On moderate deviations in Poisson approximation

In this paper we first use the distribution of the number of records to demonstrate that the right tail probabilities of counts of rare events are generally better approximated by the right tail probabilities of a Poisson distribution than those of the normal distribution. We then show that the moderate deviations in Poisson approximation generally require an adjustment and, with suitable adjustment, we establish better error estimates of the moderate deviations in Poisson approximation than those in [18]. Our estimates contain no unspecified constants and are easy to apply. We illustrate the use of the theorems via six applications: Poisson-binomial distribution, the matching problem, the occupancy problem, the birthday problem, random graphs, and 2-runs. The paper complements the works [16], [8], and [18].

The Double Occupancy Problem

Imagine a time machine travels back in time in the same way it persists into the future (i.e. by persisting backwards). For instance, this is how time machines move through time in H.G. Wells’s The Time Machine. Such a machine would have a problem: immediately upon moving back in time it would collide with its earlier self. This is the ‘Double Occupancy Problem’. Using the notions introduced in Chapters 1–3, this chapter explains how to avoid this ‘Double Occupancy Problem’: such time machines can travel back in time just as long as they are both in motion and move back in time bit by bit.

On the occupancy problem for a regime-switching model

This article studies the expected occupancy probabilities on an alphabet. Unlike the standard situation, where observations are assumed to be independent and identically distributed, we assume that they follow a regime-switching Markov chain. For this model, we (1) give finite sample bounds on the expected occupancy probabilities, and (2) provide detailed asymptotics in the case where the underlying distribution is regularly varying. We find that in the regularly varying case the finite sample bounds are rate optimal and have, up to a constant, the same rate of decay as the asymptotic result.

Paradoxes of Time Travel

Paradoxes of Time Travel is a comprehensive study of the philosophical issues raised by the possibility of time travel. The book begins, in Chapter 1, by explaining the concept of time travel and clarifying the central question to be addressed: Is time travel compatible with the laws of metaphysics and, in particular, the laws concerning time, freedom, causation, and identity? Chapter 2 then explores the various temporal paradoxes, including the double-occupancy problem, the no-destination argument, and the famous twin paradox of special relativity. Chapters 3 and 4 focus on the paradoxes of freedom, including various versions of the grandfather paradox. Chapter 5 covers causal paradoxes, including the bootstrapping paradox, the problems of backward causation, and the various puzzles raised by causal loops. Chapter 6 then concludes by looking at various paradoxes of identity. This includes a discussion of different theories of change and persistence, and an exploration of the various puzzles raised by self-visitation.

A Binomial Splitting Process in Connection with Corner Parking Problems

This paper studies a special type of binomial splitting process. Such a process can be used to model a high dimensional corner parking problem as well as determining the depth of random PATRICIA (practical algorithm to retrieve information coded in alphanumeric) tries, which are a special class of digital tree data structures. The latter also has natural interpretations in terms of distinct values in independent and identically distributed geometric random variables and the occupancy problem in urn models. The corresponding distribution is marked by a logarithmic mean and a bounded variance, which is oscillating, if the binomial parameterpis not equal to ½, and asymptotic to one in the unbiased case. Also, the limiting distribution does not exist as a result of the periodic fluctuations.

A Binomial Splitting Process in Connection with Corner Parking Problems

This paper studies a special type of binomial splitting process. Such a process can be used to model a high dimensional corner parking problem as well as determining the depth of random PATRICIA (practical algorithm to retrieve information coded in alphanumeric) tries, which are a special class of digital tree data structures. The latter also has natural interpretations in terms of distinct values in independent and identically distributed geometric random variables and the occupancy problem in urn models. The corresponding distribution is marked by a logarithmic mean and a bounded variance, which is oscillating, if the binomial parameter p is not equal to ½, and asymptotic to one in the unbiased case. Also, the limiting distribution does not exist as a result of the periodic fluctuations.