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This research work related to the realm of the "Law of iterated logarithm" for properly normalized partial sums, random sums and weighted sums of i.i.d. random variables. In brief, the classical LIL may not be possible for all integer sequence of real numbers, but for different subsequences of whole integer sequence of real numbers, which has shown by Allan Gut [1985]. His observation motivated me to study the situation for random variables which are belong to the domain of partial attraction of a Semi-Stable law. In fact, under different subsamples of whole integer sequence of real numbers,we established Chover´s form of LIL for subsequences of properly normalized partial sums, random sums and weighted sums. Also We obtained a local limit theorem for weighted sums.

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The author is presently working in the Department of Statistics at the University of Mysore India. Earlier worked in University of Botswana, Addis Ababa University and Sri Krishnadevaraya University. He got his M.Sc. from Sri Krishnadevaraya University, India and earned a Doctorate degree from the University of Mysore, Mysore, India.

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