Abstract:The almost sure convergence of weighted sums of ϕ-subgaussian m-acceptable random variables is investigated. As corollaries, the main results are applied to the case of negatively dependent and m-dependent subgaussian random variables. Finally, an application to random Fourier series is presented.
“…In section 4 the problem of convergence of the series S(ã, X ) is considered assuming that X is a d-subgaussian increments sequence. This framework enables us to deduce corresponding results for series of independent or negatively dependent increments and for conditionally subgaussian series, we thus recover some results of [1,2,3,6,9]. Finally, we examine an example of a d-subgaussian series, which is beyond the scope of the last quoted papers.…”
Section: Introductionmentioning
confidence: 53%
“…These results were extended to m-dependent subgaussian random variables by Ouy [16] and to negatively dependent subgaussian random variables by Amini et al [1,2]. More recently Guiliano et al [9] examined the convergence of the series S(ã, X ) when X is an m-acceptable sequence of φ-subgaussian random variables, they obtained positive results assuming that x → φ(|x| 1/p ) is a convex function for some p ∈ [1,2], then they deduced the corresponding results for the classical subgaussian case.…”
“…In section 4 the problem of convergence of the series S(ã, X ) is considered assuming that X is a d-subgaussian increments sequence. This framework enables us to deduce corresponding results for series of independent or negatively dependent increments and for conditionally subgaussian series, we thus recover some results of [1,2,3,6,9]. Finally, we examine an example of a d-subgaussian series, which is beyond the scope of the last quoted papers.…”
Section: Introductionmentioning
confidence: 53%
“…These results were extended to m-dependent subgaussian random variables by Ouy [16] and to negatively dependent subgaussian random variables by Amini et al [1,2]. More recently Guiliano et al [9] examined the convergence of the series S(ã, X ) when X is an m-acceptable sequence of φ-subgaussian random variables, they obtained positive results assuming that x → φ(|x| 1/p ) is a convex function for some p ∈ [1,2], then they deduced the corresponding results for the classical subgaussian case.…”
“…As is mentioned in Giuliano et al [4], a sequence of NOD random variables with a finite Laplace transform or finite moment generating function near zero (and hence a sequence of NA random variables with finite Laplace transform, too) provides us an example of acceptable random variables. For example, Xing et al [6] consider a strictly stationary NA sequence of random variables.…”
Some exponential inequalities for a sequence of acceptable random variables are obtained, such as Bernstein-type inequality, Hoeffding-type inequality. The Bernsteintype inequality for acceptable random variables generalizes and improves the corresponding results presented by Yang for NA random variables and Wang et al. for NOD random variables. Using the exponential inequalities, we further study the complete convergence for acceptable random variables. MSC (2000): 60E15, 60F15.
“…General theory and various applications of sub-Gaussian, ϕ-sub-Gaussian, and strictly ϕ-sub-Gaussian random variables and processes can be found in the book of Buldygin and Kozachenko [1] and in the papers [2,4,5,7,8,9,12,13,14].…”
We study deviations of ϕ-sub-Gaussian stochastic process from a measurable function and generalize the results of [6]. We obtain a bound for the distributions of norms in the space L p (T). As an example, the obtained result is applied for an aggregate of independent processes of generalized ϕ-sub-Gaussian fractional Brownian motions.MSC: 60G07, 60G18
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