Measurements of resonant tunneling through a localized impurity state are used to probe fluctuations in the local density of states of heavily doped GaAs. The measured differential conductance is analyzed in terms of correlation functions with respect to voltage. A qualitative picture based on the scaling theory of Thouless is developed to relate the observed fluctuations to the statistics of single-particle wave functions. In a quantitative theory correlation functions are calculated. By comparing the experimental and theoretical correlation functions, the effective dimensionality of the emitter is analyzed and the dependence of the inelastic lifetime on energy is extracted.