Local theorems for arithmetic multidimensional compound renewal processes under Cram{\'e}r's condition

A. A. Mogul'skiǐ ; E. I. Prokopenko

Matematičeskie trudy, Tome 22 (2019) no. 2, pp. 106-133.

Résumé

We continue the study of compound renewal processes (c.r.p.) under Cramér's moment condition initiated in [2–10, 12–16]. We examine two types of arithmetic multidimensional c.r.p. $\mathbf{Z}(n)$ and $\mathbf{Y}(n)$, for which the random vector $\mathbf{\xi}=(\tau,\mathbf{\zeta})$ controlling these processes ($\tau>0$ defines the distance between jumps, $\mathbf{\zeta}$ defines the value of jumps of the c.r.p.) has an arithmetic distribution and satisfies Cramér's moment condition. For these processes, we find the exact asymptotics in the local limit theorems for the probabilities $$ \mathbb{P}(\mathbf{Z}(n)=\mathbf{x}),\quad \mathbb{P}(\mathbf{Y}(n)=\mathbf{x}) $$ in the Cramér zone of deviations for $\mathbf{x}\in\mathbb{Z}^d$ (in [9, 10, 13–15], the analogous problem was solved for nonlattice c.r.p., where the vector $\mathbf{\xi}=(\tau,\mathbf{\zeta})$ has a nonlattice distribution).

Bibliographie

[1] Borovkov A. A., Asimptoticheskii analiz sluchainykh bluzhdanii. Bystro ubyvayuschie raspredeleniya priraschenii, Fizmatlit, M., 2013

[2] Borovkov A. A., “Printsipy bolshikh uklonenii v granichnykh zadachakh dlya obobschennykh protsessov vosstanovleniya”, Sib. matem. zhurn., 57:3 (2016), 562–595 | MR | Zbl

[3] Borovkov A. A., Mogulskii A. A., “Vtoraya funktsiya uklonenii i asimptoticheskie zadachi vosstanovleniya i dostizheniya granitsy dlya mnogomernykh bluzhdanii”, Sib. matem. zhurn., 37:4 (1996), 745–782 | MR | Zbl

[4] Borovkov A. A., Mogulskii A. A., “Eksponentsialnye neravenstva chebyshevskogo tipa dlya summ sluchainykh vektorov i dlya sluchainykh bluzhdanii”, TVP, 56:1 (2011), 3–29 | DOI | MR

[5] Borovkov A. A., Mogulskii A. A., “Printsipy bolshikh uklonenii dlya konechnomernykh raspredelenii obobschennykh protsessov vosstanovleniya”, Sib. matem. zhurn., 56:1 (2015), 36–64 | MR | Zbl

[6] Borovkov A. A., Mogulskii A. A., “Printsipy bolshikh uklonenii dlya traektorii obobschennykh protsessov vosstanovleniya. I”, TVP, 60:2 (2015), 227–247 | DOI | MR

[7] Borovkov A. A., Mogulskii A. A., “Printsipy bolshikh uklonenii dlya traektorii obobschennykh protsessov vosstanovleniya. II”, TVP, 60:3 (2015), 417–438 | DOI

[8] Borovkov A. A., Mogulskii A. A., “Integro-lokalnye predelnye teoremy dlya obobschennykh protsessov vosstanovleniya pri vypolnenii usloviya Kramera. I”, Sib. matem. zhurn., 59:3 (2018), 491–513 | MR | Zbl

[9] Borovkov A. A., Mogulskii A. A., “Integro-lokalnye predelnye teoremy dlya obobschennykh protsessov vosstanovleniya pri vypolnenii usloviya Kramera. II”, Sib. matem. zhurn., 59:4 (2018), 736–758 | MR | Zbl

[10] Koks D. R., Smit V. L., Teoriya vosstanovleniya, Izd-vo «Sovetskoe radio», M., 1967 | MR

[11] Mogulskii A. A., “Lokalnye teoremy dlya arifmeticheskikh obobschennykh protsessov vosstanovleniya pri vypolnenii usloviya Kramera”, Sib. elektron. matem. izv., 16 (2019), 21–40

[12] Mogulskii A. A., Prokopenko E. I., “Integro-lokalnye predelnye teoremy dlya mnogomernykh obobschennykh protsessov vosstanovleniya pri momentnom usloviya Kramera. I”, Sib. elektron. matem. izv., 15 (2018), 475–502 | Zbl

[13] Mogulskii A. A., Prokopenko E. I., “Integro-lokalnye predelnye teoremy dlya mnogomernykh obobschennykh protsessov vosstanovleniya pri momentnom usloviya Kramera. II”, Sib. elektron. matem. izv., 15 (2018), 503–527 | Zbl

[14] Mogulskii A. A., Prokopenko E. I., “Integro-lokalnye predelnye teoremy dlya mnogomernykh obobschennykh protsessov vosstanovleniya pri momentnom usloviya Kramera. III”, Sib. elektron. matem. izv., 15 (2018), 528–553 | Zbl

[15] Mogulskii A. A., Prokopenko E. I., “Funktsiya uklonenii i bazovaya funktsiya dlya mnogomernogo obobschennogo protsessa vosstanovleniya”, Sib. elektron. matem. izv., 16 (2019), 1449–1463 | Zbl

[16] Asmussen S., Albrecher H., Ruin Probabilities, 2nd ed., World Scientific, Hackensack, NJ, 2010 | MR | Zbl