The Coding Complexity of Lévy Processes

Abstract

Summary. We investigate the high resolution coding problem for general real-valued Lévy processes under L p [0, 1]-norm distortion. Tight asymptotic formulas are found under mild regularity assumptions.

High resolution quantization and entropy coding of jump processes

“…The theorems and examples presented in this subsection complement results from [2], where general real-valued Lévy processes are studied. The main result for compound Poisson processes in that paper states that, for any compound Poisson process with E log max(|Z (1) |, 1) < ∞ and all s ≥ 1,…”

“…for fractional Brownian motion, diffusions, or stable-like Lévy processes, see for instance [11,17,12,18,9,10,19,2]). Let us consider a simple example.…”

“…Since about 2000 researchers are attracted by the problem in the case where the original signal is infinitedimensional. A series of articles followed on (infinite-dimensional) random vectors X that are Gaussian (see for instance [11], [16], [12]), diffusions ( [17], [9], [10]), and Lévy processes ( [18], [2]).…”

“…Now we ask for lower bounds. Clearly, one cannot expect a non-trivial lower bound when only assuming (2). Thus, let us assume in this subsection that the jump positions constitute a Poisson point process and that condition (*) holds.…”