We report the results of several theoretical
studies into the convergence rate for certain random series
representations of α-stable random variables, which are
motivated by and find application in modelling heavy-tailed
noise in time series analysis, inference, and stochastic processes.
The use of α-stable noise distributions generally leads
to analytically intractable inference problems. The particular
version of the Poisson series representation invoked here
implies that the resulting distributions are “conditionally
Gaussian,” for which inference is relatively straightforward,
although an infinite series is still involved. Our approach is
to approximate the residual (or “tail”) part of the series from
some point, c > 0, say, to ∞, as a Gaussian random variable.
Empirically, this approximation has been found to be very
accurate for large c. We study the rate of convergence, as
c → ∞, of this Gaussian approximation. This allows the
selection of appropriate truncation parameters, so that a
desired level of accuracy for the approximate model can be
achieved. Explicit, nonasymptotic bounds are obtained for
the Kolmogorov distance between the relevant distribution
functions, through the application of probability-theoretic
tools. The theoretical results obtained are found to be in very
close agreement with numerical results obtained in earlier
work
