We study a nonparametric method for estimating the boundary measure of a compact body G ⊂ ℝd (the boundary length when d = 2 and the surface area for d = 3) in the case when this measure agrees with the corresponding Minkowski content. The estimator we consider is closely related to the one proposed in Cuevas, Fraiman and Rodríguez-Casal (2007). Our method relies on two sets of random points, drawn inside and outside the set G, with different sampling intensities. Strong consistency and asymptotic normality are obtained under some shape hypotheses on the set G. Some applications and practical aspects are briefly discussed.

Consider randomly scattered radio transceivers in ℝd , each of which can transmit signals to all transceivers in a given randomly chosen region about itself. If a signal is retransmitted by every transceiver that receives it, under what circumstances will a signal propagate to a large distance from its starting point? Put more formally, place points {x i } in ℝd according to a Poisson process with intensity 1. Then, independently for each x i , choose a bounded region A x i from some fixed distribution and let be the random directed graph with vertex set whenever x j ∈ x i + A x i . We show that, for any will almost surely have an infinite directed path, provided the expected number of transceivers that can receive a signal directly from x i is at least 1 + η, and the regions x i + A x i do not overlap too much (in a sense that we shall make precise). One example where these conditions hold, and so gives rise to percolation, is in ℝd , with each A x i a ball of volume 1 + η centred at x i , where η → 0 as d → ∞. Another example is in two dimensions, where the A x i are sectors of angle ε γ and area 1 + η, uniformly randomly oriented within a fixed angle (1 + ε)θ. In this case we can let η → 0 as ε → 0 and still obtain percolation. The result is already known for the annulus, i.e. that the critical area tends to 1 as the ratio of the radii tends to 1, while it is known to be false for the square (l∞) annulus. Our results show that it does however hold for the randomly oriented square annulus.

In this work we give precise asymptotic expressions for the probability of the existence of fixed-size components at the threshold of connectivity for random geometric graphs.

In this paper we obtain an elementary expression for the chord length distribution function of a regular polygon. The formula is derived using δ-formalism in Pleijel identity. In the particular cases of a regular triangle, a square, a regular pentagon, and a regular hexagon, our formula coincides with the results of Sulanke (1961), Gille (1988), Aharonyan and Ohanyan (2005), and Harutyunyan (2007), respectively.

Given a pure-jump subordinator (i.e. nondecreasing Lévy process with no drift) with continuous Lévy measure ν, we derive a formula for the distribution function F s (x; t) at time t of the associated subordinator whose Lévy measure is the restriction of ν to (0,s]. It will be expressed in terms of ν and the marginal distribution function F (⋅; t) of the original process. A generalization concerning an arbitrary truncation of ν will follow. Under certain conditions, an analogous formula will be obtained for the nth derivative, ∂n F s (x; t) ∂ x n . The requirement that ν is continuous is shown to have no intrinsic meaning. A number of interesting results involving the size ordered jumps of subordinators will be derived. An appropriate approximation for the small jumps of a gamma process will be considered, leading to a revisiting of the generalized Dickman distribution.

We show that, contrary to common wisdom, the cumulative input process in a fluid queue with cluster Poisson arrivals can converge, in the slow growth regime, to a fractional Brownian motion, and not to a Lévy stable motion. This emphasizes the lack of robustness of Lévy stable motions as ‘birds-eye’ descriptions of the traffic in communication networks.

In this paper we study the tail and the extremal behaviors of stationary solutions of threshold autoregressive (TAR) models. It is shown that a regularly varying noise sequence leads in general to only an O-regularly varying tail of the stationary solution. Under further conditions on the partition, it is shown however that TAR(S,1) models of order 1 with S regimes have regularly varying tails, provided that the noise sequence is regularly varying. In these cases, the finite-dimensional distribution of the stationary solution is even multivariate regularly varying and its extremal behavior is studied via point process convergence. In particular, a TAR model with regularly varying noise can exhibit extremal clusters. This is in contrast to TAR models with noise in the maximum domain of attraction of the Gumbel distribution and which is either subexponential or in ℒ(γ) with γ > 0. In this case it turns out that the tail of the stationary solution behaves like a constant times that of the noise sequence, regardless of the order and the specific partition of the TAR model, and that the process cannot exhibit clusters on high levels.

In this article we analyse the behaviour of the extremes of a random walk in a random scenery. The random walk is assumed to be in the domain of attraction of a stable law, and the scenery is assumed to be in the domain of attraction of an extreme value distribution. The resulting random sequence is stationary and strongly dependent if the underlying random walk is recurrent. We prove a limit theorem for the extremes of the resulting stationary process. However, if the underlying random walk is recurrent, the limit distribution is not in the class of classical extreme value distributions.

