Decompounding on Symmetric Spaces

Lévy processes are the left-invariant cadlag Markov processes on a Lie group or homogeneous space. This thesis deals with compound processes, the Lévy processes obtained as random walks whose walking frequency is driven by a homogeneous Poisson process. Methods of decompounding, i.e. the problem of inferring the step distribution of a compound process, will be established and analyzed. It will be shown that decompounding on symmetric spaces is possible for low-frequency sampling, with convergence in probability. The probability that the squared-norm error of the estimated probability density is greater than a given tolerance will be shown to decrease at an inverse proportional rate to the number of available observations. The method relies on a Fourier series expansion, and it will be shown that the number of approximating terms can be chosen in a manner that depends only on the number of available observations and intrinsic properties of the underlying Lie group on which the process appears.