The asymptotic behavior of the Jaccard index in G(n, p), the classical Erdös–Rényi random graph model, is studied as n goes to infinity. We first derive the asymptotic distribution of the Jaccard index of any pair of distinct vertices, as well as the first two moments of this index. Then the average of the Jaccard indices over all vertex pairs in G(n, p) is shown to be asymptotically normal under an additional mild condition that $np\to\infty$ and $n^2(1-p)\to\infty$.

The crystal growth of NH4-illite (NH4-I) from the hydrothermal system of Harghita Bãi (Eastern Carpathians) was deduced from the shapes of crystal thickness distributions (CTDs). The <2 mm and the <2-0.2 mm fractions of clay samples collected from the argillized andesite rocks consist of NH4-illite-smectite (I-S) interstratified structures (R1, R2, and R3-type ordering) with a variable smectite-layer content. The NH4-I-S (40-5% S) structures were identified underground in a hydrothermal breccia structure, whereas the K-I/NH4-I mixtures were found at the deepest level sampled (-110 m). The percentage of smectite interlayers generally decreases with increasing depth in the deposit. This decrease in smectite content is related to the increase in degree of fracturing in the breccia structure and corresponds to a general increase in mean illite crystal thickness. In order to determine the thickness distributions of NH4-I crystals (fundamental illite particles) which make up the NH4-I-S interstratified structures and the NH4-I/K-I mixtures, 27 samples were saturated with Li and aqueous solutions of PVP-10 to remove swelling and then were analyzed by X-ray diffraction. The profiles for the mean crystallite thickness (Tmean) and crystallite thickness distribution (CTD) of NH4-I crystallites were determined by the Bertaut-Warren-Averbach method using the MudMaster computer code. The Tmean of NH4-I from NH4-I-S samples ranges from 3.4 to 7.8 nm. The Tmean measured for the NH4-I/K-I mixture phase ranges from 7.8 nm to 11.7 nm (NH4-I) and from 12.1 to 24.7 nm (K-I).

The CTD shapes of NH4-I fundamental particles are asymptotic and lognormal, whereas illites from NH4-I/K-I mixtures have bimodal shapes related to the presence of two lognormal-like CTDs corresponding to NH4-I and K-I.

The crystal-growth mechanism for NH4-I samples was simulated using the Galoper code. Reaction pathways for NH4-I crystal nucleation and growth could be determined for each sample by plotting their CTD parameters on an α-ß2 diagram constructed using Galoper. This analysis shows that NH4-I crystals underwent simultaneous nucleation and growth, followed by surface-controlled growth without simultaneous nucleation.

The process of software testing usually involves the correction of a detected bug immediately upon detection. In this article, in contrast, we discuss continuous time testing of a software with periodic debugging in which bugs are corrected, instead of at the instants of their detection, at some pre-specified time points. Under the assumption of renewal distribution for the time between successive occurrence of a bug, maximum-likelihood estimation of the initial number of bugs in the software is considered, when the renewal distribution belongs to any general parametric family or is arbitrary. The asymptotic properties of the estimated model parameters are also discussed. Finally, we investigate the finite sample properties of the estimators, specially that of the number of initial number of bugs, through simulation.

We consider the problem of estimation in a partially observed discrete-time Galton-Watson branching process, focusing on the first two moments of the offspring distribution. Our study is motivated by modelling the counts of new cases at the onset of a stochastic epidemic, allowing for the facts that only a part of the cases is detected, and that the detection mechanism may affect the evolution of the epidemic. In this setting, the offspring mean is closely related to the spreading potential of the disease, while the second moment is connected to the variability of the mean estimators. Inference for branching processes is known for its nonstandard characteristics, as compared with classical inference. When, in addition, the true process cannot be directly observed, the problem of inference suffers significant further perturbations. We propose nonparametric estimators related to those used when the underlying process is fully observed, but suitably modified to take into account the intricate dependence structure induced by the partial observation and the interaction scheme. We show consistency, derive the limiting laws of the estimators, and construct asymptotic confidence intervals, all valid conditionally on the explosion set.