In this paper we obtain some equivalent conditions and sufficient conditions for the local and nonlocal asymptotics of the φ-moments of the overshoot and undershoot of a random walk, where φ is a nonnegative, long-tailed function. By the strong Markov property, it can be shown that the moments of the overshoot and undershoot and the moments of the first ascending ladder height of a random walk satisfy some renewal equations. Therefore, in this paper we first investigate the local and nonlocal asymptotics for the moments of the first ascending ladder height of a random walk, and then give some equivalent conditions and sufficient conditions for the asymptotics of the solutions to some renewal equations. Using the above results, the main results of this paper are obtained.

In this paper we first consider the expectation of the total discounted claim costs up to the time of ruin, and then, more generally, we study the expectation of the total discounted operating costs up to the time of default, which is the first passage time of a surplus process downcrossing a given level. These two quantities include the expected discounted penalty function at ruin or the Gerber–Shiu function, the expected total discounted dividends up to ruin, and other interesting quantities as special cases among a class of risk processes. As an illustration, we consider a piecewise-deterministic compound Poisson risk model. This model recovers many risk models appearing in the literature such as the compound Poisson risk models with interest, absolute ruin, dividends, multiple thresholds, and their dual models. We derive and solve the integro-differential equation for the expected present value of the total discounted operating costs up to default. The solutions to the expected present value of the total discounted operating costs up to default can be used as a unified approach to solving many ruin-related quantities. As applications, we derive explicit solutions for the expected accumulated utility up to ruin, the absolute ruin probability with varying borrowing rates, the expected total discounted claim costs up to ruin, the Gerber–Shiu function with two-sided jumps, and the price for a perpetual American put option with two-sided jumps.

Markov and semi-Markov processes with block tridiagonal transition matrices for their embedded discrete-time Markov chains are underlying stochastic models in many applied probability problems. In particular, identity-by-descent (IBD) problems for uncle-type and cousin-type relationships fall into this class. More specifically, the exact distributions of relevant IBD statistics for two individuals in either an uncle-type or cousin-type relationship are of interest. Such statistics are the amount of genome shared IBD by the two related individuals on a chromosomal segment and the number of IBD pieces on such a segment. These lead to special reward distributions associated with block tridiagonal transition matrices for continuous-time Markov chains. A method is provided for calculating explicit, closed-form expressions for Laplace transforms of general reward functions for such Markov chains. Some calculation results on the cumulative probabilities of relevant IBD statistics via a numerical inversion of the Laplace transforms are also provided for uncle/nephew and first-cousin relationships.

Parallel processing is a way to use resources efficiently by processing several jobs simultaneously on different servers. In a well-controlled environment where the status of the servers and the jobs are well known, everything is nearly deterministic and replicating jobs on different servers is obviously a waste of resources. However, in a poorly controlled environment where the servers are unreliable and/or their capacity is highly variable, it is desirable to design a system that is robust in the sense that it is not affected by the poorly performing servers. By replicating jobs and assigning them to several different servers simultaneously, we not only achieve robustness but we can also make the system more efficient under certain conditions so that the jobs are processed at a faster rate overall. In this paper we consider the option of replicating jobs and study how the performance of different ‘degrees’ of replication, ranging from no replication to full replication, affects the performance of a system of parallel servers.

The speed of extinction for some generalized Jiřina processes {X n } is discussed. We first discuss the geometric speed. Under some mild conditions, the results reveal that the sequence {c n }, where c does not equal the pseudo-drift parameter at x = 0, cannot estimate the speed of extinction accurately. Then the general case is studied. We determine a group of sufficient conditions such that X n /c n , with a suitable constant c n , converges almost surely as n → ∞ to a proper, nondegenerate random variable. The main tools used in this paper are exponent martingales and stochastic growth models.

An occupancy problem with an infinite number of bins and a random probability vector for the locations of the balls is considered. The respective sizes of the bins are related to the split times of a Yule process. The asymptotic behavior of the landscape of the first empty bins, i.e. the set of corresponding indices represented by point processes, is analyzed and convergences in distribution to mixed Poisson processes are established. Additionally, the influence of the random environment, the random probability vector, is analyzed. It is represented by two main components: an independent, identically distributed sequence and a fixed random variable. Each of these components has a specific impact on the qualitative behavior of the stochastic model. It is shown in particular that, for some values of the parameters, some rare events, which are identified, determine the asymptotic behavior of the average values of the number of empty bins in some regions.