Inversion formula for diadic wavelet representation of second-order processes

We give an inversion formula for orthogonal random measure(ORM) of diadic wavelet representation of second-order processes. The keys to the method are construction of indicator function of a desired set on (a, b)-domain and an evaluation of a principal value integral related to sinξ/ξ. Though we can not construct any such indicator functions by wavelet transform, we show that we can, in certain limit form, by using a series of wavelet transforms. But we encounter there an uncertainty lying between variablesa andb so that we can not make the indicator function of a set on (a, b)-domain which is as "small" as we like.[PUBLICATION ABSTRACT]