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Long‐Range Dependence of Markov Renewal Processes
Author(s) -
Vesilo R.A.
Publication year - 2004
Publication title -
australian and new zealand journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.434
H-Index - 41
eISSN - 1467-842X
pISSN - 1369-1473
DOI - 10.1111/j.1467-842x.2004.00321.x
Subject(s) - mathematics , moment (physics) , markov chain , markov renewal process , renewal theory , markov process , countable set , second moment of area , kernel (algebra) , markov kernel , range (aeronautics) , markov model , markov property , discrete mathematics , statistical physics , statistics , variable order markov model , physics , geometry , materials science , classical mechanics , composite material
Summary This paper examines long‐range dependence (LRD) and asymptotic properties of Markov renewal processes generalizing results of Daley for renewal processes. The Hurst index and discrepancy function, which is the difference between the expected number of arrivals in (0, t ] given a point at 0 and the number of arrivals in (0, t ] in the time stationary version, are examined in terms of the moment index. The moment index is the supremum of the set of r > 0 such that the r th moment of the first return time to a state is finite, employing the solidarity results of Sgibnev. The results are derived for irreducible, regular Markov renewal processes on countable state spaces. The paper also derives conditions to determine the moment index of the first return times in terms of the Markov renewal kernel distribution functions of the process.