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References

Published online by Cambridge University Press:  16 June 2022

A. A. Borovkov
Translated by
Alexey Alimov
A. A. Borovkov
Affiliation:
Sobolev Institute of Mathematics, Russia
Alexey Alimov
Affiliation:
Steklov Institute of Mathematics, Moscow
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Compound Renewal Processes , pp. 357 - 362
Publisher: Cambridge University Press
Print publication year: 2022

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References

Aleskjavicene, A.: Large deviations for homogeneous Markov chains (1965), Litovsk. Math. Sb., 5, 199–209.Google Scholar
Anscombe, C.J.: Large sample theory of sequentional estimation (1952), Proc. Cambridge Philos. Soc., 48, 600–607.CrossRefGoogle Scholar
Araujo, A., Giné, E.: The central limit theorem for real and Banach valued random variables (1980), Wiley.Google Scholar
Asmussen, S.: Approximations for the probability of ruin within finite time (1984), Scand. Actuarial J., 31–57.CrossRefGoogle Scholar
Asmussen, S.: Applied Probability and Queues Stochastic Modelling and Applied Probability (2003), 2nd ed., in: Applications of Mathematics (New York), vol. 51, Springer.Google Scholar
Asmussen, S., Albrecher, H.: Ruin Probabilities. (2010), 2nd ed. Advanced Series on Statistical Science & Applied Probability 14. World Scientific.Google Scholar
Athreya, K.B., Ney, P.E.: A new approach to the limit theory of recurrent Markov chains (1978), Trans. Amer. Math. Soc., 245, 493–501.CrossRefGoogle Scholar
Azencott, R.: Grandes deviations et applications (1980), École d’Été de Probabilités de Saint-Flour VIII, 1978. Lecture Notes in Math. vol. 774, pp. 1–176. Springer.Google Scholar
von Bahr, B.: Ruin probabilities expressed in terms of ladder height distributions (1974), Scand. Actuarial J., 190–204.Google Scholar
Billingsley, P.: Convergence of Probability Measures (1999), 2nd ed., John Wiley & Sons.Google Scholar
Bingham, N.H., Goldie, C.H., and Teugels, J.L.: Regular Variations (1987), Cambridge University Press.CrossRefGoogle Scholar
Borovkov, A.A.: Asymptotic Analysis of Random Walks. Light Tailed Jump Distributions (2020), Cambridge University Press.Google Scholar
Borovkov, A.A.: New limit theorems in boundary-value problems for sums of independent terms (1962), Sibirsk. Mat. Zh., 3(5), 645–694.Google Scholar
Borovkov, A.A.: Analysis of large deviations in boundary-value problems with arbitrary boundaries I, II (1964), Sibirsk. Mat. Zh., 5(2); Sibirsk. Mat. Zh., 5(4), 750–767.Google Scholar
Borovkov, A.A.: Boundary problems for random walks and large deviations in functional spaces (1967), Theory Probab. Appl., 12(4), 575–595.CrossRefGoogle Scholar
Borovkov, A.A.: The convergence of tributions of functionals on stochastic processes (1972), Russian Math. Surveys, 27(1), 1–42.CrossRefGoogle Scholar
Borovkov, A.A.: Stochastic Processes in Queueing Theory (1976), Springer.Google Scholar
Borovkov, A.A.: Asymptotic Methods in Queueing Theory (1984), Wiley.Google Scholar
Borovkov, A.A.: Ergodicity and Stability of Stochastic Processes (1998), Wiley.Google Scholar
Borovkov, A.A.: On subexponential distributions and asymptotics of the distribution of the maximum of sequential sums (2002), Siberian Math. J., 43(6), 995–1022.Google Scholar
Borovkov, A.A.: Probability Theory (2013), Springer.Google Scholar
Borovkov, A.A.: Integro-local and local theorems for normal and large deviations of sums of nonidentically distributed random variables in the scheme of series (2009), Theory Probab. Appl., 54(4), 571–587.CrossRefGoogle Scholar
Borovkov, A.A.: Asymptotic Analysis of Random Walks: Rapidly Decreasing Distributions of Jumps (2013), Fizmatlit [in Russian].Google Scholar
Borovkov, A.A.: Second order approximation for distribution of maximum of random walk with negative drift and infinite variance (2014), Theory Probab. Appl., 59(1), 3–22.CrossRefGoogle Scholar
Borovkov, A.A.: Integral theorems for the first passage time of an arbitrary boundary by a compound renewal process (2015), Siberian Math. J., 56(5), 765–782.CrossRefGoogle Scholar
