Asymptotics for recurrent diffusions with application to high frequency regression

We provide the asymptotic theory for functionals of recurrent diffusions. Our asymptotics are completely general and applicable for all cases, including positive and null recurrent diffusions, and diffusions with and without the integrability conditions. They are established directly from the representation of diffusion as time-changed Brownian motion. Our approach provides a unified framework, and combines all existing theories of diffusion asymptotics with new results that appear to be particularly relevant in many practical applications. For an illustration of our asymptotics, we employ them to analyze a class of high frequency regressions that is commonly used in empirical economics and finance.