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By Mark J. Beran

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Due to the assumed iid structure of the random walk, the differences between the successive ascending (descending) ladders are also iid, and therefore the distribution of the sum of any k of them can be handled by forming a k-fold convolution ‘power’ of the distribution of one. These convolution powers of the common distribution of the ascending ladder heights make up the ‘positive part’ of the Wiener-Hopf factorization. The ‘negative part’ stems from the incomplete excursion, by first noting that its distribution remains the same when the order of its steps is reversed and that, when considered in this manner ‘backwards in time’, the position at which the original random walk had its maximum now becomes a minimal record value.

One of the main sources of inspiration was Volume 2 of Introduction to Probability Theory and its Applications by Feller [8], which had come out sixteen years after the publication of Volume 1 [7], and was then followed five years later by an expanded Second Edition. Feller was a master in making probability theory look like it were a collection of challenging puzzles, for which one, if only sufficiently clever, could find an elegant solution by some ingenious trick that actually made the original problem look like it had been trivial.

16:185–199, 1973. Collection of articles dedicated to the memory of Hanna Neumann, II. [17] T. P. Speed and M. W. Evans. A note on commutative Baer rings. J. Aust. Math. , 13:1–6, 1972. 6 1 Algebra 1 Algebra 7 8 1 Algebra 1 Algebra 9 10 1 Algebra 1 Algebra 11 12 1 Algebra 1 Algebra 13 14 1 Algebra 1 Algebra 15 16 1 Algebra 1 Algebra 17 18 1 Algebra 1 Algebra 19 20 1 Algebra 1 Algebra 21 22 1 Algebra 1 Algebra 23 24 1 Algebra 1 Algebra 25 26 1 Algebra 1 Algebra 27 28 1 Algebra 1 Algebra 29 30 1 Algebra 1 Algebra 31 32 1 Algebra 1 Algebra 33 34 1 Algebra 1 Algebra 35 36 1 Algebra 1 Algebra 37 38 1 Algebra Chapter 2 Probability Elja Arjas Writing a brief commentary on three of Terry Speed’s papers in probability brings to mind many memories from a time now almost forty years away.

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Statistical continuum theories by Mark J. Beran

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