We study the rate-distortion problem for both scalar and vector memoryless heavy-tailed α-stable sources (0 < α < 2). Using a recently defined notion of “strength” as a power measure, we derive the rate-distortion function for α-stable sources subject to a constraint on the strength of the error and show it to be logarithmic in the strength-to-distortion ratio. We show how our framework paves the way for finding optimal quantizers for α- stable sources and other general heavy-tailed ones. In addition, we study high-rate scalar quantizers and show that uniform ones are asymptotically optimal under the error-strength distortion measure. We compare uniform Gaussian and Cauchy quantizers and show that more representation points for the Cauchy source are required to guarantee the same quantization quality. Our findings generalize the well-known results of rate-distortion and quantization of Gaussian sources (α = 2) under a quadratic distortion measure.

We propose a framework for the rate-distortion analysis that extends the Gaussian MSE setup for stable sources. Specifically, we study the rate-distortion function of$\alpha$-stable sources when the distortion constraint is specified using the recently defined notion of strength of the error variable. We also “generalize” the notion of$d$-tilted information under the new setup and use it to determine second-order terms for both fixed and variable-length codes.