Real networks often exhibit clustering, the tendency to form relatively small groups of nodes with high edge densities. This clustering property can cause large numbers of small and dense subgraphs to emerge in otherwise sparse networks. Subgraph counts are an important and commonly used source of information about the network structure and function. We study probability distributions of subgraph counts in a community affiliation graph. This is a random graph generated as an overlay of m partly overlapping independent Bernoulli random graphs (layers) $G_1,\dots,G_m$ with variable sizes and densities. The model is parameterised by a joint distribution of layer sizes and densities. When m grows linearly in the number of nodes n, the model generates sparse random graphs with a rich statistical structure, admitting a nonvanishing clustering coefficient and a power-law limiting degree distribution. In this paper we establish the normal and $\alpha$-stable approximations to the numbers of small cliques, cycles, and more general 2-connected subgraphs of a community affiliation graph.

The generalized perturbative approach is an all-purpose variant of Stein’s method used to obtain rates of normal approximation. Originally developed for functions of independent random variables, this method is here extended to functions of the realization of a hidden Markov model. In this dependent setting, rates of convergence are provided in some applications, leading, in each instance, to an extra log-factor vis-à-vis the rate in the independent case.

Explicit bounds are given for the Kolmogorov and Wasserstein distances between a mixture of normal distributions, by which we mean that the conditional distribution given some $\sigma$ -algebra is normal, and a normal distribution with properly chosen parameter values. The bounds depend only on the first two moments of the first two conditional moments given the $\sigma$ -algebra. The proof is based on Stein’s method. As an application, we consider the Yule–Ornstein–Uhlenbeck model, used in the field of phylogenetic comparative methods. We obtain bounds for both distances between the distribution of the average value of a phenotypic trait over n related species, and a normal distribution. The bounds imply and extend earlier limit theorems by Bartoszek and Sagitov.

It is well known that assumptions of monotonicity in size-bias couplings may be used to prove simple, yet powerful, Poisson approximation results. Here we show how these assumptions may be relaxed, establishing explicit Poisson approximation bounds (depending on the first two moments only) for random variables which satisfy an approximate version of these monotonicity conditions. These are shown to be effective for models where an underlying random variable of interest is contaminated with noise. We also state explicit Poisson approximation bounds for sums of associated or negatively associated random variables. Applications are given to epidemic models, extremes, and random sampling. Finally, we also show how similar techniques may be used to relax the assumptions needed in a Poincaré inequality and in a normal approximation result.

In this paper we derive the exact asymptotics of the probability of Parisian ruin for self-similar Gaussian risk processes. Additionally, we obtain the normal approximation of the Parisian ruin time and derive an asymptotic relation between the Parisian and the classical ruin times.

We study a type of nonnormal small jump approximation of infinitely divisible distributions. By incorporating compound Poisson, gamma, and normal distributions, the approximation has a higher order of cumulant matching than its normal counterpart, and, hence, in many cases a higher rate of approximation error decay as the cutoff for the jump size tends to 0. The parameters of the approximation are easy to fix, and its random sampling has the same order of computational complexity as the normal approximation. An error bound of the approximation in terms of the total variation distance is derived. Simulations empirically show that the nonnormal approximation can have a significantly smaller error than its normal counterpart.

In the so-called lightbulb process, on days r = 1,…,n, out of n lightbulbs, all initially off, exactly r bulbs, selected uniformly and independent of the past, have their status changed from off to on, or vice versa. With X the number of bulbs on at the terminal time n, an even integer, and μ = n/2, σ2 = var(X), we have supz ∈ R | P((X - μ)/σ ≤ z) - P(Z ≤ z) | ≤ nΔ̅0/2σ2 + 1.64n/σ3 + 2/σ, where Z is a standard normal random variable and Δ̅0 = 1/2√n + 1/2n + e−n/2/3 for n ≥ 6, yielding a bound of order O(n−1/2) as n → ∞. A similar, though slightly larger bound, holds for odd n. The results are shown using a version of Stein's method for bounded, monotone size bias couplings. The argument for even n depends on the construction of a variable Xs on the same space as X that has the X-size bias distribution, that is, which satisfies E[Xg(X)] = μE[g(Xs)] for all bounded continuous g, and for which there exists a B ≥ 0, in this case B = 2, such that X ≤ Xs ≤ X + B almost surely. The argument for odd n is similar to that for even n, but one first couples X closely to V, a symmetrized version of X, for which a size bias coupling of V to Vs can proceed as in the even case. In both the even and odd cases, the crucial calculation of the variance of a conditional expectation requires detailed information on the spectral decomposition of the lightbulb chain.

