  Mathematical formulations and proofs for a wavelet based statistic employed
in functional data analysis is elaborately discussed in this report. The
propositions and derivations discussed here apply to a wavelet based statistic
with hard thresholding. The proposed analytic distribution is made feasible
only due to the assumption of normality. Since the statistic is developed for
applications in high dimensional data analysis, the assumption holds true in
most practical situations. In the future, the work here could be extended to
address data that are non-Gaussian. Aside from establishing a rigorous
mathematical foundation for the distribution of the statistic, the report also
explores a few approximations for the proposed statistic.
