Limit theorems for von Mises statistics of a measure preserving transformation

For a measure preserving transformation $T$ of a probability space $(X,\mathcal F,\mu)$ we investigate almost sure and distributional convergence of random variables of the form $$x \to \frac{1}{C_n} \sum_{i_1<n,...,i_d<n} f(T^{i_1}x,...,T^{i_d}x),\, n=1,2,...,$$ where $f$ (called the \emph{kernel}) is a function from $X^d$ to $\R$ and $C_1, C_2,...$ are appropriate normalizing constants. We observe that the above random variables are well defined and belong to $L_r(\mu)$ provided that the kernel is chosen from the projective tensor product $$L_p(X_1,\mathcal F_1, \mu_1) \otimes_{\pi}...\otimes_{\pi} L_p(X_d,\mathcal F_d, \mu_d)\subset L_p(\mu^d)$$ with $p=d\,r,\, r\ \in [1, \infty).$ We establish a form of the individual ergodic theorem for such sequences. Next, we give a martingale approximation argument to derive a central limit theorem in the non-degenerate case (in the sense of the classical Hoeffding's decomposition). Furthermore, for $d=2$ and a wide class of canonical kernels $f$ we also show that the convergence holds in distribution towards a quadratic form $\sum_{m=1}^{\infty} \lambda_m\eta^2_m$ in independent standard Gaussian variables $\eta_1, \eta_2,...$. Our results on the distributional convergence use a $T$--\,invariant filtration as a prerequisite and are derived from uni- and multivariate martingale approximations

Insurance loss coverage and demand elasticities

Restrictions on insurance risk classification may induce adverse selection, which is usually perceived as a bad outcome. We suggest a counter-argument to this perception in circumstances where modest levels of adverse selection lead to an increase in `loss coverage', defined as expected losses compensated by insurance for the whole population. This happens if the shift in coverage towards higher risks under adverse selection more than offsets the fall in number of individuals insured. The possibility of this outcome depends on insurance demand elasticities for higher and lower risks. We state elasticity conditions which ensure that for any downward-sloping insurance demand functions, loss coverage when all risks are pooled at a common price is higher than under fully risk-differentiated prices. Empirical evidence suggests that these conditions may be realistic for some insurance markets

Temperature limitation of hydrogen turnover and methanogenesis in anoxic paddy soil

The shift of incubation temperature in anoxic paddy soil from 30 to 15 degrees C resulted in a reversible decrease of the methane production rate and of the H2 steady state partial pressure. Only at 30 degrees C but not at 17 degrees C, total CH4 production rates were enhanced by the addition of H2, acetate, or cellulose compared to the control. Apparent activation energies which were calculated from the temperature dependence of CH4 production were higher in presence than in absence of excess H2. Decrease of temperature caused a decrease of the H2 turnover rate constant and of the Gibbs free energy of H2-dependent methanogenesis, and also resulted in a smaller contribution of H2 to total methanogenesis. However, H2-dependent methanogenesis was significantly stimulated by excess H2 and slightly inhibited by acetate at low as well as high temperature. The results show that H2-producing bacteria were limited by temperature to a greater extent than the methanogens so that the methanogenic microbial community in paddy soil was limited by the supply of H2. At lows as well as high temperatures, excess H2 apparently enabled part of the methanogenic community to shift from acetate-dependent to H2-dependent CH4 had only this effect, but with increasing temperature, excess H2 additionally stimulated total methanogenic activity and eventually even growth. (IFU