Weak Convergence to Stochastic Integrals

The main object of this chapter is to prove the convergence of the covariances of stochastic processes with their increments to stochastic integrals with respect to Brownian motion. Some preliminary theory is given relating to random functionals on C, stochastic integrals, and the important Itô isometry. The main result is first proved for the tractable special cases of martingale difference increments and linear processes. The final section is devoted to proving the more difficult general case, of NED functions of mixing processes.

Evolution of the housing market under the framework of adaptive market hypothesis and martingale difference hypothesis: a case of India

PurposeThe purpose of this study to evaluate the evolving market efficiency of the housing market under the framework of adaptive market hypothesis and martingale difference hypothesis taking a case of India.Design/methodology/approachThe study used a wild bootstrap version of the generalized spectral (GS) test in the rolling window framework to measure possible time-varying linear and non-linear dependence in the housing market.FindingsThe study finds that the Indian housing market, in general, is not efficient, and this efficiency is dynamic, which changes with time lending support to the adaptive market hypothesis. The study confirms that the evolutionary model of individuals adapting to a changing environment via behavioural biases affects the efficiency of the housing market, which leads to the evolving efficiency of the housing market prices.Research limitations/implicationsThe study believes that the potential implications go beyond evolutionary forces and the adaptive market hypothesis , which, does not only depend on an individual's decision-making process but also on social psychology. Thus, a further attempt in this line, taking into account the social psychology and quantitative rigour towards drivers of evolving efficiency is suggested for future research.Practical implicationsThe study suggests that there is a possibility of extra returns for market players, but not always. The Indian housing market has witnessed several landmark reforms in recent years, so it is believed that these reforms would decrease the inefficiency level of this market. Contrary to this, the study’s findings reveal an increase in the inefficiency level in recent years. As the Indian housing market shows evolving efficiency, it is believed that the increased inefficiency is temporary. The increased inefficiency can be regarded as the settlement stage of the various policy and technical reforms.Originality/valueConfirming the presence or absence of adaptive efficiency in the housing market under possible non-linear dependence will be a significant addition to the existing literature.

Strong consistency of regression function estimator with martingale difference errors

In this paper, we consider the regression model with fixed design: Y i = g ( x i ) + ε i {Y}_{i}=g\left({x}_{i})+{\varepsilon }_{i} , 1 ≤ i ≤ n 1\le i\le n , where { x i } \left\{{x}_{i}\right\} are the nonrandom design points, and { ε i } \left\{{\varepsilon }_{i}\right\} is a sequence of martingale, and g g is an unknown function. Nonparametric estimator g n ( x ) {g}_{n}\left(x) of g ( x ) g\left(x) will be introduced and its strong convergence properties are established.