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

Passage time and fluctuation calculations for subexponential L\'evy processes

We consider the passage time problem for L\'evy processes, emphasising heavy tailed cases. Results are obtained under quite mild assumptions, namely, drift to $-\infty$ a.s. of the process, possibly at a linear rate (the finite mean case), but possibly much faster (the infinite mean case), together with subexponential growth on the positive side. Local and functional versions of limit distributions are derived for the passage time itself, as well as for the position of the process just prior to passage, and the overshoot of a high level. A significant connection is made with extreme value theory via regular variation or maximum domain of attraction conditions imposed on the positive tail of the canonical measure, which are shown to be necessary for the kind of convergence behaviour we are interested in.Comment: Published at http://dx.doi.org/10.3150/15-BEJ700 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Distributional representations and dominance of a L\'{e}vy process over its maximal jump processes

Distributional identities for a L\'evy process $X_t$, its quadratic variation process $V_t$ and its maximal jump processes, are derived, and used to make "small time" (as $t\downarrow0$) asymptotic comparisons between them. The representations are constructed using properties of the underlying Poisson point process of the jumps of $X$. Apart from providing insight into the connections between $X$, $V$, and their maximal jump processes, they enable investigation of a great variety of limiting behaviours. As an application, we study "self-normalised" versions of $X_t$, that is, $X_t$ after division by $\sup_{0<s\le t}\Delta X_s$, or by $\sup_{0<s\le t}| \Delta X_s|$. Thus, we obtain necessary and sufficient conditions for $X_t/\sup_{0<s\le t}\Delta X_s$ and $X_t/\sup_{0<s\le t}| \Delta X_s|$ to converge in probability to 1, or to $\infty$, as $t\downarrow0$, so that $X$ is either comparable to, or dominates, its largest jump. The former situation tends to occur when the singularity at 0 of the L\'evy measure of $X$ is fairly mild (its tail is slowly varying at 0), while the latter situation is related to the relative stability or attraction to normality of $X$ at 0 (a steeper singularity at 0). An important component in the analyses is the way the largest positive and negative jumps interact with each other. Analogous "large time" (as $t\to \infty$) versions of the results can also be obtained.Comment: Published at http://dx.doi.org/10.3150/15-BEJ731 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm