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Dependent phenomena, such as relational, spatial and temporal phenomena, tend to be characterized by local dependence in the sense that units which are close in a well‐defined sense are dependent. In contrast with spatial and temporal phenomena, though, relational phenomena tend to lack a natural neighbourhood structure in the sense that it is unknown which units are close and thus dependent. Owing to the challenge of characterizing local dependence and constructing random graph models with local dependence, many conventional exponential family random graph models induce strong dependence and are not amenable to statistical inference. We take first steps to characterize local dependence in random graph models, inspired by the notion of finite neighbourhoods in spatial statistics and M‐dependence in time series, and we show that local dependence endows random graph models with desirable properties which make them amenable to statistical inference. We show that random graph models with local dependence satisfy a natural domain consistency condition which every model should satisfy, but conventional exponential family random graph models do not satisfy. In addition, we establish a central limit theorem for random graph models with local dependence, which suggests that random graph models with local dependence are amenable to statistical inference. We discuss how random graph models with local dependence can be constructed by exploiting either observed or unobserved neighbourhood structure. In the absence of observed neighbourhood structure, we take a Bayesian view and express the uncertainty about the neighbourhood structure by specifying a prior on a set of suitable neighbourhood structures. We present simulation results and applications to two real world networks with ‘ground truth’.

The Berry-Esseen bound provides an upper bound on the Kolmogorov distance between a random variable and the normal distribution. In this paper, we establish Berry-Esseen bounds with optimal rates for self-normalized sums of locally dependent random variables, assuming only a second-moment condition. Our proof leverages Stein’s method and introduces a novel randomized concentration inequality, which may also be of independent interest for other applications. Our main results have applied to self-normalized sums of m -dependent random variables and graph dependency models.

Let {Xn: n ϵ ℤd] be a weakly dependent stationary random field with maxima MA := sup{Xi: i ϵ A} for finite A ⊂ ℤd and Mn := sup{Xi: 1 ≤ i ≤ n } for n ϵ ℕd. In a general setting we prove that ℙ(M(N₁(n),N₂(n),...,Nd(n))≤vn) = exp(-ndℙ(X₀ > vn, ${M_{{A_n}}}\\, \\leqslant \\,{v_n}$)) + o(1) for some increasing sequence of sets An of size o(nd), where (N₁(n), N₂(n),..., Nd(n)) → (∞, ∞, . . . , ∞) and N₁(n)N₂(n) · · · Nd(n) ~ nd. The sets An are determined by a translation-invariant total order ≼ on ℤd. For a class of fields satisfying a local mixing condition, including m-dependent ones, the main theorem holds with a constant finite A replacing An. The above results lead to new formulas for the extremal index for random fields. The new method for calculating limiting probabilities for maxima is compared with some known results and applied to the moving maximum field.

In the paper we solve the limit problem for partial maxima of m-dependent stationary random fields and we extend the obtained solution to fields satisfying some local mixing conditions. New methods for describing the limitting distribution of maxima are proposed. A notion of a phantom distribution function for a random field is investigated. As an application, several original formulas for calculation of the extremal index are provided. Moving maxima and moving averages as well as Gaussian fields satisfying the Berman condition are considered.

We establish Nagaev and Rosenthal-type inequalities for dependent random variables. The imposed dependence conditions, which are expressed in terms of functional dependence measures, are directly related to the physical mechanisms of the underlying processes and are easy to work with. Our results are applied to nonlinear time series and kernel density estimates of linear processes.

A flexible list sequential Tips sampling method is introduced and studied. It can reproduce any given sampling design without replacement, of fixed or random sample size. The method is a splitting method and uses successive updating of inclusion probabilities. The main advantage of the method is in real-time sampling situations where it can be used as a powerful alternative to Bernoulli and Poisson sampling and can give any desired second-order inclusion probabilities and thus considerably reduce the variability of the sample size.

The present paper continues the research started in the works of I. S. Borisov, A. A. Bystrov, and the author [Siberian Math. J., 47 (2006), pp. 980-989; Siberian Adv. Math., 18 (2008), pp. 244-259], where some limit theorems were proved for canonical U - and V -statistics based on stationary observations under ... , or ... mixing. However, the conditions in the papers mentioned insure a certain limit behavior of these statistics and include either essential restrictions on finite dimensional distributions of the initial stationary sequence or some regularity conditions for kernels of the statistics under consideration. In the present work, it is shown that, in the case of stationary sequences of m-dependent observations, it is possible to remove the above-mentioned additional conditions while describing the limit behavior of U - and V -statistics. (ProQuest: ... denotes formulae/symbols omitted.)

Wasssily Hoeffding's terminal illness and untimely death in 1991 put an end to efforts that were made to interview him for Statistical Science. An account of his scientific work is given in Fisher and Sen [The Collected Works of Wassily Hoeffding (1994) Springer], but the present authors felt that the statistical community should also be told about the life of this remarkable man. He contributed much to statistical science, but will also live on in the memory of those who knew him as a kind and modest teacher and friend, whose courage and learning were matched by a wonderful sense of humor.

In this paper, a theorem on the moderate deviation principle for random arrays under m -dependence with unbounded m is established. This partially extends the results of Chen (Stat. Probab. Lett. 35:123–134, 1997 ). As an application, the moderate deviation principle for the truncation estimator of the variance in the analysis of time series is obtained.

m-dependent stationary infinitely divisible sequences are characterized as a class of generalized finite moving average sequences via the structure of the associated Lévy measure. This characterization is used to find necessary and sufficient conditions for the weak convergence of centered and normalized partial sums of m-dependent stationary infinitely divisible sequences. Partial sum convergence for stationary infinitely divisible sequences that can be approximated by m-dependent ones is then studied.