Borovkov, A.A.: Large deviation principles in boundary problems for compound renewal processes (2016), Siberian Math. J., 57(3), 442–469.Google Scholar
Borovkov, A.A.: On the distribution of the first passage time of an arbitrary remote boundary by random walk (2016), Theory Probab. Appl., 61(2), 235–254.Google Scholar
Borovkov, A.A.: Stability theorems and the second-order asymptotics in threshold phenomena for boundary functionals of random walks (2016), Siberian Adv. Math., 26(4), 231–246.Google Scholar
Borovkov, A.A.: Some Boundary Problems of Probability Theory (2016), Saarbrucken: Palmarium Academic Publishing.Google Scholar
Borovkov, A.A.: Integro-local limit theorems for compound renewal processes (2017), Theory Probab. Appl., 62(2), 175–195.CrossRefGoogle Scholar
Borovkov, A.A.: Functional limit theorems for compound renewal processes (2019), Siberian Math. J., 60(1), 27–40.CrossRefGoogle Scholar
Borovkov, A.A.: Moderately large deviation principles for trajectories of compound renewal processes (2019), Theory Probab. Appl., 64(2), 324–333.CrossRefGoogle Scholar
Borovkov, A.A.: Extension of the invariance principle for compound renewal processes to the zones of moderately large and small deviations (2021), Theory Probab. Appl., 65(4), 651–670.Google Scholar
Borovkov, A.A.: On large deviation principles for compound renewal processes (2019), Math. Notes, 106(6), 864–871.Google Scholar
Borovkov, A.A.: Integro-local theorems in boundary problems for compound renewal processes (2019), Sibirsk. Mat. Zh., 60(6), 1229–1246.Google Scholar
Borovkov, A.A.: Boundary problems for compound renewal processes (2020), Sibirsk. Mat. Zh., 61(1), 29–59.Google Scholar
Borovkov, K.A.: Rate of convergence in the invariance principle for generalized renewal processes (1983), Theory Probab. Appl., 27(3), 461–471.Google Scholar
Borovkov, A.A., Borovkov, K.A.: Asymptotic Analysis of Random Walks. Vol. I: Slowly Decaying Jump Distributions (2008), Moscow: Fizmatlit (in Russian).Google Scholar
Borovkov, A.A., Mogul’skii, A.A.F.: The second rate function and the asymptotic problems of renewal and hitting the boundary for multidimensional random walks (1996), Siberian Math. J., 37(4), 647–682.CrossRefGoogle Scholar
Borovkov, A.A., Mogul’skii, A.A.F.: Integro-local limit theorems including large deviations for sums of random vectors (1999, 2001), Theory Probab. Appl., 43(1), 1–12; Theory Probab. Appl., 45(1), 3–22.Google Scholar
Borovkov, A.A., Mogul’skii, A.A.: Limit theorems in the boundary hitting problem for a multidimensional random walk (2001), Siberian Math. J., 42(2), 245–270.CrossRefGoogle Scholar
Borovkov, A.A., Mogul’skii, A.A. Integro-local and integral theorems for sums of random variables with semiexponential distributions (2006), Siberian Math. J., 47(6), 990–1026.Google Scholar
Borovkov, A.A., Mogul’skii, A.A.F.: On large deviation principles for random walk trajectories. I, II, III (2011, 2012, 2013), Theory Probab. Appl., 56(4), 538–561; Theory Probab. Appl., 58(1), 1–27; Theory Probab. Appl., 58(1), 25–37.Google Scholar
Borovkov, A.A., Mogul’skii, A.A.: Large deviation principles for sums of random vectors and the corresponding renewal functions in the inhomogeneous case (2015), Siberian Adv. Math., 25(4), 255–267.Google Scholar
Borovkov, A.A., Mogul’skii, A.A.: Large deviation principles for the finite-dimensional distributions of compound renewal processes (2015), Siberian Math. J., 56(1), 28–53.Google Scholar
Borovkov, A.A., Mogul’skii, A.A.: Large deviation principles for trajectories of compound renewal processes, I, II (2016), Theory Probab. Appl., 60(2), 207–221; Theory Probab. Appl., 60(3), 349–366.Google Scholar
Borovkov, A.A., Mogul’skii, A.A.F.: Integro-local limit theorems for compound renewal processes under Cramér’s condition. I, II (2018), Siberian Math. J., 59(3), 383–402, Siberian Math. J., 59(4), 578–597.Google Scholar
Borovkov, A.A., Mogul’skii, A.A.F., Prokopenko, E.I.: Properties of the deviation rate function and the asymptotics for the Laplace thansform of the distribution of a compound renewal process (2019), Theory Probab. Appl., 64(4), 499–512Google Scholar
