We prove strong convergence and asymptotic normality for the record and the weak record rate of observations of the form Yn = Xn + Tn, n ≥ 1, where (Xn)n∈Z is a stationary ergodic sequence of random variables and (Tn)n≥1 is a stochastic trend process, with stationary ergodic increments. The strong convergence result follows from the Dubins-Freedman law of large numbers and Birkhoff’s ergodic th...

Confidence intervals and joint confidence sets are constructed for the nonparametric calibration of exponential Lévy models based on prices of European options. This is done by showing joint asymptotic normality for the estimation of the volatility, the drift, the intensity and the Lévy density at finitely many points in the spectral calibration method. Furthermore, the asymptotic normality res...

L-functionals summarize numerous statistical parameters and actuarial risk measures. Their sample estimators are linear combinations of order statistics L-statistics . There exists a class of heavy-tailed distributions for which the asymptotic normality of these estimators cannot be obtained by classical results. In this paper we propose, by means of extreme value theory, alternative estimators...

This paper considers the construction of prediction intervals for future observations in high dimensional regression models. We propose a new approach to evaluate the uncertainty for estimating the mean parameter based on the widely-used penalization/regularization methods. The proposed method is then applied to construct prediction intervals for sparse linear models as well as sparse additive ...

In this paper we consider the problem of parameter estimation for the multi-component harmonic signals in multiplicative and additive noise. The least squares estimators (LSEs) are proposed for the estimation of the frequencies and the means of random amplitudes, and some statistical results of the LSEs are obtained, including the strong consistency, the strong convergence rate and the asymptot...

The consistency and asymptotic normality of a linear least squares estimate of the form (X,X)-X’Y when the mean is not X/I is investigated in this paper. The least squares estimate is a consistent estimate of the best linear approximation of the true mean function for the design chosen. The asymptotic normality of the least squares estimate depends on the design and the asymptotic mean may not ...

In the paper we develop an approach to asymptotic normality through factorial cumulants. Factorial cumulants arise in the same manner from factorial moments as do (ordinary) cumulants from (ordinary) moments. Another tool we exploit is a new identity for 'moments' of partitions of numbers. The general limiting result is then used to (re-)derive asymptotic normality for several models including ...

We study the convergence and asymptotic normality of a generalized form of stochastic approximation algorithm with deterministic perturbation sequences. Both one-simulation and two-simulation methods are considered. Assuming a special structure of deterministic sequence, we establish sufficient condition on the noise sequence for a.s. convergence of the algorithm. Construction of such a special...

The asymptotic normality of the sample mean of iid rv 's is equivalent to the well known conditions of Levy or Feller. More recently, additional equivalences have been developed in terms of the quantile function (qf). And other useful probabilistic equivalences could be cited. The emphasis here is on equivalences, and many other useful and informative equivalences will be developed. Many depend...

1. BASIC PROBABILITY THEORY 1.1 Measure-theoretical foundation of probability theory 1.2 Independence 1.3 Borel measurable functions 1.4 Mathematical expectatlon 1.5 Characteristic functions 1.6 Random functions 2. CONVERGENCE 2.1 Heak and strong convergence of random variables 2.2 Convergence of mathematical expectations 2.3 Convergence of distributions 2.4 Central limit theoreras 2.5 Further ...