Bootstrapping the empirical distribution of a stationary process with change-point

© 2019, Institute of Mathematical Statistics. All rights reserved. When detecting a change-point in the marginal distribution of a stationary time series, bootstrap techniques are required to determine critical values for the tests when the pre-change distribution is unknown. In this paper, we propose a sequential moving block bootstrap and demonstrate its validity under a converging alternative. Furthermore, we demonstrate that power is still achieved by the bootstrap under a non-converging alternative. We follow the approach taken by Peligrad in [14], and avoid assumptions of mixing, association or near epoch dependence. These results are applied to a linear process and are shown to be valid under very mild conditions on the existence of any moment of the innovations and a corresponding condition of summability of the coefficients

Change-point detection in the marginal distribution of a linear process

© 2016, Institute of Mathematical Statistics. All rights reserved. The subject of this paper is the detection of a change in the marginal distribution of a stationary linear process. By considering the marginal distribution, the change-point model can simultaneously incorporate any change in the coefficients and/or the innovations of the linear process. Furthermore, the change point can be random and data dependent. The key is an analysis of the asymptotic behaviour of the sequential empirical process, both with and without a change point. Our results hold under very mild conditions on the existence of any moment of the innovations and a corresponding condition of summability of the coefficients

The multitype branching diffusion

The multitype branching diffusion (MBD) is considered. A review of the general theory of multitype point processes is given in Section 2, and spatial central limit theorems for homogeneous infinitely divisible processes are proven in Section 3. In Section 4, the MBD is defined, and equations for its first four factorial moment density functions are found. The behaviour of the mean and covariance functionals as time approaches infinity is studied. The MBD with immigration (MBDI) is introduced in Section 5. The existence of a steady state is proven, and spatial central limit theorems are developed for the MBDI.Single-type point process multitype point process branching diffusion probability generating funcional infinitely divisible factorial moments factorial cumulants mixing central limit theorem Skorokhod topology positive regular branching process

Stopping times and tightness for multiparameter martingales

Two major results on stopping times and tightness of Aldous are generalized to strong martingales with a multidimensional time parameterPrimary 60B10 Secondary 60G48 Weak convergence Stopping time Tightness Strong martingale

Stopping and set-indexed local martingales

Set-indexed local martingales are defined and studied. We present some optional sampling theorems for strong martingales, martingales and weak martingales. The class of set-indexed processes which are locally of class (D) is introduced. A Doob-Meyer decomposition is obtained: any local weak submartingale has a unique decomposition into the sum of a local weak martingale and a local predictable increasing process. Finally some examples are given.Lattice Set-indexed martingale Submartingale Predictable [sigma]-algebra Stopping set Class (D) Optional sampling Local martingale Doob-Meyer decomposition