The paper concerns the image, level and sojourn time sets associated with sample paths of the Rosenblatt process. We obtain results regarding the Hausdorff (both classical and macroscopic), packing and intermediate dimensions, and the logarithmic and pixel densities. As a preliminary step we also establish the time inversion property of the Rosenblatt process, as well as some technical points regarding the distribution of Z.

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 to stochastic geometry, leading, in each instance, to an extra log-factor vis a vis the rate in the independent case.

Let Z := {Z(t) ,t >= 0} be a stationary Gaussian process. We study two estimators of E[Z(0)(2)], namely (f) over cap (T)(Z) := 1/T integral(T)(0) Z(t)(2)dt, and (f) over tilde (n) (Z) := 1/n Sigma(n)(i=1) Z(ti)(2), where t(i) = i Delta(n), i = 0, 1, ..., n, Delta -> 0 and T-n := n Delta(n) -> infinity. We prove that the two estimators are strongly consistent and establish Berry-Esseen bounds for a central limit theorem involving (f) over cap (T)(Z) and (f) over tilde (n)(Z). We apply these results to asymptotically stationary Gaussian processes and estimate the drift parameter for Gaussian Ornstein-Uhlenbeck processes.

Let Z=(Zt)t≥0 be the Rosenblatt process with Hurst index H∈(1∕2,1). We prove joint continuity for the local time of Z, and establish Hölder conditions for the local time. These results are then used to study the irregularity of the sample paths of Z. Based on analogy with similar known results in the case of fractional Brownian motion, we believe our results are sharp. A main ingredient of our proof is a rather delicate spectral analysis of arbitrary linear combinations of integral operators, which arise from the representation of the Rosenblatt process as an element in the second chaos.

AbstractLet (X, Y) = (Xn, Yn)n≥1 be the output process generated by a hidden chain Z = (Zn)n≥1, where Z is a finite-state, aperiodic, time homogeneous, and irreducible Markov chain. Let LCn be the length of the longest common subsequences of X1,..., Xn and Y1,..., Yn. Under a mixing hypothesis, a rate of convergence result is obtained for E[LCn]/n.

This paper concerns the associative lower central series ideals Mi of the free algebra An on n generators. Namely, we study the successive quotients Ni=Mi/Mi+1, which admit an action of the Lie algebra Wn of vector fields on Cn. We bound the degree |λ| of tensor field modules Fλ appearing in the Jordan–Hölder series of each Ni, confirming a recent conjecture of Arbesfeld and Jordan. As an application, we compute these decompositions for small n and i.

This aspect of soil quality consideration is especially significant and actual for now modern mechanised agriculture. At this type of agriculture soils are subjected to an "aggressive" form of use. Often this leads to loss of organic matter and nutrient content, acidification and severe destruction of soil aggregates. The consequences are soil compaction and deterioration of other significant soil properties. Most of the Bulgarian cultivated soils are characterised by poor soil structure of the arable layer. This accounts for unstable physical conditions for root development. The main reason is the destruction of the soil aggregates by human-induced activities that cause the diminishing of the soil organic matter. These effects are aggravated by the mechanical impacts of the heavy machinery and cultivation at unsuitable soil moisture conditions (Dilkova et al., 1998). The influence of these two factors on the soil physical properties of the arable layer is expressed practically through the instability of the soil structure during the vegetation and hence in deterioration of soil porosity determining drainage aeration and plant available water capacity. Both these mean deterioration of soil quality and the productive capacity. To maintain good aggregate stability of the arable layer usually it is recommended to enrich the soil with organic matter by manuring and crop incorporating residues (Lal, 1998; Swindale, 1998). Many investigations show that the soil aggregation and water stability of the soil aggregates depends on the banding of mineral particles with organic matter through R O and especially through amorphous oxalate-extractable iron (Kemper and Kosh, 1966; 23