Arindam Sengupta

Asymptotic properties of sums of upper records

Arnold and Villaseñor (1999) raised several questions for upper records, including characterizing all limit distributions of normalized partial sums of upper records. We provide some answers in the case when the distribution from which the samples are drawn is bounded above. When the distribution is not bounded above, we give sufficient conditions on the distribution for the properly normalized partial sums to converge to a standard normal distribution. We show that our conditions are general enough so that the examples provided by Arnold and Villaseñor (1999) are covered by our results.

Time-space polynomial martingales cenerated by a discrete-time martingale

We investigate, for a given martingaleM={Mn: n≥0}, the conditions for the existence of polynomialsP(·,·) of two variables, “time” and “space,” and of arbitrary degree in the latter, such that{P(n, Mn)} is a martingale for the natural filtration ofM. Denoting by ℘ the vector space of all such polynomials, we ask, in particular, when such a sequence can be chosen so as to span ℘. A complete necessary and sufficient condition is obtained in the case whenM has independent increments. For generalM, we obtain a necessary condition which entails, under mild additional hypotheses, thatM is necessarily Markovian. Considering a slightly more general class of polynomials than ℘ we obtain necessary and sufficient conditions in the case of general martingales also. It is moreover observed that in most of the cases, the set ℘ determines the law of the martingale in a certain sense.

Ban Estimates of Class Boundaries Under Likert-Type Scaling

Abtsrcat We prove that under very mild assumptions, the estimated class boundaries for the method of Likert-type scaling are Best Asymptotic Normal estimators of the true values. Also, we show that in the situation where the number of categories increases with sample size, the estimates are uniformly consistent under suitable conditions.

Option Pricing Under Compound Bernoulli and Poisson Perturbations of AR(1) Log-Price Process

Compound Poisson and compound Bernoulli perturbations are introduced in addition to normal errors in the AR(1) model for discrete time version of log-price process. The formula for expected present value of European Call Option is extended to these cases. Other than European option, the valuation of Asian option is also considered and formulas for Geometric Asian option for AR(1) log-price process incorporating compound Poisson and compound Bernoulli jumps also obtained.

Moments of escape times of random walk

Extending an idea of Spitzer [2], a way to compute the moments of the time of escape from (−N,L) by a symmetric simple random walk is exhibited. It is shown that all these moments depend polynomially onL andN.