the generalizations of concepts of mean, convex function and majorization, ...Using the axiomatic method,abstract concepts such as abstract mean, abstract convex function and abstract majorization are proposed. They are the generalizations of concepts of mean, convex function and majorization, respectively. Through the logical deduction, the fundamental theorems about abstract majorization inequalities are established as follows: for arbitrary abstract mean Σ and Σ , and abstract Σ→Σ strict convex function f(x) on the interval I, if xi, yi ∈ I (i = 1, 2, . . . , n) satisfy that (x1, x2, . . . , xn) <nΣ (y1, y2, . . . , yn), then Σ {f(x1), f(x2), . . . , f(xn)} ≥Σ {f(y1), f(y2), . . . , f(yn)}. This class of inequalities extends and generalizes the fundamental theorem of majorization inequalities. Moreover, concepts such as abstract vector mean are proposed, the fundamental theorems about abstract majorization inequalities are generalized to n-dimensional vector space. The fundamental theorem of majorization inequalities about the abstract vector mean are established as follows: for arbitrary symmetrical convex set S Rn, and n-variable abstract symmetrical Σ→Σ strict convex function φ() on S, if , ■∈S satisfy nΣ■, then φ() 〈(■); if vector group i, ■i∈ S (i = 1, 2, . . . , m) satisfy {1, 2, . . . , m} 〈Σn {■1, ■2, . . . , ■m}, then Σ {φ(1), φ(2), . . . , φ(m)} Σ {φ(■1), φ(■2), . . . , φ(■m)}.展开更多
基金supported by National Natural Science Foundation of China (Grant Nos. 10871001, 60803059)the Innovation Group Foundation of Anhui University
文摘In this paper, we establish some maximal inequalities for demimartingales which generalize and improve the results of Christofides. The maximal inequalities for demimartingales are used as key inequalities to establish other results including Doob’s type maximal inequality for demimartingales, strong laws of large numbers and growth rate for demimartingales and associated random variables. At last, we give an equivalent condition of uniform integrability for demisubmartingales.
基金supported by National Natural Science Foundation of China (Grant No.10571167)National Basic Research Program of China (973 Program) (Grant No.2007CB814902)Science Fund for Creative Research Groups (Grant No.10721101)
文摘In this paper,we give an overview of representation theorems for various static risk measures:coherent or convex risk measures, risk measures with comonotonic subadditivity or convexity, law-invariant coherent or convex risk measures, risk measures with comonotonic subadditivity or convexity and respecting stochastic orders.
基金the National Natural Science Fbundation of China (Grant Nos. 10161004, 70221001, 70331001)the Natural Science Foundation of Guangxi Province of China (Grant No. 04047033)
文摘Some exponential inequalities for partial sums of associated random variables are established. These inequalities improve the corresponding results obtained by Ioannides and Roussas (1999), and Oliveira (2005). As application, some strong laws of large numbers are given. For the case of geometrically decreasing covariances, we obtain the rate of convergence n-1/2(log log n)1/2(logn) which is close to the optimal achievable convergence rate for independent random variables under an iterated logarithm, while Ioannides and Roussas (1999), and Oliveira (2005) only got n-1/3(logn)2/3 and n-1/3(logn)5/3, separately.
基金supported by the National Key Basic Research Project of China (Grant No. 2004CB318003)the Foundation of the Education Department of Sichuan Province of China (Grant No. 07ZA087)
文摘Using the axiomatic method,abstract concepts such as abstract mean, abstract convex function and abstract majorization are proposed. They are the generalizations of concepts of mean, convex function and majorization, respectively. Through the logical deduction, the fundamental theorems about abstract majorization inequalities are established as follows: for arbitrary abstract mean Σ and Σ , and abstract Σ→Σ strict convex function f(x) on the interval I, if xi, yi ∈ I (i = 1, 2, . . . , n) satisfy that (x1, x2, . . . , xn) <nΣ (y1, y2, . . . , yn), then Σ {f(x1), f(x2), . . . , f(xn)} ≥Σ {f(y1), f(y2), . . . , f(yn)}. This class of inequalities extends and generalizes the fundamental theorem of majorization inequalities. Moreover, concepts such as abstract vector mean are proposed, the fundamental theorems about abstract majorization inequalities are generalized to n-dimensional vector space. The fundamental theorem of majorization inequalities about the abstract vector mean are established as follows: for arbitrary symmetrical convex set S Rn, and n-variable abstract symmetrical Σ→Σ strict convex function φ() on S, if , ■∈S satisfy nΣ■, then φ() 〈(■); if vector group i, ■i∈ S (i = 1, 2, . . . , m) satisfy {1, 2, . . . , m} 〈Σn {■1, ■2, . . . , ■m}, then Σ {φ(1), φ(2), . . . , φ(m)} Σ {φ(■1), φ(■2), . . . , φ(■m)}.
