Establishing conditions for weak convergence to stochastic integrals

Jiangyan Peng and Qiying Wang

Abstract

Limit theory involving stochastic integrals plays a major role in time series econometrics. In earlier contributions on weak convergence to stochastic integrals, the literature commonly uses martingale and semimartingale structures. Liang, et al (2015) (see also Wang (2015), Chapter 4.5) currently extended the weak convergence to stochastic integrals by allowing for the linear process in the innovations. While these martingale and linear processes structures have wild relevance, they are not sufficiently general to cover many econometric applications where endogeneity and nonlinearity are present. This paper provides new conditions for weak convergence to stochastic integrals. Our frameworks allow for long memory processes, causal processes and near-epoch dependence in the innovations, which can be applied to a wild range of areas in econometrics, such as GARCH, TAR, bilinear and other nonlinear models.

Keywords: Stochastic integral, convergence, long memory process, near-epoch dependence, linear process, causal process, TAR model, bilinear model, GARCH model.