One possible explanation for the substantial organismal differences between humans and chimpanzees is that there have been changes in gene regulation. Given what is known about transcription factor binding sites, this motivates the following probability question: given a 1000 nucleotide region in our genome, how long does it take for a specified six to nine letter word to appear in that region in some individual? Stone and Wray [Mol. Biol. Evol. 18 (2001) 1764–1770] computed 5,950 years as the answer for six letter words. Here, we will show that for words of length 6, the average waiting time is 100,000 years, while for words of length 8, the waiting time has mean 375,000 years when there is a 7 out of 8 letter match in the population consensus sequence (an event of probability roughly 5/16) and has mean 650 million years when there is not. Fortunately, in biological reality, the match to the target word does not have to be perfect for binding to occur. If we model this by saying that a 7 out of 8 letter match is good enough, the mean reduces to about 60,000 years.

We consider random variables of the form F=f(V₁, …, V_n), where f is a smooth function and V_i, i∈ℕ, are random variables with absolutely continuous law p_i(y) dy. We assume that p_i, i=1, …, n, are piecewise differentiable and we develop a differential calculus of Malliavin type based on ∂lnp_i. This allows us to establish an integration by parts formula E(∂_iϕ(F)G)=E(ϕ(F)H_i(F, G)), where H_i(F, G) is a random variable constructed using the differential operators acting on F and G. We use this formula in order to give numerical algorithms for sensitivity computations in a model driven by a Lévy process.

This paper provides a construction of a Fleming–Viot measure valued diffusion process, for which the transition function is known, by extending recent ideas of the Gibbs sampler based Markov processes. In particular, we concentrate on the Chapman–Kolmogorov consistency conditions which allows a simple derivation of such a Fleming–Viot process, once a key and apparently new combinatorial result for Pólya-urn sequences has been established.

Let {s_ij: i, j=1, 2, …} consist of i.i.d. random variables in ℂ with $\mathsf{E}s_{11}=0$, $\mathsf{E}|s_{11}|^{2}=1$. For each positive integer N, let s_k=s_k(N)=(s_1k, s_2k, …, s_Nk)^T, 1≤k≤K, with K=K(N) and K/N→c>0 as N→∞. Assume for fixed positive integer L, for each N and k≤K, α_k=(α_k(1), …, α_k(L))^T is random, independent of the s_ij, and the empirical distribution of (α₁, …, α_K), with probability one converging weakly to a probability distribution H on ℂ^L. Let β_k=β_k(N)=(α_k(1)s_k^T, …, α_k(L)s_k^T)^T and set C=C(N)=(1/N)∑_k=2^K β_k β_k^*. Let σ²>0 be arbitrary. Then define SIR₁=(1/N)β₁^*(C+σ²I)⁻¹ β₁, which represents the best signal-to-interference ratio for user 1 with respect to the other K−1 users in a direct-sequence code-division multiple-access system in wireless communications. In this paper it is proven that, with probability 1, SIR₁ tends, as N→∞, to the limit ∑_ℓ,ℓ'=1^Lα̅₁(ℓ)α₁(ℓ')a_ℓ,ℓ', where A=(a_ℓ,ℓ') is nonrandom, Hermitian positive definite, and is the unique matrix of such type satisfying $A=\bigl(c\,\mathsf{E}\frac{\mathbf{\alpha}\mathbf{\alpha}^{*}}{1+\mathbf{\alpha}^{*}A\mathbf{\alpha}}+\sigma^{2}I_{L}\bigr)^{-1}$, where α∈ℂ^L has distribution H. The result generalizes those previously derived under more restricted assumptions.

The biological theory of adaptive dynamics proposes a description of the long-term evolution of a structured asexual population. It is based on the assumptions of large population, rare mutations and small mutation steps, that lead to a deterministic ODE describing the evolution of the dominant type, called the “canonical equation of adaptive dynamics.” Here, in order to include the effect of stochasticity (genetic drift), we consider self-regulated randomly fluctuating populations subject to mutation, so that the number of coexisting types may fluctuate. We apply a limit of rare mutations to these populations, while keeping the population size finite. This leads to a jump process, the so-called “trait substitution sequence,” where evolution proceeds by successive invasions and fixations of mutant types. Then we apply a limit of small mutation steps (weak selection) to this jump process, that leads to a diffusion process that we call the “canonical diffusion of adaptive dynamics,” in which genetic drift is combined with directional selection driven by the gradient of the fixation probability, also interpreted as an invasion fitness. Finally, we study in detail the particular case of multitype logistic branching populations and seek explicit formulae for the invasion fitness of a mutant deviating slightly from the resident type. In particular, second-order terms of the fixation probability are products of functions of the initial mutant frequency, times functions of the initial total population size, called the invasibility coefficients of the resident by increased fertility, defence, aggressiveness, isolation or survival.

In this paper we consider the optimal dividend problem for an insurance company whose risk process evolves as a spectrally negative Lévy process in the absence of dividend payments. The classical dividend problem for an insurance company consists in finding a dividend payment policy that maximizes the total expected discounted dividends. Related is the problem where we impose the restriction that ruin be prevented: the beneficiaries of the dividends must then keep the insurance company solvent by bail-out loans. Drawing on the fluctuation theory of spectrally negative Lévy processes we give an explicit analytical description of the optimal strategy in the set of barrier strategies and the corresponding value function, for either of the problems. Subsequently we investigate when the dividend policy that is optimal among all admissible ones takes the form of a barrier strategy.

