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We introduce a betting game where the gambler aims to guess the last success epoch in a series of inhomogeneous Bernoulli trials paced randomly in time. At a given stage, the gambler may bet on either the event that no further successes occur, or the event that exactly one success is yet to occur, or may choose any proper range of future times (a trap). When a trap is chosen, the gambler wins if the last success epoch is the only one that falls in the trap. The game is closely related to the sequential decision problem of maximising the probability of stopping on the last success. We use this connection to analyse the best-choice problem with random arrivals generated by a Pólya-Lundberg process.

Suppose N independent Bernoulli trials with success probabilities$$p_1, p_2,\\ldots $$p 1 , p 2 , … are observed sequentially at times of a mixed binomial process. The task is to maximise, by using a nonanticipating stopping strategy, the probability of stopping at the last success. The case$$p_k=1/k$$p k = 1 / k has been studied by many authors as a version of the familiar best choice problem, where both N and the observation times are random. We consider a more general profile$$p_k=\\theta /(\\theta +k-1)$$p k = θ / ( θ + k - 1 ) and assume that the prior distribution of N is negative binomial with shape parameter$$\\nu $$ν , so the arrivals occur at times of a mixed Poisson process. The setting with two parameters offers a high flexibility in understanding the nature of the optimal strategy, which we show is intrinsically related to monotonicity properties of the Gaussian hypergeometric function. Using this connection, we find that the myopic stopping strategy is optimal if and only if$$\\nu \\ge \\theta $$ν ≥ θ . Furthermore, we derive formulas to assess the winning probability and discuss limit forms of the problem for large N .

Suppose N independent Bernoulli trials with success probabilities p 1 , p 2 , … are observed sequentially at times of a mixed binomial process. The task is to maximise, by using a nonanticipating stopping strategy, the probability of stopping at the last success. The case p k = 1 / k has been studied by many authors as a version of the familiar best choice problem, where both N and the observation times are random. We consider a more general profile p k = θ / ( θ + k - 1 ) and assume that the prior distribution of N is negative binomial with shape parameter ν , so the arrivals occur at times of a mixed Poisson process. The setting with two parameters offers a high flexibility in understanding the nature of the optimal strategy, which we show is intrinsically related to monotonicity properties of the Gaussian hypergeometric function. Using this connection, we find that the myopic stopping strategy is optimal if and only if ν ≥ θ . Furthermore, we derive formulas to assess the winning probability and discuss limit forms of the problem for large N .

We consider an infinite balls-in-boxes occupancy scheme with boxes organised in nested hierarchy, and random probabilities of boxes defined in terms of iterated fragmentation of a unit mass. We obtain a multivariate functional limit theorem for the cumulative occupancy counts as the number of balls approaches infinity. In the case of fragmentation driven by a homogeneous residual allocation model our result generalises the functional central limit theorem for the block counts in Ewens’ and more general regenerative partitions.

The Bernoulli sieve is a recursive construction of a random composition (ordered partition) of an integer n. This composition can be induced by sampling from a random discrete distribution which has frequencies equal to the sizes of components of a stick-breaking interval partition of [0, 1]. We exploit the Markov property of the composition and its renewal representation to study the number of its parts. We derive asymptotics of the moments and prove a central limit theorem.

We consider an occupancy scheme in which \"balls\" are identified with n points sampled from the standard exponential distribution, while the role of \"boxes\" is played by the spacings induced by an independent random walk with positive and nonlattice steps. We discuss the asymptotic behavior of five quantities: the index $K_{n}^{*}$ of the last occupied box, the number $K_{n}$ of occupied boxes, the number $K_{n,0}$ of empty boxes whose index is at most $K_{n}^{*}$ , the index $W_{n}$ of the first empty box and the number of balls $Z_{n}$ , in the last occupied box. It is shown that the limiting distribution of properly scaled and centered $K_{n}^{*}$ coincides with that of the number of renewals not exceeding log n. A similar result is shown for $K_{n}$ and $W_{n}$ under a side condition that prevents occurrence of very small boxes. The condition also ensures that $K_{n,0}$ converges in distribution. Limiting results for $Z_{n}$ are established under an assumption of regular variation.

For τ, a stopping rule adapted to a sequence of n independent and identically distributed observations, we define the loss to be E[q(R τ)], where R j is the rank of the jth observation and q is a nondecreasing function of the rank. This setting covers both the best-choice problem, with q(r) = 1(r > 1), and Robbins' problem, with q(r) = r. As n tends to ∞, the stopping problem acquires a limiting form which is associated with the planar Poisson process. Inspecting the limit we establish bounds on the stopping value and reveal qualitative features of the optimal rule. In particular, we show that the complete history dependence persists in the limit; thus answering a question asked by Bruss (2005) in the context of Robbins' problem.

A two-parameter family of random permutations of$[n]$is introduced, with distribution conditionally uniform given the counts of upper and lower records. The family interpolates between two versions of Ewens' distribution. A distinguished role of the family is determined by the fact that every sequence of coherent permutations$(π _n,n=1,2,\\ldots)$with the indicated kind of sufficiency is obtainable by randomisation of the parameters. Generating algorithms and asymptotic properties of the permutations follow from the representation via initial ranks.

The full-information best choice problem asks one to find a strategy maximising the probability of stopping at the minimum (or maximum) of a sequence X1,⋯,Xn of i.i.d. random variables with continuous distribution. In this paper we look at more general models, where independent Xj’s may have different distributions, discrete or continuous. A central role in our study is played by the running minimum process, which we first employ to re-visit the classic problem and its limit Poisson counterpart. The approach is further applied to two explicitly solvable models: in the first the distribution of the jth variable is uniform on {j,⋯,n}, and in the second it is uniform on {1,⋯,n}.

For positive q ≠ 1, the q-exchangeability of an infinite random word is introduced as quasi-invariance under permutations of letters, with a special cocycle which accounts for inversions in the word. This framework allows us to extend the q-analog of de Finetti's theorem for binary sequences—see Greschonig and Schmidt [Colloq. Math. 84/85 (2000) 495–514]—to general real-valued sequences. In contrast to the classical case of exchangeability (q = 1), the order on ${\\Bbb R}$ plays a significant role for the q-analogs. An explicit construction of ergodic q-exchangeable measures involves random shuffling of ${\\Bbb N}$ = {1, 2,...} by iteration of the geometric choice. Connections are established with transient Markov chains on q-Pascal pyramids and invariant random flags over the Galois fields.