Studies of several curious probabilistic phenomena: Unobservable tail exponents in random difference equations, and confusion between models of long-range dependence and changes in regime

The dissertation is centered on two research topics. The first topic concerns reduction of bias in estimation of tail exponents in random difference equations (RDEs). The bias is due to deviations from the exact power-law tail, which are quantified by proving a weaker form of the so-called second-order regular variation of distribution tails of RDEs. In particular, the latter suggests that the distribution tails of RDEs have an explicitly known second-order power-law term. By taking this second-order term into account, a number of successful bias-reduced tail exponent estimators are proposed and examined. The second topic concerns the confusion between long-range dependent (LRD) time series and several nonstationary alternatives, such as changes in local mean level superimposed by short-range dependent series. Exploratory and informal tools based on the so-called unbalanced Haar transformation are first suggested and examined to assess the adequacy of LRD models in capturing changes in local mean in real time series. Second, formal statistical procedures are proposed to distinguish between LRD and alternative models, based on estimation of LRD parameter in time series after removing changes in local mean level. Basic asymptotic properties of the tests are studied and applications to several real time series are also discussed.