On the Berry-Esseen theorem

Summary The paper presents a simple derivation of a generalized Berry-Esseen theorem not requiring moments.Research connected partly with a Project for research in probability at Princeton University, supported by the Army Research Office.     

Sequential multi-hypothesis testing for compound Poisson processes

Suppose that there are finitely many simple hypotheses about the unknown arrival rate and mark distribution of a compound Poisson process, and that exactly one of them is correct. The objective is to determine the correct hypothesis with minimal error probability and as soon as possible after the observation of the process starts. This problem is formulated in a Bayesian framework, and its solution is presented. Provably convergent numerical methods and practical near-optimal strategies are described and illustrated on various examples.     

Density estimation for compound Poisson processes from discrete data

In this article we investigate the nonparametric estimation of the jump density of a compound Poisson process from the discrete observation of one trajectory over [0,T]$[0,T]$. We consider the case where the sampling rate Δ=_ΔT→0$\Delta ={\Delta }_T\to 0$ as T→∞$T\to \infty$. We propose an adaptive wavelet threshold density estimator and study its performance for _Lp$L_p$ losses, p≥1$p\ge 1$, over Besov spaces. The main novelty is that we achieve minimax rates of convergence for sampling rates _ΔT${\Delta }_T$ that vanish slowly. The estimation procedure is based on the explicit inversion of the operator giving the law of the increments as a nonlinear transformation of the jump density.     

Edgeworth expansions for compound Poisson processes and the bootstrap

One-term Edgeworth Expansions for the studentized version of compound Poisson processes are developed. For a suitably defined bootstrap in this context, the so called one-term Edgeworth correction by bootstrap is also established. The results are applicable for constructing second-order correct confidence intervals (which make correction for skewness) for the parameter “mean reward per unit time”. Research work of Gutti Jogesh Babu was supported in part by NSF grants DMS-9626189 and DMS-0101360.     

双复合Poisson风险模型

研究了保费收取过程是复合Po isson过程,索赔总额是复合Po isson过程的风险模型,给出了不破产概率的积分表示,以及在特殊情况下不破产概率的具体表达式,并用鞅方法得出了破产概率满足的Lundberg不等式和一般公式.     

The Berry-Esseen theorem for functions of uniform spacings

Summary A Berry-Esseen bound of order is established for suitably normalized sums of nonlinear measurable functions of uniform spacings under the natural moment assumptions.Some of this study was carried out at the Mathematical Centre, Amsterdam, and some of it when the second author was a member of the Mathematical Sciences Research Institute, Berkeley     

The Berry-Esseen theorem for functionals of discrete Markov chains

Summary The error bound O(1/n) is derived in the central limit theorem for partial sums where _j is a recurrent discrete Markov chain and f is a real valued function on the state space. In particular it is shown that for bounded f and starting distribution dominated by some multiple of the stationary one, it is sufficient for the chain to have recurrence times with third moments on order to get this bound.     

Polynomials of binomial type and compound Poisson processes

Let q_nand s_n, n ε N, respectively, be a set of polynomials of binomial type and a Sheffer set related to it, both having positive coefficients. Then q_n(x), x > 0 is connected with the probability that a compound Poisson process starting at zero is in state n at time τx andq_n(x)/q_n(1) is the probability generating function of the number of jumps of this process in [0, τ] given that it is in state n at time τ. The s_n admit similar interpretations when the initial distribution of the compound Poisson process is not concentrated at zero. The possible limits for n → ∞ ofq_n(x)/q_n(1)ands_n(x)/s_n(1) are studied.     

A Berry-Esseen theorem for hypergeometric probabilities under minimal conditions

In this paper, we consider simple random sampling without replacement from a dichotomous finite population and derive a necessary and sufficient condition on the finite population parameters for a valid large sample Normal approximation to Hypergeometric probabilities. We then obtain lower and upper bounds on the difference between the Normal and the Hypergeometric distributions solely under this necessary and sufficient condition.

     

Sequential testing of simple hypotheses about compound Poisson processes

One of two simple hypotheses for the unknown arrival rate and jump distribution of a compound Poisson process is correct. We start observing the process, and the problem is to decide on the correct hypothesis as soon as possible and with the smallest probability of wrong decision. We find a Bayes-optimal sequential decision rule and describe completely how to calculate its parameters without any restrictions on the arrival rate and the jump distribution.     

On a multi-threshold compound Poisson process perturbed by diffusion

We consider a multi-layer compound Poisson surplus process perturbed by diffusion and examine the behaviour of the Gerber–Shiu discounted penalty function. We derive the general solution to a certain second order integro-differential equation. This permits us to provide explicit expressions for the Gerber–Shiu function depending on the current surplus level. The advantage of our proposed approach is that if the diffusion term converges to zero, the above-mentioned explicit expressions converge to those under the classical compound Poisson model, provided that the same initial conditions apply. This is subsequently illustrated by an extended example related to the probability of ultimate ruin.     

On compound Poisson processes arising in change-point type statistical models as limiting likelihood ratios

Different change-point type models encountered in statistical inference for stochastic processes give rise to different limiting likelihood ratio processes. In a previous paper of one of the authors it was established that one of these likelihood ratios, which is an exponential functional of a two-sided Poisson process driven by some parameter, can be approximated (for sufficiently small values of the parameter) by another one, which is an exponential functional of a two-sided Brownian motion. In this paper we consider yet another likelihood ratio, which is the exponent of a two-sided compound Poisson process driven by some parameter. We establish, that similarly to the Poisson type one, the compound Poisson type likelihood ratio can be approximated by the Brownian type one for sufficiently small values of the parameter. We equally discuss the asymptotics for large values of the parameter and illustrate the results by numerical simulations.     

On a multivariate central limit theorem for stationary bilinear processes

In 1957, Parzen proved a central limit theorem for a class of scalar processes which he called multilinear processes. In the present paper only stationary bilinear processes are considered, but the theory is generalized to the multivariate case.     

On some properties of Poisson processes

The Poisson process, i.e., the simple stream, is defined by Khintchine as a stationary, orderly and finite stream without after-effects. A necessary and sufficient condition for a stream to be a simple stream is that the interarrival times are independent random variables with identical exponential distributions. This paper gives a simple and rigorous proof of the necessary and sufficient condition, and discusses the other necessary and sufficient conditions for a renewal process to be a Poisson process.Institute of Applied Mathematics, Academia Sinica     

Uniform asymptotics for compound Poisson processes with regularly varying jumps and vanishing drift

This paper addresses heavy-tailed large-deviation estimates for the distribution tail of functionals of a class of spectrally one-sided Lévy processes. Our contribution is to show that these estimates remain valid in a near-critical regime. This complements recent similar results that have been obtained for the all-time supremum of such processes. Specifically, we consider local asymptotics of the all-time supremum, the supremum of the process until exiting $[0,\infty )$, the maximum jump until that time, and the time it takes until exiting $[0,\infty )$. The proofs rely, among other things, on properties of scale functions.