AN ALMOST SURE RECURRENCE THEOREM WITH DISTORTION FOR STATIONARY ERGODIC SOURCES

Citation
F. Kanaya et J. Muramatsu, AN ALMOST SURE RECURRENCE THEOREM WITH DISTORTION FOR STATIONARY ERGODIC SOURCES, IEICE transactions on fundamentals of electronics, communications and computer science, E80A(11), 1997, pp. 2264-2267
Citations number
9
ISSN journal
09168508
Volume
E80A
Issue
11
Year of publication
1997
Pages
2264 - 2267
Database
ISI
SICI code
0916-8508(1997)E80A:11<2264:AASRTW>2.0.ZU;2-Z
Abstract
Let (X-k)(k=-infinity)(infinity) be a stationary and ergodic informati on source, where each X-k takes values in a standard alphabet A with a distance function d: A x A --> [0, infinity) defined on it. For each sample sequence x = (...,x(-1),x(0),x(1),...) and D > 0 let the approx imate D-match recurrence time be defined by R-n(x, D) = min(m greater than or equal to n : d(n)(X-1(n), X-m+1(m+n)) less than or equal to D} where X-i(j) denotes the string x(i)x(i+1) ... x(j) and d(n) : A(n) x A(n) --> [0, infinity) is a metric of A(n) induced by d for each n. L et R(D) be the rate distortion function of the source (X-k)(k=-infinit y)(infinity) relative to the fidelity criterion (d(n)). Then it is sho wn that lim sup(n-->infinity) 1/n log R-n (X, D) less than or equal to R(D/2) a.s.