We study a collector's problem with K renewal arrival processes for different type items, where the objective is to collect complete sets. In particular, we derive the asymptotic distribution of the sequence of interarrival times between set completions.

Let X1,...,Xn1 be arandom sample from a population with mean µ 1 and variance $\sigma_1^2$ , and X1,...,Xn1 be a random sample fromanother population with mean µ 2 and variance $\sigma_2^2$ independent of{Xi,1 ≤ i ≤ n1 }.Consider the twosample t-statistic $ T={{\bar X-\bar Y-(\mu_1-\mu_2)} \over\sqrt{s_1^2/n_1+s_2^2/n_2}}$ . This paper shows thatln P(T ≥ x) ~ -x²/2 for any x := x(n1,n2) satisfying x → ∞, x = o(n1 + n2)1/2 as n1,n2 → ∞ provided 0 < c1 ≤ n1/n2 ≤ c2 < ∞. If, inaddition, E|X1|3 < ∞, E|Y1|3 < ∞, then $\frac{P(T \geq x)}{1-\Phi(x)} \to 1 $ holds uniformly in x ∈ (O,o((n1 + n2)1/6))

The maximum drawdown at time T of a random process on [0,T] can be defined informally as the largest drop from a peak to a trough. In this paper, we investigate the behaviour of this statistic for a Brownian motion with drift. In particular, we give an infinite series representation of its distribution and consider its expected value. When the drift is zero, we give an analytic expression for the expected value, and for nonzero drift, we give an infinite series representation. For all cases, we compute the limiting (T → ∞) behaviour, which can be logarithmic (for positive drift), square root (for zero drift) or linear (for negative drift).

We study the properties of sums of lower records from a distribution on [0,∞) which is either continuous, except possibly at the origin, or has support contained in the set of nonnegative integers. We find a necessary and sufficient condition for the partial sums of lower records to converge almost surely to a proper random variable. An explicit formula for the Laplace transform of the limit is derived. This limit is infinitely divisible and we show that all infinitely divisible random variables with continuous Lévy measure on [0,∞) originate as infinite sums of lower records.

A sequence of first-order integer-valued autoregressive (INAR(1)) processes is investigated, where the autoregressive-type coefficient converges to 1. It is shown that the limiting distribution of the conditional least squares estimator for this coefficient is normal and the rate of convergence is n3/2. Nearly critical Galton–Watson processes with unobservable immigration are also discussed.

Although forest tree pattern-dynamics has long been a focus for ecological theory, many aspects of basic analysis remain problematic. This paper describes, examines and illustrates an ‘extreme-value’ approach to clump detection. Simulations demonstrate that the approach, though simple, is sensitive and well suited to identifying aggregation, even in small data sets. The test is complementary to the χ2 approach and significantly outperforms it when high-density clusters form a relatively small fraction of the total area. This difference holds true as density declines, though the power of both tests decreases and the χ2 test eventually fails. Though powerful, the extreme-value tests are slightly conservative. The approach is adaptable to other null distributions and applications. An illustration uses tree data from a Ugandan forest plot with records from 1939 to 1992. One plausible explanation for observed stem increases in this plot is an unusually high incidence of large tree-fall events. Evidence for this is sought through spatial localization of various stem populations. Chi-square tests detect non-random density patterns in the populations that comprise ‘all stems > 10 cm diameter recorded in the 1992 survey’, and ‘all recruits in 1976’. In the extreme-value test three populations are found to be clumped, these are ‘pioneers > 10 cm diameter recorded in the 1992 survey’, ‘all recruits in 1992’ and again ‘all recruits in 1976’. These patterns, particularly the pioneer trees and stem recruitment, signal a ‘gap-dynamic process’ of tree regeneration. Despite this, the trends observed in the Ugandan plot appear unlikely to be caused by tree-falls alone. Various technical and ecological aspects of the extreme-value approach and tree spatial analyses are discussed.

