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The first-order autoregressive Mittag-Leffler process. (English) Zbl 0777.60063
The authors discuss the first-order autoregressive semi-Mittag-Leffler (SMLAR(1)) process \(X_ n\) introduced first by the second author [Opsearch 26, 57 (1988)]. Particularly a distribution of the process and its Laplace transform is discussed as well as its deviation from the exponential one is analysed. An important aspect of the stationary sequence \(X_ n\) is the distribution of sums \(T_ r=X_ n+X_{n+1}+\dots+X_{n+r-1}\) and joint distribution of \((X_ n,X_{n+1})\). The Laplace transforms of these distributions are presented and time-irreversibility of the SMLAR(1) is proved. The first- order autoregressive Mittag-Leffler process as an example of SMLAR(1) is considered.

MSC:
60J05 Discrete-time Markov processes on general state spaces
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
60E05 Probability distributions: general theory
60E10 Characteristic functions; other transforms
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