Geodesic
On exact large deviation principles for compound renewal processes
Teoriâ veroâtnostej i ee primeneniâ, Tome 66 (2021) no. 2, pp. 214-230 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider two large deviation principles (LDPs): the “ordinary” LDP (when the “strong” Cramér condition is met) and the “extended” LDP when only the standard Cramér condition is met and the deviation functional may be finite also for discontinuous trajectories. The standard formulation of these principles involves two asymptotic (upper and lower) estimates for the logarithms of the probabilities that the normalized trajectory of the process lies in a given set $B$. We obtain conditions on a set $B$ such that these estimates coincide and the large deviation principles take the form of exact asymptotic equalities. Such LDPs are called exact. We show that the estimating interval of an ordinary LDP is contained in the estimating interval of the extended LDP. Hence the fulfillment of the exact extended LDP implies that of the exact ordinary LDP. The results obtained in the present paper are also fully valid and relevant for random walks (a special case of compound recovery processes).
Keywords: large deviation principle, extended large deviation principle, exact large deviation principle, most probable trajectory, deviation functional, random walks. 
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A. A. Borovkov. On exact large deviation principles for compound renewal processes. Teoriâ veroâtnostej i ee primeneniâ, Tome 66 (2021) no. 2, pp. 214-230. http://geodesic.mathdoc.fr/item/TVP_2021_66_2_a0/

[1] A. A. Borovkov, Obobschennye protsessy vosstanovleniya, RAN, M., 2020, 455 pp.

[2] A. A. Borovkov, Asymptotic analysis of random walks. Light-tailed distributions, Encyclopedia Math. Appl., 176, Cambridge Univ. Press, Cambridge, 2020, xvi+420 pp. | DOI | Zbl | Zbl

[3] A. Dembo, O. Zeitouni, Large deviations techniques and applications, Appl. Math. (N. Y.), 38, 2nd ed., Springer-Verlag, New York, 1998, xvi+396 pp. | DOI | MR | Zbl

[4] Jin Feng, T. G. Kurtz, Large deviations for stochastic processes, Math. Surveys Monogr., 131, Amer. Math. Soc., Providence, RI, 2006, xii+410 pp. | DOI | MR | Zbl

[5] A. A. Borovkov, A. A. Mogulskii, “Printsipy bolshikh uklonenii dlya traektorii sluchainykh bluzhdanii. I”, Teoriya veroyatn. i ee primen., 56:4 (2011), 627–655 ; II, 57:1 (2012), 3–34 ; III, 58:1 (2013), 37–52 ; A. A. Borovkov, A. A. Mogulskii, “Large deviation principles for random walk trajectories. I”, Theory Probab. Appl., 56:4 (2011), 538–561 ; II, 57:1 (2013), 1–27 ; III, 58:1 (2014), 25–37 | DOI | MR | Zbl | DOI | Zbl | DOI | Zbl | DOI | DOI | MR | DOI | MR

[6] A. V. Skorokhod, “Limit theorems for stochastic processes”, Theory Probab. Appl., 1:3 (1956), 261–290 | DOI | MR | Zbl

[7] A. A. Mogulskii, “Rasshirennyi printsip bolshikh uklonenii dlya traektorii obobschennykh protsessov vosstanovleniya”, Matem. tr., 24 (2021) (to appear)