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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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