Start with a compact set K ⊂ Rd. This has a random number of daughter sets, each of which is a (rotated and scaled) copy of K and all of which are inside K. The random mechanism for producing daughter sets is used independently on each of the daughter sets to produce the second generation of sets, and so on, repeatedly. The random fractal set F is the limit, as n goes to ∞, of the union of the nth generation sets. In addition, K has a (suitable, random) mass which is divided randomly between the daughter sets, and this random division of mass is also repeated independently, indefinitely. This division of mass will correspond to a random self-similar measure on F. The multifractal spectrum of this measure is studied here. Our main contributions are dealing with the geometry of realisations in Rd and drawing systematically on known results for general branching processes. In this way we generalise considerably the results of Arbeiter and Patzschke (1996) and Patzschke (1997).

The so-called STIT tessellations form a class of homogeneous (spatially stationary) tessellations in Rd which are stable under the nesting/iteration operation. In this paper we establish the mixing property for these tessellations and give the decay rate of P(A ∩ M = ∅, ThB ∩ M = ∅) / P(A ∩ Y = ∅)P(B ∩ Y = ∅) − 1, where A and B are both compact connected sets, h is a vector of Rd, Th is the corresponding translation operator, and M is a STIT tessellation.

Suppose that {Xt} is a Markov chain such as the state space model for a threshold GARCH time series. The regularity assumptions for a drift condition approach to establishing the ergodicity of {Xt} typically are ϕ-irreducibility, aperiodicity, and a minorization condition for compact sets. These can be very tedious to verify due to the discontinuous and singular nature of the Markov transition probabilities. We first demonstrate that, for Feller chains, the problem can at least be simplified to focusing on whether the process can reach some neighborhood that satisfies the minorization condition. The results are valid not just for the transition kernels of Markov chains but also for bounded positive kernels, opening the possibility for new ergodic results. More significantly, we show that threshold GARCH time series and related models of interest can often be embedded into Feller chains, allowing us to apply the conclusions above.

In this paper we consider a risk model where claims arrive according to a Markovian arrival process (MAP). When the surplus becomes negative or the insurer is in deficit, the insurer could borrow money at a constant debit interest rate to repay the claims. We derive the integro-differential equations satisfied by the discounted penalty functions and discuss the solutions. A matrix renewal equation is obtained for the discounted penalty function provided that the initial surplus is nonnegative. Based on this matrix renewal equation, we present some asymptotic formulae for the discounted penalty functions when the claim size distributions are heavy tailed.

Within an anticipative stochastic calculus framework, we study a market game with asymmetric information and feedback effects. We derive necessary and sufficient criteria for the existence of Nash equilibria and study how general welfare is affected by the level of information. In particular, we show that, under certain conditions in a competitive environment, an increased level of information may in fact lower the level of general welfare, leading to the so-called Hirshleifer effect (see Hirshleifer (1971)). Finally, we determine equilibrium prices for particular pieces of information, by extending our market game with a pre-stage, in which information is traded.

In this paper we study the fighter problem with discrete ammunition. An aircraft (fighter) equipped with n anti-aircraft missiles is intercepted by enemy airplanes, the appearance of which follows a homogeneous Poisson process with known intensity. If j of the n missiles are spent at an encounter, they destroy an enemy plane with probability a(j), where a(0) = 0 and {a(j)} is a known, strictly increasing concave sequence, e.g. a(j) = 1 - qj, 0 < q < 1. If the enemy is not destroyed, the enemy shoots the fighter down with known probability 1 - u, where 0 ≤ u ≤ 1. The goal of the fighter is to shoot down as many enemy airplanes as possible during a given time period [0, T]. Let K(n, t) be the smallest optimal number of missiles to be used at a present encounter, when the fighter has flying time t remaining and n missiles remaining. Three seemingly obvious properties of K(n, t) have been conjectured: (A) the closer to the destination, the more of the n missiles one should use; (B) the more missiles one has; the more one should use; and (C) the more missiles one has, the more one should save for possible future encounters. We show that (C) holds for all 0 ≤ u ≤ 1, that (A) and (B) hold for the ‘invincible fighter’ (u = 1), and that (A) holds but (B) fails for the ‘frail fighter’ (u = 0); the latter is shown through a surprising counterexample, which is also valid for small u > 0 values.

