David Steinsaltz

Understanding Mortality Rate Deceleration and Heterogeneity

ABSTRACT Generic relationships between heterogeneity in population frailty and flattening of aggregate population hazard functions at extreme ages are drawn from classical mathematical results on the limiting behavior of Laplace transforms. In particular, it shows that the population hazard function converges to a constant precisely when the distribution of unobserved heterogeneity in initial mortalities behaves asymptotically as a polynomial near zero.

Re-evaluating a test of the heterogeneity explanation for mortality plateaus.

[Drapeau, M.D., Gass, E.K., Simison, M.D., Mueller, L.D., Rose, M.R., 2000. Testing the heterogeneity theory of late-life mortality plateaus by using cohorts of Drosophila melanogaster, Experimental Gerontology, 35 71-84.] tested, in populations of Drosophila melanogaster, a prediction of the heterogeneity explanation for mortality plateaus. They concluded that heterogeneity could not explain their results. We contend here that the statistical analysis was flawed. It was declared that there was no difference between the mortality plateaus of three different strains, on the basis of averaged outcomes. In fact, the results for the different strains were quite different. Most trials showed the expected lowering of the mortality plateaus for the flies selected for robustness, but these effects were washed out by a small number of very large opposing deviations. There is ample reason to believe that the opposing deviations are artifacts of fitting an overly restrictive hazard-rate model. When we fit more appropriate models, the evidence points toward a rejection of the null hypothesis (of identical plateaus), hence toward modest support for the heterogeneity explanation.

Random logistic maps and Lyapunov exponents

Abstract We prove that under certain basic regularity conditions, a random iteration of logistic maps converges to a random point attractor when the Lyapunov exponent is negative, and does not converge to a point when the Lyapunov exponent is positive.

Quasilimiting behavior for one-dimensional diffusions with killing

This paper extends and clarifies results of Steinsaltz and Evans [Trans. Amer. Math. Soc. 359 (2007) 1285–1234], which found conditions for convergence of a killed one-dimensional diffusion conditioned on survival, to a quasistationary distribution whose density is given by the principal eigenfunction of the generator. Under the assumption that the limit of the killing at infinity differs from the principal eigenvalue we prove that convergence to quasistationarity occurs if and only if the principal eigenfunction is integrable. When the killing at ∞ is larger than the principal eigenvalue, then the eigenfunction is always integrable. When the killing at ∞ is smaller, the eigenfunction is integrable only when the unkilled process is recurrent; otherwise, the process conditioned on survival converges to 0 density on any bounded interval.

Instability of Baroclinic Waves with Bottom Slope

Abstract Pedloskys theory explaining the behavior of unstable baroclinic waves in the β-plane is modified to include a sloped bottom (although the β effect is ignored). The result found is the same sort of nonlinear oscillatory behavior described by Pedlosky, except in the case of short wavelengths for negative shears. In that case, the theory predicts an initial explosive growth of the wave amplitude, so that it will reach amplitudes that are very large compared with its initial scale. This suggests a possible mechanism for small-scale current fluctuations in the oceans.

Convergence of moments in a Markov-chain central limit theorem

Let (Xi)i=0∞ be a V-uniformly ergodic Markov chain on a general state space, and let π be its stationary distribution. For g : χ → R, define It is shown that if |g| ≤ V1/n for a positive integer n, then ExWk(g)n converges to the n-th moment of a normal random variable with expectation 0 and variance This extends the existing Markov-chain central limit theorems, according to which expectations of bounded functionals of Wk(g) converge. We also derive nonasymptotic bounds for the error in approximating the moments of Wk (g) by the normal moments. These yield easy bounds of all feasible polynomial orders, and exponential bounds as well under some circumstances, for the probabilities of large deviations by the empirical measure along the Markov chain path Xi.

Fluctuation bounds for sock-sorting and other stochastic processes

Abstract Using standard methods from empirical-process theory, in particular symmetrization, we derive exponential bounds on the fluctuations of stochastic processes which may be represented as the averages of many small functions. As examples, self-service queueing and storage problems are analyzed. We eliminate some of the very large constants or polynomial factors which have appeared in other, more asymptotically oriented results. The range of applications is extended by our replacing the more usual covering-number bounds by a dependence on the total variation of the component functions. While this restricts our results to the one-dimensional context, it allows the bounds to be applied to cases in which the component functions are not jump-functions and not of a uniform shape. We also dispense with the assumption that the functions are identically distributed.