The present study was undertaken to estimate the (co)variance components and genetic parameters of body weights recorded in Landlly piglets from birth to weaning at weekly intervals (w0 to w6). The data pertained to body weights of 2462 piglets, born to 91 sires and 159 dams across different generations during a 7-year period from 2014 to 2020. Five animal models (I–V), differentiated by inclusion or exclusion of maternal effects with or without covariance between maternal and direct genetic effects, were fitted on the data using the Bayesian algorithm. The analyses were implemented by Gibbs sampling in the BLUPF90 program and Markov chain Monte Carlo (MCMC) methodology was used to draw samples of posterior distribution pertaining to (co)variance components. Based on deviance information criteria (DIC), model V with inclusion of direct additive genetic, direct maternal genetic and permanent environmental effect of dam as random factors along with covariance between direct additive and maternal effects best fitted the data on pre-weaning traits (w0 to w5). Whereas, model I incorporating only the direct additive genetic effect best fitted the weaning weight (w6) data in Landlly piglets. The posterior mean estimates of direct heritability under the best models for W0 to W6 were 0.13, 0.19, 0.29, 0.13, 0.26, 0.32 and 0.46, respectively. Inclusion of the maternal component helped in better partitioning of variance for different body weights in Landlly piglets. The maternal heritability ranged from 0.06 to 0.14, while the litter heritability ranged from 0.11 to 0.15 for pre-weaning weights (W0 to W5) under the best-fit models. The influence of maternal environment was greater than maternal genetic effect from birth to 4th week of age. The results implied that variations in body weight of Landlly pigs were genetically controlled to moderate levels (especially w2 and w4) with contributions from direct additive and maternal genotype that can be exploited by designing efficient breeding programmes.

We study the size and the external path length of random tries and show that they are asymptotically independent in the asymmetric case but strongly dependent with small periodic fluctuations in the symmetric case. Such an unexpected behavior is in sharp contrast to the previously known results on random tries, that the size is totally positively correlated to the internal path length and that both tend to the same normal limit law. These two dependence examples provide concrete instances of bivariate normal distributions (as limit laws) whose components have correlation either zero or one or periodically oscillating. Moreover, the same type of behavior is also clarified for other classes of digital trees such as bucket digital trees and Patricia tries.

The Marcinkiewicz strong law, limn→∞(1 / n1/p)∑k=1n(Dk - D) = 0 almost surely with p ∈ (1, 2), is studied for outer products Dk = {XkX̅kT}, where {Xk} and {X̅k} are both two-sided (multivariate) linear processes (with coefficient matrices (Cl), (C̅l) and independent and identically distributed zero-mean innovations {Ξ} and {Ξ̅}). Matrix sequences Cl and C ̅l can decay slowly enough (as |l| → ∞) that {Xk,X ̅k} have long-range dependence, while {Dk} can have heavy tails. In particular, the heavy-tail and long-range-dependence phenomena for {Dk} are handled simultaneously and a new decoupling property is proved that shows the convergence rate is determined by the worst of the heavy tails or the long-range dependence, but not the combination. The main result is applied to obtain a Marcinkiewicz strong law of large numbers for stochastic approximation, nonlinear function forms, and autocovariances.

For a time series, a plot of sample covariances is a popular way to assess its dependence properties. In this paper we give a systematic characterization of the asymptotic behavior of sample covariances of long-memory linear processes. Central and noncentral limit theorems are obtained for sample covariances with bounded as well as unbounded lags. It is shown that the limiting distribution depends in a very interesting way on the strength of dependence, the heavy-tailedness of the innovations, and the magnitude of the lags.

