اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدOn universal estimates for binary renewal processes
358
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
A binary renewal process is a stochastic process ${X_n}$ taking values in ${0,1}$ where the lengths of the runs of 1s between successive zeros are independent. After observing ${X_0,X_1,...,X_n}$ one would like to predict the future behavior, and the problem of universal estimators is to do so without any prior knowledge of the distribution. We prove a variety of results of this type, including universal estimates for the expected time to renewal as well as estimates for the conditional distribution of the time to renewal. Some of our results require a moment condition on the time to renewal and we show by an explicit construction how some moment condition is necessary.
قيم البحث
اقرأ أيضاً
We review the theory of renewal reward processes, which describes renewal processes that have some cost or reward associated with each cycle. We present a new simplified proof of the renewal reward theorem that mimics the proof of the elementary rene
wal theorem and avoids the technicalities in the proof that is presented in most textbooks. Moreover, we mention briefly the extension of the theory to partial rewards, where it is assumed that rewards are not accrued only at renewal epochs but also during the renewal cycle. For this case, we present a counterexample which indicates that the standard conditions for the renewal reward theorem are not sufficient; additional regularity assumptions are necessary. We present a few examples to indicate the usefulness of this theory, where we prove the inspection paradox and Littles law through the renewal reward theorem.
We show how a description of Brownian exponential functionals as a renewal series gives access to the law of the hitting time of a square-root boundary by a Bessel process. This extends classical results by Breiman and Shepp, concerning Brownian moti
on, and recovers by different means, extensions for Bessel processes, obtained independently by Delong and Yor.
We refine some previous results concerning the Renewal Contact Processes. We significantly widen the family of distributions for the interarrival times for which the critical value can be shown to be strictly positive. The result now holds for any sp
atial dimension $d geq 1$ and requires only a moment condition slightly stronger than finite first moment. We also prove a Complete Convergence Theorem for heavy tailed interarrival times. Finally, for heavy tailed distributions we examine when the contact process, conditioned on survival, can be asymptotically predicted knowing the renewal processes. We close with an example of an interarrival time distribution attracted to a stable law of index 1 for which the critical value vanishes, a tail condition uncovered by previous results.
We consider the problem of determining escape probabilities from an interval of a general compound renewal process with drift. This problem is reduced to the solution of a certain integral equation. In an actuarial situation where only negative jumps
arise we give a general solution for escape and survival probabilities under Erlang$(n)$ and hypo-exponential arrivals. These ideas are generalized to the class of arrival distributions having rational Laplace transforms. In a general situation with two-sided jumps we also identify important families of solvable cases. A parallelism with the scale function of diffusion processes is drawn.
We consider a real-valued diffusion process with a linear jump term driven by a Poisson point process and we assume that the jump amplitudes have a centered density with finite moments. We show upper and lower estimates for the density of the solutio
n in the case that the jump amplitudes follow a Gaussian or Laplacian law. The proof of the lower bound uses a general expression for the density of the solution in terms of the convolution of the density of the continuous part and the jump amplitude density. The upper bound uses an upper tail estimate in terms of the jump amplitude distribution and techniques of the Malliavin calculus in order to bound the density by the tails of the solution. We also extend the lower bounds to the multidimensional case.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
