The distribution of the sequence of ranked maximum and minimum values attained during excursions of a standard Brownian bridge (B br t ; 0 t 1) is described. The height M br+ j of the jth highest maximum over a positive excursion of the bridge has the same distribution as M br+ 1 =j, where the distribution of M br+ 1 = sup 0t1 B br t is given by L'evy's formula P (M br+ 1 ? x) = e \Gamma2x 2 . The probability density of the height M br j of the jth highest maximum of excursions of the reflecting Brownian bridge (jB br t j; 0 t 1) is given by a modification of the known `-function series for the density of M br 1 = sup 0t1 jB br t j. These results are obtained from a more general description of the distribution of ranked values of a homogeneous functional of excursions of the standardized bridge of a self-similar recurrent Markov process. Keywords: Brownian bridge, Brownian excursion, Brownian scaling, local time, selfsimilar recurrent Markov process, Bessel p...

We provide an O(N) algorithm for a non-sequential semi-predictive encoder whose pointwise redundancy with respect to any (unbounded depth) tree source is O(1) bits per state above Rissanen’s lower bound. This is achieved by using the Burrows Wheeler transform (BWT), an invertible permutation transform that has been suggested for lossless data compression. First, we use the BWT only as an efficient computational tool for pruning context trees, and encode the input sequence rather than the BWT output. Second, we estimate the minimum description length (MDL) source by incorporating suffix tree methods to construct the unbounded depth context tree that corresponds to the input sequence in O(N) time. Third, we point out that a variety of previous source coding methods required superlinear complexity for determining which tree source state generated each of the symbols of the input. We show how backtracking from the BWT output to the input sequence enables to solve this problem in O(N) worst-case complexity.

The theory of the incomplete gamma functions, as part of the theory of conuent hypergeometric functions, has received its rst systematic exposition by Tricomi in the early 1950s. His own contributions, as well as further advances made thereafter, are surveyed here with particular emphasis on asymptotic expansions, zeros, inequalities, computational methods, and applications.

Abstract. In the present paper, necessary and sufficient conditions are established for a function involving divided differences of the digamma and trigamma functions to be completely monotonic. Consequently, necessary and sufficient conditions are derived for a function involving the ratio of two gamma functions to be logarithmically completely monotonic, and some double inequalities are deduced for bounding divided differences of polygamma functions. 1.

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Abstract. We give a probabilistic interpretation for the Barnes G-function which appears in random matrix theory and in analytic number theory in the important moments conjecture due to Keating-Snaith for the Riemann zeta function, via the analogy with the characteristic polynomial of random unitary matrices. We show that the Mellin transform of the characteristic polynomial of random unitary matrices and the Barnes G-function are intimately related with products and sums of gamma, beta and log-gamma variables. In particular, we show that the law of the modulus of the characteristic polynomial of random unitary matrices can be expressed with the help of products of gamma or beta variables, and that the reciprocal of the Barnes G-function has a Lévy-Khintchin type representation. These results lead us to introduce the so called generalized gamma convolution variables.