No abstract

Histopathological alterations in the heart are often reported in fish as a result of exposure to a variety of chemical compounds. However, researchers presently lack a standardized method for the evaluation of histopathological alterations in the cardiovascular system of fish and the calculation of an 'organ index'. Therefore, we designed a method for a standardized assessment and evaluation of histopathological alterations in the heart of fish. As a model species, we used rainbow trout Oncorhynchus mykiss, but the protocol was also successfully applied to other fish species belonging to different taxonomic orders. To test the protocol, we re-evaluated sections of atenolol-exposed and unexposed rainbow trout obtained in a previous study. The results were in accordance with those previously published, demonstrating the applicability of the protocol. The protocol provides a universal method for the comparative evaluation of histopathological changes in the heart of fish.

Let (X i ) be a stationary process adapted to a filtration (F i ), E(X i ) = 0, E(X 2 i ) < ∞; by S n = n−1 i=0 X i we denote the partial sums and σ 2 n = S n 2 2 . Wu and Woodroofe [Wei Biao Wu, M. Woodroofe, Martingale approximation for sums of stationary processes, Ann. Probab. 32 (2004) 1674-1690] have shown that if E(S n | F 0 ) 2 = o(σ n ) then there exists an array of row-wise stationary martingale difference sequences approximating the partial sums S n . If ∞ n=1 E(S n |F 0 ) 2 n 3/2 < ∞ then by [M. Maxwell, M. Woodroofe, Central limit theorems for additive functionals of Markov chains, Ann. Probab. 28 (2000) 713-724] there exists a stationary martingale difference sequence approximating the partial sums S n , and the central limit theorem holds. We will show that the process (X i ) can be found so that E(S n | F 0 ) 2 = O √ n log 1/2 n , σ 2 n /n → constant but the central limit theorem does not hold. The linear growth of the variances σ 2 n is a substantial source of complexity of the construction.

Recently, invariance principles for partial sums of Bernoulli random fields over rectangular index sets have been proved under Hannan's condition. In this note we complement previous results by establishing limit theorems for weighted Bernoulli random fields, including central limit theorems for partial sums over arbitrary index sets and invariance principles for Gaussian random fields. Most results improve earlier ones on Bernoulli random fields under Wu's condition, which is stronger than Hannan's condition.