Geodesic
The invariance principle type theorems for recurrent semi-Markov processes with a~general state space
Teoriâ veroâtnostej i ee primeneniâ, Tome 24 (1979) no. 3, pp. 529-541

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The invariance principle type theorems for processes of sums of random variables defined on a recurrent semi-Markov process with a general state space are proved. The results are obtained in assumption that for a fixed subset of state space the recurrent times have a stable limit distribution $L_\alpha$, $\alpha\in(0,1]$ (the case $\alpha=1$ corresponds to the ergodic process). 
@article{TVP_1979_24_3_a6,
     author = {E. I. Kaplan and D. S. Sil'vestrov},
     title = {The invariance principle type theorems for recurrent {semi-Markov} processes with a~general state space},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {529--541},
     publisher = {mathdoc},
     volume = {24},
     number = {3},
     year = {1979},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1979_24_3_a6/}
}
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E. I. Kaplan; D. S. Sil'vestrov. The invariance principle type theorems for recurrent semi-Markov processes with a~general state space. Teoriâ veroâtnostej i ee primeneniâ, Tome 24 (1979) no. 3, pp. 529-541. http://geodesic.mathdoc.fr/item/TVP_1979_24_3_a6/