Curve crossing for random walks reflected at their maximum

Let $R_n=\max_{0\leq j\leq n}S_j-S_n$ be a random walk $S_n$ reflected in its maximum. Except in the trivial case when $P(X\ge0)=1$, $R_n$ will pass over a horizontal boundary of any height in a finite time, with probability 1. We extend this by giving necessary and sufficient conditions for finiteness of passage times of $R_n$ above certain curved (power law) boundaries, as well. The intuition that a degree of heaviness of the negative tail of the distribution of the increments of $S_n$ is necessary for passage of $R_n$ above a high level is correct in most, but not all, cases, as we show. Conditions are also given for the finiteness of the expected passage time of $R_n$ above linear and square root boundaries.Comment: Published at http://dx.doi.org/10.1214/009117906000000953 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Resolving the Spin Crisis: Mergers and Feedback

We model in simple terms the angular momentum (J) problem of galaxy formation in CDM, and identify the key elements of a scenario that can solve it. The buildup of J is modeled via dynamical friction and tidal stripping in mergers. This reveals how over-cooling in incoming halos leads to transfer of J from baryons to dark matter (DM), in conflict with observations. By incorporating a simple recipe of supernova feedback, we match the observed J distribution in disks. Gas removal from small incoming halos, which make the low-J component of the product, eliminates the low-J baryons. Partial heating and puffing-up of the gas in larger incoming halos, combined with tidal stripping, reduces the J loss of baryons. This implies a higher baryonic spin for lower mass halos. The observed low baryonic fraction in dwarf galaxies is used to calibrate the characteristic velocity associated with supernova feedback, yielding v_fb sim 100 km/s, within the range of theoretical expectations. The model then reproduces the observed distribution of spin parameter among dwarf and bright galaxies, as well as the J distribution inside these galaxies. This suggests that the model captures the main features of a full scenario for resolving the spin crisis.Comment: 8 pages, Latex, svmult.cls, subeqnar.sty, sprmindx.sty, physprbb.sty, cropmark.sty, in The Mass of Galaxies at Low and High Redshift, eds. R. Bender & A. Renzini (Springer-Verlag, ESO Astrophysics Symposia

Animals and the Problem of Evil in Recent Theodicies

This paper critically evaluates the theodicies of John Hick, Richard Swinburne and process theism regarding animal suffering and evils. The positions of Hick and Swinburne are based on false empirical assumptions, e.g., animals do not suffer. Process theism’s claim that God is not omnipotent is an unsatisfactory answer inconsistent with the traditional concept of God. These positions cannot fully explain the mass suffering and unnecessary deaths of animals throughout time. My positive position is that God’s putative love for all sentient beings does not necessarily entail that he loves every individual human and animal. Humans do not interfere with the suffering and deaths of animals in the wild, and God has no obligation to interfere with human evils. It is very possible that God acts similarly with humans and animals regarding evils. This theory partly explains human tragedies such as the Holocaust and much unnecessary animal and human suffering