The Shortest Possible Return Time of ss-Mixing Processes

Author(s)

Abadi, Miguel ^[1] ; Gallo, Sandro ^[2] ; Rada-Mora, Erika Alejandra ^[3]

Affiliation

^[1] Univ Sao Paulo, Inst Matemat & Estat, Sao Paulo - Brazil

^[2] Univ Fed Sao Carlos, Dept Estat, Sao Paulo - Brazil

^[3] Univ Fed ABC, Ctr Matemat Computacao & Cognicao, Sao Bernardo Do Campo, SP - Brazil

• Document type: Journal article
• Source: IEEE TRANSACTIONS ON INFORMATION THEORY; v. 64, n. 7, p. 4895-4906, JUL 2018.
• Web of Science Citations: 1

Abstract

We consider a stochastic process and a given n-string. We study the shortest possible return time (or shortest return path) of the string over all the realizations of process starting from this string. For a beta-mixing process having complete grammar, and for each size n of the strings, we approximate the distribution of this short return (properly re-scaled) by a non-degenerated distribution. Under mild conditions on the beta coefficients, we prove the existence of the limit of this distribution to a non-degenerated distribution. We also prove that ergodicity is not enough to guaranty this convergence. Finally, we present a connection between the shortest return and the Shannon entropy, showing that maximum of the re-scaled variables grow as the matching function of Wyner and Ziv. (AU)

• FAPESP's process: 14/19805-1 - Statistics of extreme events and dynamics of recurrence
• Grantee: Miguel Natalio Abadi
• Support type: Regular Research Grants
• FAPESP's process: 15/09094-3 - Stochastic chains with long memory
• Grantee: Alexsandro Giacomo Grimbert Gallo
• Support type: Regular Research Grants
• FAPESP's process: 13/07699-0 - Research, Innovation and Dissemination Center for Neuromathematics - NeuroMat
• Grantee: Jefferson Antonio Galves
• Support type: Research Grants - Research, Innovation and Dissemination Centers - RIDC