In this article we propose a bootstrap test for the probability of ruin in the compound Poisson risk process. We adopt the P-value approach, which leads to a more complete assessment of the underlying risk than the probability of ruin alone. We provide second-order accurate P-values for this testing problem and consider both parametric and nonparametric estimators of the individual claim amount distribution. Simulation studies show that the suggested bootstrap P-values are very accurate and outperform their analogues based on the asymptotic normal approximation.

Let X1, X2,… and Y1, Y2,… be two sequences of absolutely continuous, independent and identically distributed (i.i.d.) random variables with equal means E(Xi)=E(Yi), i=1,2,… In this work we provide upper bounds for the total variation and Kolmogorov distances between the distributions of the partial sums ∑i=1nXi and ∑i=1nYi. In the case where the distributions of the Xis and the Yis are compared with respect to the convex order, the proposed upper bounds are further refined. Finally, in order to illustrate the applicability of the results presented, we consider specific examples concerning gamma and normal approximations.

We describe a stochastic susceptible–infective–susceptible (SIS) model for transmission of infectious disease through a population, incorporating both direct host–host transmission and indirect transmission via free-living infectious stages (e.g. environmental bacteria). Existence of a quasi-stationary distribution conditional upon nonextinction of infection is established. A bivariate Ornstein–Uhlenbeck approximation is used to investigate the long-term behaviour of the process conditional upon nonextinction of infection. We show that indirect transmission leads to lower variability in the number of infected hosts present in quasi-stationarity and, consequently, to a greater tendency of infection to persist, compared with a model with direct transmission only and the same average individual infectivity. Some numerical work illustrating these results is presented.

In this paper, we give necessary and sufficient conditions to ensure the validity of confidence intervals, based on the central limit theorem, in simulations of highly reliable Markovian systems. We resort to simulations because of the frequently huge state space in practical systems. So far the literature has focused on the property of bounded relative error. In this paper we focus on ‘bounded normal approximation’ which asserts that the approximation of the normal law, suggested by the central limit theorem, does not deteriorate as the reliability of the system increases. Here we see that the set of systems with bounded normal approximation is (strictly) included in the set of systems with bounded relative error.

This paper is devoted to the study of the initial reserve, as a function of the retention limit, needed to assure that the probability of ruin, at the end of a certain period of time, is not higher than an agreed value, for an excess of loss treaty. To assess the probability of ruin, the normal and the normal power approximation are used. It is shown that the initial reserve is not in general an increasing function of the retention, having a minimum under fair assumptions.

Consider the number Xm of comparisons made in a sequence of comparisons between two opponents, which terminates as soon as one opponent wins m comparisons. The convergence of Xm to the normal variable is completely characterized. The normal approximations to the probability function and to the distribution function of Xm are obtained for any sufficiently large m, together with estimates of the errors in these approximations. Similar results are obtained for the negative binomial distribution as well. Finally, some simple estimates of the mean, variance and the incomplete beta function with equal arguments are constructed.

A general method based on a limit theorem for generation of random numbers from infinitely divisible distributions with essentially given Lévy measure is studied. Some classes of infinitely divisible distributions that appear in a natural way in this context are paid particular attention.

It is well known that the number of renewals in the time interval [0, t] for an ordinary renewal process is approximately normally distributed under general conditions. We give a remainder term estimate for this normal distribution approximation.

Several simple stochastic models are given for a finite closed system of individuals existing in clusters which may come together to form larger clusters which may in turn split up. Some of these models are analysed and compared in equilibrium. Several of the models fit into a general framework established and investigated by Whittle (1965a); it is shown that these models have an equilibrium solution of a particularly simple form, deduced by Whittle, if and only if the models are stochastically reversible. A normal approximation to two of the models in equilibrium is found to give the same mean value as a deterministic approximation.