Stable non-Gaussian random processes , by G. Samorodnitsky and M. S. Taqqu. Pp. 632. £49.50. 1994. ISBN 0-412-05171-0 (Chapman and Hall).

distribution, queuing theory, random walks, and so on. On many topical issues he is prepared to admit that there are no definitive answers, considering, inter alia, the following questions: how convinced are we that the trends in climate change over the past thirty years are an indication of global warming rather than just random fluctuations? how much belief can there be in miracles? is the movement of share prices better explained by chaos theory than by statistics? He also emphasizes that issues such as psychology and economic efficiency sometimes have as much of a bearing on eventual decisions as purely statistical considerations.

Foundations of modern probability (2nd edn), by Olav Kallenberg. Pp. 638. £49 (hbk). 2002. ISBN 0 387 95313 2 (Springer-Verlag).

We introduce the notions of scaling transition and distributional long-range dependence for stationary random fields $Y$ on $\mathbb {Z}^2$ whose normalized partial sums on rectangles with sides growing at rates $O(n)$ and $O(n^{\gamma})$ tend to an operator scaling random field $V_{\gamma}$ on $\mathbb {R}^2$, for any $\gamma>0$. The scaling transition is characterized by the fact that there exists a unique $\gamma_0>0$ such that the scaling limits $V_{\gamma}$ are different and do not depend on $\gamma$ for $\gamma>\gamma_0$ and $\gamma<\gamma_0$. The existence of scaling transition together with anisotropic and isotropic distributional long-range dependence properties is demonstrated for a class of $\alpha$-stable $(1<\alpha\le2)$ aggregated nearest-neighbor autoregressive random fields on $\mathbb{Z}^2$ with a scalar random coefficient $A$ having a regularly varying probability density near the "unit root" $A=1$.

Aggregation of autoregressive random fields and anisotropic long-range dependence

Scaling transition for long-range dependent Gaussian random fields

Scaling transition for nonlinear random fields with long-range dependence

Joint temporal and contemporaneous aggregation of random-coefficient AR(1) processes

We obtain a complete description of anisotropic scaling limits of random grain model on the plane with heavy tailed grain area distribution. The scaling limits have either independent or completely dependent increments along one or both coordinate axes and include stable, Gaussian and some `intermediate' infinitely divisible random fields. Asymptotic form of the covariance function of the random grain model is obtained. Application to superposed network traffic is included.

Vytautė Pilipauskaitė

Papers

Joint temporal and contemporaneous aggregation of random-coefficient AR(1) processes

Scaling transition for nonlinear random fields with long-range dependence

Anisotropic scaling of random grain model with application to network traffic

Anisotropic scaling of the random grain model with application to network traffic

Joint aggregation of random-coefficient AR(1) processes with common innovations