Large deviation asymptotics and control variates for simulating large functions

Consider the normalized partial sums of a real-valued function $F$ of a Markov chain, $$\phi_n:=n^{-1}\sum_{k=0}^{n-1}F(\Phi(k)),\qquad n\ge1.$$ The chain $\{\Phi(k):k\ge0\}$ takes values in a general state space $\mathsf {X}$, with transition kernel $P$, and it is assumed that the Lyapunov drift condition holds: $PV\le V-W+b\mathbb{I}_C$ where $V:\mathsf {X}\to(0,\infty)$, $W:\mathsf {X}\to[1,\infty)$, the set $C$ is small and $W$ dominates $F$. Under these assumptions, the following conclusions are obtained: 1. It is known that this drift condition is equivalent to the existence of a unique invariant distribution $\pi$ satisfying $\pi(W)<\infty$, and the law of large numbers holds for any function $F$ dominated by $W$: $$\phi_n\to\phi:=\pi(F),\qquad{a.s.}, n\to\infty.$$ 2. The lower error probability defined by $\mathsf {P}\{\phi_n\le c\}$, for $c<\phi$, $n\ge1$, satisfies a large deviation limit theorem when the function $F$ satisfies a monotonicity condition. Under additional minor conditions an exact large deviations expansion is obtained. 3. If $W$ is near-monotone, then control-variates are constructed based on the Lyapunov function $V$, providing a pair of estimators that together satisfy nontrivial large asymptotics for the lower and upper error probabilities. In an application to simulation of queues it is shown that exact large deviation asymptotics are possible even when the estimator does not satisfy a central limit theorem.Comment: Published at http://dx.doi.org/10.1214/105051605000000737 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Indian Microfinance Sector: A Case Study

In developing areas of the world with very little economic structure, many activities are not monetized. In other words, money is not used to carry out these tasks because the people in these areas do not have the expendable funds required. In order to combat this problem, micro financing has become increasingly more apparent in these areas of distress. Microfinance is an economic development strategy that allows for those in need to borrow actual money in order to start a business, go to school, or even gain access to everyday living requirements. Microfinance has made tremendous strides over the years, but still faces several obstacles including regulation, loan strategies, and loan consumption. This study will attempt to analyze the microfinance industry in India by challenging certain aspects of its use, as well as offer suggestions that could have beneficial effects upon the industry

A Review of Marcus Borg\u27s Meeting Jesus Again for the First Time

In lieu of an abstract, below is the essay\u27s first paragraph. Meeting Jesus Again for the First Time was written by Marcus J. Borg and was first published in 1994. Much of the information and theories found within the book came from and annual lecture Borg gave at the Northern California Conference of the United Church of Christ at Asilomar, California in 1992. The main idea of Borg’s novel is that “there is a strong connection between images of Jesus and images of the Christian life, between how we think of Jesus and how we think of the Christian life “(Howard 1-2)

An AFSCME Local Transforms Local Politics

[Excerpt] In the late 1980s, AFSCME Local 2703, in Merced, California, found itself in a position familiar to many union members at that time: fighting hard to keep pay and benefits from being slashed. The realization that these confrontations could occur at every contract led local members to look for ways to make local government more responsive to the working men and women of the community. Through commitment and determination — and two bitter strikes — Local 2703 held its own against a local government determined to render it powerless. Through community education and coalition, the local to (sic) achieved its first small victories. The lessons the local members learned enabled them to build an effective and influential political action committee that reaches beyond the limits of their own union contracts to pursue political change, furthers community awareness and promote coalitions across racial, ethnic, and economic barriers. Their continuing commitment to the notion that working men and women are brothers and sisters, who deserve the respect of their peers and their government leaders, has them poised to be a visible influence in the politics of Merced County

Properties of mass-loading shocks: 1. Hydrodynamic considerations

The one-dimensional hydrodynamics of flows subjected to mass loading are considered anew, with particular emphasis placed on determining the properties of mass-loading shocks. This work has been motivated by recent observations of the outbound Halley bow shock (Neubauer et al., 1990), which cannot be understood in terms of simple hydrodynamical or magnetohydrodynamical descriptions. By including mass injection at the shock, we have investigated the properties of the Rankine-Hugoniot conditions on the basis of a geometric formulation of the entropy condition. Such a condition, which is more powerful than the usual thermodynamical formulation, serves to determine those solutions to the Rankine-Hugoniot conditions which correspond to a physically realizable downstream state. On this basis a concise theoretical description of hydrodynamic mass-loading shocks is obtained. We show that mass-loading shocks have more in common with combustion shocks than with ordinary nonreacting gas dynamical shocks. It is shown that for decelerated solutions to the Rankine-Hugoniot conditions to exist, the upstream flow speed u0 must satisfy u0 > ucrit > cs, where cs is the sound speed. Besides the usual supersonic-subsonic transition, mass-loading fronts can also admit a decelerating supersonic-supersonic transition, the structure of which consists of a sharp decrease in the flow velocity preceding a recovery and an increase in the final downstream flow speed. We suggest the possibility that such structures may describe the inbound Halley bow shock (Coates et al., 1987a). Both parallel and oblique shocks are considered, the primary difference being that oblique shocks are subjected to a shearing stress due to mass loading. It is conjectured that such a shearing may destabilize the shock