Some asymptotic formulars for a Brownian motion whose dimension is variable with a regular variation from a parabolic domain to research the variation of biological species

Consider a Brownian motion with a regular variation starting at an interior point of a domain D in R d+1 ,d ≥ 1, let τ D denote the first time the Brownian motion exits from D. Estimates with exact constants for the asymptotics of logP(τ D > T) are given for T → ∞, depending on the shape of the domain D and the order of the regular variation. Furthermore, the asymptotically equivalence are obtained. The problem is motivated by the early results of Lifshits and Shi, Li in the first exit time and Karamata in the regular variation. The methods of proof are based on their results and the calculus of variations.

Asymptotically ideal invariant equivalence

In this paper, the concepts of asymptotically \mathcal{I}_{\sigma}-equivalence, \sigma-asymptotically equivalence, st\-rong\-ly \sigma-asymptotically equivalence and strongly \sigma-asymptotically p-equivalence for real number sequences are de\-fi\-ned. Also, we give relationships among these new type equivalence concepts and the concept of \linebreak S_{\sigma}-asymptotically equivalence which is studied in [Savaş, E. and Patterson, R. F., \sigma-asymptotically lacunary statistical equivalent sequences, Cent. Eur. J. Math., 4 (2006), No. 4, 648–655].

On asymptotically deferred statistical equivalence of sequences

In this study, combining the definition of asymptotically equivalence of sequences and deferred density, the concepts of asymptotically deferred statistical equivalence and strong deferred asymptotically equivalence of nonnegative sequences are introduced. Besides, the main properties of asymptotically deferred statistical equivalence and strong deferred asymptotically equivalence, some inclusion and equivalence results are given.

Asymptotically Lacunary Statistical Equivalence of Double Sequences of Sets

The concepts of Wijsman asymptotically equivalence, Wijsman asymptotically statistically equivalence, Wijsman asymptotically lacunary equivalence and Wijsman asymptotically lacunary statistical equivalence for sequences of sets were studied by Ulusu and Nuray [24]. In this paper, we get analogous results for double sequences of sets.

On -Asymptotically Statistical Equivalence of Sequences of Sets

This paper presents the notion of -asymptotically statistical equivalence, which is a natural combination of asymptotic -equivalence, and -statistical equivalence for sequences of sets. We find its relations to -asymptotically statistical convergence, strong -asymptotically equivalence, and strong Cesaro -asymptotically equivalence for sequences of sets.