Abstract
In the study of fluctuations of certain random sequences combinatorial methods have been found to be extremely effective, and as such it is likely that the enumeration of paths will be encountered. Some particular sequences are such that the resulting paths, when cyclically or simply permuted, exhibit certain properties that are invariant with respect to the sequence selected. These properties lead to profound probabilistic results in fluctuation theory. The chapter reviews some invariance properties and their applications in fluctuation theory. The chapter discusses the urn problem and a precise formulation of the penetrating analysis. Besides a natural generalization of the urn problem, a different type of generalization, leading to certain combinatorial identities, is discussed. A well-known equivalence principle of Sparre Andersen in fluctuation theory is re-examined in the context of permutations of a given set of lattice paths, yielding a refinement of the principle.
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