Geodesic
On the convergence of semi-Markov walks to a~continuous semi-Markov process
Teoriâ veroâtnostej i ee primeneniâ, Tome 21 (1976) no. 3, pp. 497-511

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A sufficient condition of weak convergence of probability measures on the space $D[0,\infty)$ in terms of hitting probabilities is obtained. This condition is applied to prove the convergence of semi-Markov walks to a continuous semi-Markov process. 
@article{TVP_1976_21_3_a2,
     author = {B. P. Harlamov},
     title = {On the convergence of {semi-Markov} walks to a~continuous {semi-Markov} process},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {497--511},
     publisher = {mathdoc},
     volume = {21},
     number = {3},
     year = {1976},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1976_21_3_a2/}
}
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B. P. Harlamov. On the convergence of semi-Markov walks to a~continuous semi-Markov process. Teoriâ veroâtnostej i ee primeneniâ, Tome 21 (1976) no. 3, pp. 497-511. http://geodesic.mathdoc.fr/item/TVP_1976_21_3_a2/