We analyze a class of signal-to-interference-and-noise-ratio (SINR) random graphs. These random graphs arise in the modeling packet transmissions in wireless networks. In contrast to previous studies on SINR graphs, we consider both a space and a time dimension. The spatial aspect originates from the random locations of the network nodes in the Euclidean plane. The time aspect stems from the random transmission policy followed by each network node and from the time variations of the wireless channel characteristics. The combination of these random space and time aspects leads to fluctuations of the SINR experienced by the wireless channels, which in turn determine the progression of packets in space and time in such a network. In this paper we study optimal paths in such wireless networks in terms of first passage percolation on this random graph. We establish both ‘positive’ and ‘negative’ results on the associated time constant. The latter determines the asymptotics of the minimum delay required by a packet to progress from a source node to a destination node when the Euclidean distance between the two tends to ∞. The main negative result states that this time constant is infinite on the random graph associated with a Poisson point process under natural assumptions on the wireless channels. The main positive result states that, when adding a periodic node infrastructure of arbitrarily small intensity to the Poisson point process, the time constant is positive and finite.

In this paper we study the number of random records in an arbitrary split tree (or, equivalently, the number of random cuttings required to eliminate the tree). We show that a classical limit theorem for the convergence of sums of triangular arrays to infinitely divisible distributions can be used to determine the distribution of this number. After normalization the distributions are shown to be asymptotically weakly 1-stable. This work is a generalization of our earlier results for the random binary search tree in Holmgren (2010), which is one specific case of split trees. Other important examples of split trees include m-ary search trees, quad trees, medians of (2k + 1)-trees, simplex trees, tries, and digital search trees.

We analyze the mean cost of the partial match queries in random two-dimensional quadtrees. The method is based on fragmentation theory. The convergence is guaranteed by a coupling argument of Markov chains, whereas the value of the limit is computed as the fixed point of an integral equation.

We show how the extremal behavior of d-variate Archimedean copulas can be deduced from their stochastic representation as the survival dependence structure of an ℓ1-symmetric distribution (see McNeil and Nešlehová (2009)). We show that the extremal behavior of the radial part of the representation is determined by its Williamson d-transform. This leads in turn to simple proofs and extensions of recent results characterizing the domain of attraction of Archimedean copulas, their upper and lower tail-dependence indices, as well as their associated threshold copulas. We outline some of the practical implications of their results for the construction of Archimedean models with specific tail behavior and give counterexamples of Archimedean copulas whose coefficient of lower tail dependence does not exist.

A two-dimensional random vector in the domain of attraction of an extreme value distribution G is said to be asymptotically independent (i.e. in the tail) if G is the product of its marginal distribution functions. Ledford and Tawn (1996) discussed a form of residual dependence in this case. In this paper we give a characterization of this phenomenon (see also Ramos and Ledford (2009)), and offer extensions to higher-dimensional spaces and stochastic processes. Systemic risk in the banking system is treated in a similar framework.

We consider the asymptotic variance of the departure counting process D(t) of the GI/G/1 queue; D(t) denotes the number of departures up to time t. We focus on the case where the system load ϱ equals 1, and prove that the asymptotic variance rate satisfies limt→∞varD(t) / t = λ(1–2/π)(ca2 + cs2), where λ is the arrival rate, and ca2 and cs2 are squared coefficients of variation of the interarrival and service times, respectively. As a consequence, the departures variability has a remarkable singularity in the case in which ϱ equals 1, in line with the BRAVO (balancing reduces asymptotic variance of outputs) effect which was previously encountered in finite-capacity birth-death queues. Under certain technical conditions, our result generalizes to multiserver queues, as well as to queues with more general arrival and service patterns. For the M/M/1 queue, we present an explicit expression of the variance of D(t) for any t.

We propose an approximation for the inverse first passage time problem. It is similar in spirit and method to the tangent approximation for the original first passage time problem. We provide evidence that the technique is quite accurate in many cases. We also identify some cases where the approximation performs poorly.

We consider a branching process with Poissonian immigration where individuals have inheritable types. At rate θ, new individuals singly enter the total population and start a new population which evolves like a supercritical, homogeneous, binary Crump-Mode-Jagers process: individuals have independent and identically distributed lifetime durations (nonnecessarily exponential) during which they give birth independently at a constant rate b. First, using spine decomposition, we relax previously known assumptions required for almost-sure convergence of the total population size. Then, we consider three models of structured populations: either all immigrants have a different type, or types are drawn in a discrete spectrum or in a continuous spectrum. In each model, the vector (P1, P2,…) of relative abundances of surviving families converges almost surely. In the first model, the limit is the GEM distribution with parameter θ / b.