Drago Špoljarić

On Statistical Properties of Palindromes in DNA

The main aim of this article is to determine the distribution of the number of palindromes of a fixed length in a DNA sequence. Palindrome in DNA is a part of the DNA sequence which is equal to its complementary sequence read backwards (C ∼ G, A ∼ T). We examine the general case where the distribution of the nucleobases is arbitrary. Further, we determine conditions in which Normal approximation can be used. Special attention is given to coding and non-coding regions in DNA and to the distribution of bases in those parts of DNA.

Limiting distribution of the number of clumps of palindromes in DNA

ABSTRACT An important aspect in the analysis of long DNA sequences is to identify whether palindromes are over- or under-represented. The essential step in that direction is the analysis of the limiting distribution of the number of clumps of palindromes where clump is defined as the overlapping occurrence of palindromes. Using the Chen–Stein method, it is shown in this paper that the limiting distribution, under suitable conditions and a type of heterogeneous sequence, is the Poisson distribution. Moreover, error bounds and the rate of convergence are derived in terms of total variation distance between two probability distributions.

On randomly spaced observations and continuous-time random walks

We consider random variables observed at arrival times of a renewal process, which possibly depends on those observations and has regularly varying steps with infinite mean. Due to the dependence and heavy tailed steps, the limiting behavior of extreme observations until a given time

Extremal behaviour of random variables observed in renewal times

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Extremes of random variables observed in renewal times

tends to be rather involved. We describe this asymptotics and generalize several partial results which appeared in this setting. In contrast to the earlier studies, our analysis is based in the point processes theory. The theory is applied to determine the asymptotic distribution of maximal excursions and sojourn times for continuous time random walks.