Log-fractional stable processes

The first problem attacked in this paper is answering the question whether all 1/-self-similar -stable processes with stationary increments are -stable motions. The answer is yes for = 2, no for 1<2 and unknown for 0<<1. We single out the log-fractional stable processes for 1<2, different from -stable motions for ≠2. They can be regarded as the limit of fractional stable processes as the exponent in the kernel tends to 0. The paper ends with a limit theorem for partial sum processes of moving averages of iid random variables in the domain of attraction of a strictly stable law, with log-fractional stable processes as limits in law. The conditions involve de Haan's class Π of slowly varying functions.     

Robustness of the R / S Statistic for Fractional Stable Noises

The R / S statistic is used to detect long-range dependence in a time series and to estimate its intensity. One of its virtues is robustness against different distributions. We show here that the R / S statistic continues to be robust if the time series is a moving average with long-range dependence with innovations that are in the domain of attraction of an infinite variance stable process.     

Integral analogues of almost sure limit theorems

Summary An integral analogue of the general almost sure limit theorem is presented. In the theorem, instead of a sequence of random elements, a continuous time random process is involved, moreover, instead of the logarithmical average, the integral of delta-measures is considered. Then the general theorem is applied to obtain almost sure versions of limit theorems for semistable and max-semistable processes, moreover for processes being in the domain of attraction of a stable law or being in the domain of geometric partial attraction of a semistable or a max-semistable law.     

Fractional Hermite degenerate kernel method for linear Fredholm integral equations involving endpoint weak singularities

In this article, the Fredholm integral equation of the second kind with endpoint weakly singular kernel is considered and suppose that the kernel possesses fractional Taylor''s expansions about the endpoints of the interval. For this type kernel, the fractional order interpolation is adopted in a small interval involving the singularity and piecewise cubic Hermite interpolation is used in the remaining part of the interval, which leads to a kind of fractional degenerate kernel method. We discuss the condition that the method can converge and give the convergence order. Furthermore, we design an adaptive mesh adjusting algorithm to improve the computational accuracy of the degenerate kernel method. Numerical examples confirm that the fractional order hybrid interpolation method has good computational results for the kernels involving endpoint weak singularities.