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Let {(ξni, ηni), 1 ≤ i ≤ n, n ≥ 1} be a triangular array of independent bivariate elliptical random vectors with the same distribution function as (S1,ρnS1+1−ρn2S2),ρn∈(0,1), where (S1, S2) is a bivariate spherical random vector. For the distribution function of radius S12+S22 belonging to the max-domain of attraction of the Weibull distribution, the limiting distribution of maximum of this triangular array is known as the convergence rate of ρn to 1 is given. In this paper, under the refinement of the rate of convergence of ρn to 1 and the second-order regular variation of the distributional tail of radius, precise second-order distributional expansions of the normalized maxima of bivariate elliptical triangular arrays are established. [ABSTRACT FROM AUTHOR]

We give expansions about the Gumbel distribution in inverse powers of n and log n for M, the maximum of a sample size n or n+1 when the j-th observation is $\mu (\tfrac{j} {n}) + e_j $, µ is any smooth trend function and the residuals { e} are independent and identically distributed with as x→∞. We illustrate practical value of the expansions using simulated data sets. [ABSTRACT FROM AUTHOR]

Let { X( t), t ≥ 0} be a standard (zero-mean, unit-variance) stationary Gaussian process with correlation function r(·) and continuous sample paths. In this paper, we consider the maxima M( T) = max{ X( t), ∀ t ∈ [0, T]} with random index T, where T/ T converges to a non-degenerate distribution or to a positive random variable in probability, and show that the limit distribution of M( T) exists under some additional conditions related to the correlation function r(·). [ABSTRACT FROM AUTHOR]

Let M denote the partial maximum of a strictly stationary sequence ( X). Suppose that some of the random variables of ( X) can be observed and let $\tilde M_n$ stand for the maximum of observed random variables from the set { X, ..., X}. In this paper, the almost sure limit theorems related to random vector ( $\tilde M_n$ , M) are considered in terms of i.i.d. case. The related results are also extended to weakly dependent stationary Gaussian sequence as its covariance function satisfies some regular conditions. [ABSTRACT FROM AUTHOR]

In this paper we construct models obtained by suitably combining Brownian motions and telegraphs in such a way that their transition functions satisfy higher-order parabolic or hyperbolic equations of different types. Equations with time-varying coefficients are also derived by considering processes endowed either with drift or with suitable modifications of their structure. Finally the distribution of the maximum of the iterated Brownian motion (along with some other properties) is presented. [ABSTRACT FROM AUTHOR]