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Almost sure limit points of lightly trimmed sums when the distribution function belongs to the domain of partial attraction of a positive semi-stable law

Vasudeva R., Srilakshminarayana G.

Probability and Mathematical Statistics

2015

Vol. 35, Fasc. 1

129--142

In this paper, we obtain the set of all almost sure limit points of lightly trimmed sums, properly normalized, when the underlying distribution function belongs to the domain of partial attraction of a semi-stable law supported on the positive half axis.

Kiefer’s law of the iterated logarithm for the vector of upper order statistics

Vasudevan R., Moridani A. Y.

Probability and Mathematical Statistics

2011

Vol. 31, Fasc. 2

331--347

Let {Xn} be a sequence of independent identically distributed random variables with a common continuous distribution function and let Mj;n denote the jth upper order statistic among X1,X2, . . . ,Xn, n ≥ j. For a large class of distributions, we obtain the law of the iterated logarithm for {M1,n,M2,n}, properly normalized. As a consequence, we establish a law of the iterated logarithm for the spacings {M1,n −M2,n}.

On global maxima in multiphase queues

Minkevicius S., Steisunas S.

Control and Cybernetics

2005

Vol. 34, no 2

575-588

The target of this research in the queueing theory is to prove the law of the iterated logarithm (LIL) under the conditions of heavy traffic in multiphase queueing systems. In this paper, the LIL for global maxima is proved in the phases of a queueing system studied for an important probability characteristic of the system (total waiting time of a customer and waiting time of a customer).

A generalized Law of the Iterated Logarithm for the largest observation of a triangular array

Adler A.

Probability and Mathematical Statistics

2001

Vol. 21, Fasc. 1

149--157

Consider independent and identically distributed random variables {X, Xkj1≤j≤k, k≥1} from a particular distribution with EX=∞.Weshow that there exists an unusual generalized Law of the Iterated Logarithm involving max 1≤j≤kXkj.

Pseudo-martingales

Jajte R., Paszkiewicz A.

Probability and Mathematical Statistics

1999

Vol. 19, Fasc. 1

181--201

For a probability space (Ω, ℱ, P)and a filtration (Un) in Ω,we consider the sequences (Xn) of random variables satisfying the condition E (Xn+1 − Xn | Un) = 0, n = 1, 2, ... In general, the proces (Xn) is not required to be (Un) adapted and it is called a pseudo-martingale. We indicate simple and natural conditions implying a good asymptotic behaviour of pseudo-martingales. For example: let (Xn, Un) be a uniformly integrable pseudo-martingale with Un ↗ ℱ. Then Xn → X weakly in L1 where [formula]. Some approximation results for σ-fields are obtained with implications to pseudo-martingales. A number of examples is collected.