The effects of low-level aircraft over-flights on behaviour of harlequin ducks (Histrionicus histrionicus) breeding in central Labrador were quantified during 2000–2002. The Canadian Department of National Defence supports a low-level training programme in the 130 000 km2 Military Training Area of Labrador involving military jets. The Institute for Environmental Monitoring and Research (IEMR) undertakes scientific research into environmental impacts of low-level military jet over-flights. A suite of 17 behavioural categories of paired male and female harlequin ducks was modelled, and a canonical variable representing alert behaviour, inactivity on the water and decreased inactivity out of water in response to over-flights represented 73.1% of the variance in the data cluster and provided marked separation of disturbed and undisturbed groups. Behavioural responses of harlequin ducks to military jets were 23 times stronger than their responses to floatplanes, helicopters and military cargo planes, and the significant interaction of aircraft type and noise indicated that noise may be the primary stressor affecting behaviour. A quadratic response of the canonical variable to noise generated from aircraft during standardized 30-minute observation periods was defined. The multivariate analyses were more robust because they indicated covariance in behavioural categories associated with disturbance that was not originally detected in univariate analyses, suggesting the importance of integrating behaviours other than overt responses. The significant effects of military jet over-flights on harlequin duck behaviour emphasize the need to evaluate potential population consequences of aircraft disturbance.

Variograms and covariance functions are the fundamental tools for modeling dependent data observed over time, space, or space-time. This paper aims at constructing nonseparable spatio-temporal variograms and covariance models. Special attention is paid to an intrinsically stationary spatio-temporal random field whose covariance function is of Schoenberg-Lévy type. The correlation structure is studied for its increment process and for its partial derivative with respect to the time lag, as well as for the superposition over time of a stationary spatio-temporal random field. As another approach, we investigate the permissibility of the linear combination of certain separable spatio-temporal covariance functions to be a valid covariance, and obtain a subclass of stationary spatio-temporal models isotropic in space.

This paper studies a class of stationary covariance models, in both the discrete- and the continuous-time domains, which possess a simple functional form γ(τ + τ0)+γ(τ − τ0)− 2γ(τ), where τ 0 is a fixed lag andγ(τ) is an intrinsically stationary variogram, and include the fractional Gaussian noise of Kolmogorov (1940) and a stochastic volatility model of Barndorff-Nielsen and Shephard (2001), (2002) as special cases. Properties of the class, and interesting special cases with long memory, are studied. We also characterize the covariance function via the variogram.

The covariance C(r), r ≥ 0, of a stationary isotropic random closed set Ξ is typically complicated to evaluate. This is the reason that an exponential approximation formula for C(r) has been widely used in the literature, which matches C(0) and C(1)(0), and in many cases also limr→∞C(r). However, for 0 < r < ∞, the accuracy of this approximation is not very high in general. In the present paper, we derive representation formulae for the covariance C(r) and its derivative C(1)(r) using Palm calculus, where r ≥ 0 is arbitrary. As a consequence, an explicit expression is obtained for the second derivative C(2)(0). These results are then used to get a refined exponential approximation for C(r), which additionally matches the second derivative C(2)(0).

A survey of the development of statistical techniques of field experimentation refers to the unified approach of R. A. Fisher, in which analysis and design were closely linked. Randomization of treatment positions bypassed the need for a precise stochastic model, but sometimes at the cost of diminished accuracy.

Alternative covariance methods associated with the name of J. S. Papadakis, taking note of the yields of neighbouring plots, have stimulated much research in recent years. The results to date suggest that, while improved accuracy can often be achieved, the assessment of accuracy is more dependent on the stochastic model. Robust methods for such assessment still need to be finalized; with these alternative methods a closer link between analysis and design is also advocated.

Consider three independent Poisson processes of point events of rates λ1, λ 2 and λ12. Suppose there are two electronic counters, the first recording events from the first and third Poisson processes, and the second recording events from the second and third Poisson processes. Both counters are subject to dead times of Type II, i.e. where all events, whether recorded or not, cause dead times. The covariance of the numbers of events recorded by the two counters in a suitable time period is derived. This extends earlier work of Cox and Isham (1977) and Kingman (1977) who considered the same problem for Type I counters, i.e. where only recorded events cause dead times.

Let {W(t), 0 ≦ t < ∞} be the standard Wiener process. The main purpose of this paper is to present ways of obtaining probabilities of Wiener paths crossing certain curves on various intervals. The results are extended to other kinds of Gaussian processes.

This paper surveys recent results on the covariance structure of processes generated by queues and related stochastic processes. The generated processes include the number X(t) of customers at time t, the waiting time Wn of the nth customer and the inter-departure interval Dn, i.e., the interval between departures of the nth and (n+1)th customers. The importance of these results is discussed, particularly in the field of estimation.