Absolutely Continuous Approximations

Abstract

For absolutely continuous distributions it is more convenient to write all results in terms of distribution functions. We recall that the distribution function of F is $F(x) = F\{(-\infty,x]\}$ and $\displaystyle{ F(x) =\int \limits _{ -\infty }^{x}f(y)\,\mbox{ d}y,\qquad \widehat{F}(t) =\int \limits _{ -\infty }^{\infty }\mathrm{e}^{\mathrm{i}tx}f(x)\,\mathrm{d}x. }$ Here f(x) is a nonnegative function integrable on the real line, called the density of F.