Geodesic
Large deviation principle for multidimensional first compound renewal processes in the phase space
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 16 (2019), pp. 1464-1477 Cet article a éte moissonné depuis la source Math-Net.Ru

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We obtain the large deviation principles for multidimensional first compound renewal processes $\mathbf{Z}(t)$ in the phase space $\mathbb{R}^d$, for this we find and investigate the rate function $D_Z(\alpha)$. Also we find asymptotics for the Laplace transform of this process when the time goes to infinity, for this we find and investigate the so-called fundamental function $A_Z(\mu)$.
Keywords: compound multidimensional renewal process, large deviations, renewal measure, Cramer's condition, deviation (rate) function, second deviation (rate) function, fundamental function. 
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A. A. Mogulskii; E. I. Prokopenko. Large deviation principle for multidimensional first compound renewal processes in the phase space. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 16 (2019), pp. 1464-1477. http://geodesic.mathdoc.fr/item/SEMR_2019_16_a42/

[1] A.A. Borovkov, A.A. Mogulskii, “Integro-local limit theorems for compound renewal processes with Cramer's condition. I”, Siberian Mathematical Journal, 59:3 (2018), 491–514 | MR

[2] A.A. Borovkov, A.A. Mogulskii, “Integro-local limit theorems for compound renewal processes with Cramer's condition. II”, Siberian Mathematical Journal, 59:4 (2018), 731–750 | MR

[3] A.A. Mogulskii, E.I. Prokopenko, “Integro-local theorems for multidimensional compound renewal processes, when Cramer's condition holds. I”, Siberian Electronic Mathematical Reports, 15 (2018), 475–502 | MR | Zbl

[4] A.A. Mogulskii, E.I. Prokopenko, “Integro-local theorems for multidimensional compound renewal processes, when Cramer’s condition holds. II”, Siberian Electronic Mathematical Reports, 15 (2018), 503–527 | MR | Zbl | Zbl

[5] A.A. Mogulskii, E.I. Prokopenko, “Integro-local theorems for multidimensional compound renewal processes, when Cramer's condition holds. III”, Siberian Electronic Mathematical Reports, 15 (2018), 528–553 | MR | Zbl

[6] A.A. Mogulskii, E.I. Prokopenko, “The rate function and the fundamental function for multidimensional renewal process”, Siberian Electronic Mathematical Reports, 16 (2019), 1449–1463 | Zbl

[7] R. Lefevere, M. Mariani, L. Zambotti, “Large deviations for renewal processes”, Stochastic Processes and their Applications, 121:10 (2011), 2243–2271 | MR | Zbl

[8] A.A. Borovkov, A.A. Mogulskii, “Large deviation principles for the finite-dimensional distributions of compound renewal processes”, Sib. Math. J., 56:1 (2015), 28–53 | MR | Zbl

[9] A.A. Borovkov, A.A. Mogulskii, E.I. Prokopenko, “Properties of the deviation rate function and the asimptotics for the Laplace transform of the distribution of a compound renewal process”, Teor. Veroyatnost. i Primenen., 64:4 (2019), 625–641 | MR

[10] B. Tsirelson, “From uniform renewal theorem to uniform large and moderate deviations for renewal-reward processes”, Electron. Commun. Probab., 18:52 (2013), 1–13 | MR

[11] A.A. Borovkov, Asymptotic analysis of random walks. Rapidly decreasing distributions of increments, Fizmatlit, M., 2013 | Zbl

[12] A. Dembo, O. Zeitouni, Large Deviations Techniques and Applications, Springer-Verlag, Berlin, 2010 | MR | Zbl

[13] Rockafellar R. T., Convex Analysis, Princeton University Press, Princeton, 1970 | MR | Zbl

[14] A.A. Borovkov, Probability Theory, Springer-Verlag, London, 2013 | MR | Zbl

[15] A.A. Borovkov, A.A. Mogulskii, “The second rate function and the asymptotic problems of renewal and hitting the boundary for multidimensional random walks”, Sib. Math. J., 37:4 (1996), 647–682 | MR | Zbl