Consider a janken game (scissors-paper-rock game) started by n players such that (1) the first round is played by n players, (2) the losers of each round (if any) retire from the rest of the game, and (3) the game ends when only one player (winner) is left. Let Wn be the number of rounds played through the game. Among other things, it is proved that (2/3)nWn is asymptotically (as n → ∞) distributed according to the exponential distribution with mean ⅓, provided that each player chooses one of the three strategies (scissors, paper, rock) with equal probability and independently from other players in any round.

Consider a classical compound Poisson model. The safety loading can be positive, negative or zero. Explicit expressions for the distributions of the surplus prior and at ruin are given in terms of the ruin probability. Moreover, the asymptotic behaviour of these distributions as the initial capital tends to infinity are obtained. In particular, for positive safety loading the Cramer case, the case of subexponential distributions and some intermediate cases are discussed.

Markovian algorithms for estimating the global maximum or minimum of real valued functions defined on some domain Ω ⊂ ℝd are presented. Conditions on the search schemes that preserve the asymptotic distribution are derived. Global and local search schemes satisfying these conditions are analysed and shown to yield sharper confidence intervals when compared to the i.i.d. case.

A criterion is given for the existence of a stationary and causal multivariate integer-valued autoregressive process, MGINAR(p). The autocovariance function of this process being identical to the autocovariance function of a standard Gaussian MAR(p), we deduce that the MGINAR(p) process is nothing but a MAR(p) process. Consequently, the spectral density is directly found and gives good insight into the stochastic structure of a MGINAR(p). The estimation of parameters of the model, as well as the forecasting of the series, is discussed.

We investigate the asymptotic behaviour of the distribution of the number of comparisons needed by a quicksort-style selection algorithm that finds the lth smallest in a set of n numbers. Letting n tend to infinity and considering the values l = 1, ···,n simultaneously we obtain a limiting stochastic process. This process admits various interpretations: it arises in connection with a representation of real numbers induced by nested random partitions and also in connection with expected path lengths of a random walk in a random environment on a binary tree.

Weighted mortality rates are commonly used in actuarial work, but the inter-relationship between the weights used and the underlying mortality rates seems not to have been widely investigated.

Calculation of the ratio of weighted mortality rates to conventional mortality rates provides a simple means for an insurance company to track changes in the underlying mortality of its portfolio over time, and acts as an early warning system for possible deterioration of underwriting results. Asymptotic distributions are found for this ratio, and for the mortality rates themselves. It is suggested that insurance companies commence to gather data for the calculation of this ratio for the insurance sector as a whole, for the main annuity and assurance classes.

Consider N urns into which n balls are dropped independently with equal probability of hitting each urn and constant probability p of staying in the urn. We find the characteristic function of the joint distribution of KN and LN where KN is the number of urns that have not been hit, and LN is the number of urns where all balls have fallen through. Furthermore, we study the asymptotic distribution of (KN, LN) as n, N → ∞ at various rates.

The asymptotic final size distribution of a multitype Reed–Frost process, a chain-binomial model for the spread of an infectious disease in a finite, closed multitype population, is derived, as the total population size grows large. When all subgroups are of comparable size, the infection pattern irreducible and the epidemic started by a small number of initial infectives, the classical threshold behaviour is obtained, depending on the basic reproduction rate of the disease in the population, and the asymptotic distributions for small and large outbreaks can be found. The same techniques can then be used to study other asymptotic situations, e.g. small groups in an otherwise large population, large numbers of initial infectives and reducible infection patterns.

The concept of strong mixing is used to obtain a generalization of results on the asymptotic distribution of finite-parameter estimates of linear processes and extend them for stationary sequences and random fields.

We derive asymptotic forms for the distributions of k-point scan statistics as the interval L under study becomes infinite, while k and the window length are held fixed. In the Poisson case the intensity is also held fixed. In the uniform case the number of points N becomes infinite and N/L tends to a limit, representing a limiting intensity. These results are made explicit for k = 3, and in the Poisson case provide approximations which are typically accurate to six or seven figures, even for small L.