We introduce a definition of long range dependence of random processes and fields on an (unbounded) index space $T\subseteq \mathbb{R}^d$ in terms of integrability of the covariance of indicators that a random function exceeds any given level. This definition is specifically designed to cover the case of random functions with infinite variance. We show the value of this new definition and its connection to limit theorems via some examples including subordinated Gaussian as well as random volatility fields and time series.

We discuss the joint temporal and contemporaneous aggregation of N independent copies of random-coefficient AR(1) processes driven by independent and identically distributed innovations in the domain of normal attraction of an $\alpha$ -stable distribution, $0< \alpha \le 2$ , as both N and the time scale n tend to infinity, possibly at different rates. Assuming that the tail distribution function of the random autoregressive coefficient regularly varies at the unit root with exponent $\beta > 0$ , we show that, for $\beta < \max (\alpha, 1)$ , the joint aggregate displays a variety of stable and non-stable limit behaviors with stability index depending on $\alpha$ , $\beta$ and the mutual increase rate of N and n. The paper extends the results of Pilipauskaitė and Surgailis (2014) from $\alpha =2$ to $0 < \alpha < 2$ .

Contemporaneous aggregation of N independent copies of a random-coefficient AR(1) process with random coefficient a ∈ (−1, 1) and independent and identically distributed innovations belonging to the domain of attraction of an α-stable law (0 < α < 2) is discussed. We show that, under the normalization N1/α, the limit aggregate exists, in the sense of weak convergence of finite-dimensional distributions, and is a mixed stable moving average as studied in Surgailis, Rosiński, Mandrekar and Cambanis (1993). We focus on the case where the slope coefficient a has probability density vanishing regularly at a = 1 with exponent b ∈ (0, α − 1) for α ∈ (1, 2). We show that in this case, the limit aggregate {X̅t} exhibits long memory. In particular, for {X̅t}, we investigate the decay of the codifference, the limit of partial sums, and the long-range dependence (sample Allen variance) property of Heyde and Yang (1997).

Let , be a recurrent Markov renewal process and Mik(t) be the expected value of Nk(t) provided that at the initial moment the system is in state i. It is shown that when the mean recurrence times μ ii are finite, the differences μ ij Mki (t) – t behave asymptotically the same for all states i and k.

We derive the limit behaviour of the distribution tail of the global maximum of a critical Galton–Watson process and also of the expectations of partial maxima of the process, when the offspring law belongs to the domain of attraction of a stable law. Thus the Lindvall (1976) and Athreya (1988) results are extended to the infinite variance case. It is shown that in the general case these two asymptotics are closely related to each other, and the latter follows readily from the former. We also discuss a related problem from the theory of general branching processes.

We consider an L1 analogue of the least squares estimator for the parameters of stationary, finite-order autoregressions. This estimator, the least absolute deviation (LAD), is shown to be strongly consistent via a result that may have independent interest. The striking feature is that the conditions are so mild as to include processes with infinite variance, notably the stationary, finite autoregressions driven by stable increments in Lα, α > 1. Finally, sampling properties of LAD are compared to those of least squares. Together with a known convergence rate result for least squares, the Monte Carlo study provides evidence for a conjecture on the convergence rate of LAD.