Let $\mathbf{s}_{k}=\frac{1}{\sqrt{N}}(v_{1k},\ldots,v_{Nk})^{T}$, k=1, …, K, where {v_ik, i, k=1, …} are independent and identically distributed random variables with Ev₁₁=0 and Ev₁₁²=1. Let S_k=(s₁, …, s_k−1, s_k+1, …, s_K), P_k=diag (p₁, …, p_k−1, p_k+1, …, p_K) and β_k=p_ks_k^T(S_kP_kS_k^T+σ²I)⁻¹s_k, where p_k≥0 and the β_k is referred to as the signal-to-interference ratio (SIR) of user k with linear minimum mean-square error (LMMSE) detection in wireless communications. The joint distribution of the SIRs for a finite number of users and the empirical distribution of all users’ SIRs are both investigated in this paper when K and N tend to infinity with the limit of their ratio being positive constant. Moreover, the sum of the SIRs of all users, after subtracting a proper value, is shown to have a Gaussian limit.

This work focuses on time-inhomogeneous Markov chains with two time scales. Our motivations stem from applications in reliability and dependability, queueing networks, financial engineering and manufacturing systems, where two-time-scale scenarios naturally arise. One of the important questions is: As the rate of fluctuation of the Markov chain goes to infinity, if the limit distributions of suitably centered and scaled sequences of occupation measures exist, what can be said about the convergence rate? By combining singular perturbation techniques and probabilistic methods, this paper addresses the issue by concentrating on sequences of centered and scaled functional occupation processes. The results obtained are then applied to treat a queueing system example.

Suppose that m_n observations are made from the distribution R and n−m_n from the distribution S. Associate with each pair, x from R and y from S, a nonnegative score ϕ(x, y). An optimal reading policy is one that yields a sequence m_n that maximizes $\mathbb{E}(M(n))$, the expected sum of the (n−m_n)m_n observed scores, uniformly in n. The alternating policy, which switches between the two sources, is the optimal nonadaptive policy. In contrast, the greedy policy, which chooses its source to maximize the expected gain on the next step, is shown to be the optimal policy. Asymptotics are provided for the case where the R and S distributions are discrete and ϕ(x, y)=1 or 0 according as x=y or not (i.e., the observations match). Specifically, an invariance result is proved which guarantees that for a wide class of policies, including the alternating and the greedy, the variable M(n) obeys the same CLT and LIL. A more delicate analysis of the sequence $\mathbb{E}(M(n))$ and the sample paths of M(n), for both alternating and greedy, reveals the slender sense in which the latter policy is asymptotically superior to the former, as well as a sense of equivalence of the two and robustness of the former.

We present sufficient conditions for sums of dependent point processes to converge in distribution to a Poisson process. This extends the classical result of Grigelionis [Theory Probab. Appl. 8 (1963) 172–182] for sums of uniformly null point processes that have Poisson limits. Included is an application in which a particle-survivor point process converges to a Poisson process. This result sheds light on the “surprising” Poisson limit of the species competition process of Durrett and Limic [Stochastic Process. Appl. 102 (2002) 301–309].

We compare convergence rates of Metropolis–Hastings chains to multi-modal target distributions when the proposal distributions can be of “local” and “small world” type. In particular, we show that by adding occasional long-range jumps to a given local proposal distribution, one can turn a chain that is “slowly mixing” (in the complexity of the problem) into a chain that is “rapidly mixing.” To do this, we obtain spectral gap estimates via a new state decomposition theorem and apply an isoperimetric inequality for log-concave probability measures. We discuss potential applicability of our result to Metropolis-coupled Markov chain Monte Carlo schemes.

We analyze a class of spatial random spanning trees built on a realization of a homogeneous Poisson point process of the plane. This tree has a simple radial structure with the origin as its root.

We first use stochastic geometry arguments to analyze local functionals of the random tree such as the distribution of the length of the edges or the mean degree of the vertices. Far away from the origin, these local properties are shown to be close to those of a variant of the directed spanning tree introduced by Bhatt and Roy.

We then use the theory of continuous state space Markov chains to analyze some nonlocal properties of the tree, such as the shape and structure of its semi-infinite paths or the shape of the set of its vertices less than k generations away from the origin.

This class of spanning trees has applications in many fields and, in particular, in communications.

In many situations, the decision maker observes items in sequence and needs to determine whether or not to retain a particular item immediately after it is observed. Any decision rule creates a set of items that are selected. We consider situations where the available information is the rank of a present observation relative to its predecessors. Certain “natural” selection rules are investigated. Theoretical results are presented pertaining to the evolution of the number of items selected, measures of their quality and the time it would take to amass a group of a given size.

This paper shows the existence of independent random matching of a large (continuum) population in both static and dynamic systems, which has been popular in the economics and genetics literatures. We construct a joint agent-probability space, and randomized mutation, partial matching and match-induced type-changing functions that satisfy appropriate independence conditions. The proofs are achieved via nonstandard analysis. The proof for the dynamic setting relies on a new Fubini-type theorem for an infinite product of Loeb transition probabilities, based on which a continuum of independent Markov chains is derived from random mutation, random partial matching and random type changing.