Borovkov, A.A., Rogozin, B.A.: Boundary problems in some two-dimensional random walks (1964), Theory Probab. Appl., 9(3), 361–388.CrossRefGoogle Scholar
Chung, K.L.: A Course in Probability Theory (1974), 2nd ed., Academic Press.Google Scholar
Çinlar, E.: Markov additive processes. I (1972), Z. Warsch. verw. Gebiete, 24, 84–94.Google Scholar
Çinlar, E.: Markov additive processes. II (1972), Z. Warsch. verw. Gebiete, 24, 94–121.Google Scholar
Cox, D.R., Smith, W.L. Renewal Theory (1962), Wiley.Google Scholar
Cramér, H.: Collective Risk Theory (1955), Stockholm: Erselte.Google Scholar
Csörgö, M., Horváth, L., and Steinebach, J.: Invariance principles for renewal processes (1987), Ann. Probability, 15(4), 1441–1460.Google Scholar
Csörgö, M., Deheuvels, P., and Hervatt, L.: An approximation of stopped sums with applications in queueing theory (1987), Adv. Appl. Probability, 19, 674–690.Google Scholar
Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications (1998), 2nd ed., Springer.Google Scholar
Doney, R.A.: A large deviation local limit theorem (1989), Math. Proc. Cambridge Phil. Soc., 105(3), 575–577.Google Scholar
Donsker, M.D. and Varadhan, S.R.S.: Asymptotic evaluation of certain Markov process expectations for large time. I, (1975) Comm. Pure Math., 28, 1–47; (1975) 28(1), 279–301; (1976) 29, 389–461; (1983) 36, 183–212.Google Scholar
Donsker, M.D., and Varadhan, S.R.S.: Asymptotic evaluation of certain Markov process expectations for large time. II // Comm. Pure Math., 1975. V. 28(2). P. 279–301.Google Scholar
Donsker, M.D., and Varadhan, S.R.S.: Asymptotic evaluation of certain Markov process expectations for large time. III // Comm. Pure Math., 1976. V. 29(4). P. 389–461.Google Scholar
Donsker, M.D., and Varadhan, S.R.S.: Asymptotic evaluation of certain Markov process expectations for large time. IV // Comm. Pure Math., 1983. V. 36(2). P. 183–212.CrossRefGoogle Scholar
Drekic, S., and Willmot, G.E.: On the density and moments of the time of ruin with exponential claims (2003), ASTIN Bull., 33(1), 11–21.Google Scholar
Feng, J., Kurtz, T.G.: Large Deviations for Stochastic Processes (2006), Mathematical Surveys and Monographs, v. 131, AMS.Google Scholar
Foss, S. G., Puhalskii, A. A.: On the limit law of a random walk conditioned to reach a high level (2011), Stoch. Proc. and Appl., 121(2), 288–313.Google Scholar
Frolov, A.N.: Limit theorems for increments of compound renewal processes (2007), J. Math. Sci. (N. Y.), 152(6), 944–957.Google Scholar
Gamkrelidze, N.G.: On local limit theorem for integer random vectors (2015), Theory Probab. Appl., 59(3), 494–499.Google Scholar
Gikhman, I.I., Skorokhod, A.V.: Theory of Random Processes. I (1974), Springer.Google Scholar
Gnedenko, B.V., Kolmogorov, A.N.: Limit Distributions for Sums of Independent random Variables (1954), Addison-Wesley.Google Scholar
Gnedenko, B.V., Sherif, A.: Limit theorems for extreme terms of a variational series (1983), Dokl. Akad. Nauk SSSR, 270(3), 523–525.Google Scholar
Gudinas, P.P.: Large deviations for sums of random variables connected in a Markov chain (1986), + Litovsk. Math. Sb., 26(2), P.246–258.Google Scholar
Gut, Allan: Stopped Random Walks. Limit Theorem and Applications (2009), 2nd ed., Springer.Google Scholar
Hervé, M.: Several Complex Variables (1963), Oxford University Press.Google Scholar
Keilson, J., Wishart, D.: Addenda to processes defined on a finite Markov chain (1964), Proc. Cambridge Philos. Soc., 63 (1), 187–195.CrossRefGoogle Scholar
Keilson, J., Wishart, D.: A central limit theorem for processes defined on a Markov chain (1964), Proc. Cambridge Philos. Soc., 60(3), 547–567.CrossRefGoogle Scholar
Khintchine, A.: Zwei Sätze über stochastische Prozesse mit stabilen Verteilungen (1938), Rec. Math. [Mat. Sbornik] N.S., 3(45), 577–583.Google Scholar
Kingman, F. G.: On queues in heavy traffic (1962), J.R. Statist. Soc., Ser. B., 24(2), 383–392.Google Scholar
Komlós, J., Major, P., and Tusnády, : An approximation of partial sums of independent R.V.’s and the sample D.F. I (1976), Z. Warsch. Verw. Gebiete, 32(1), 111–131; (1976) 34. 33–58.Google Scholar
Korshunov, D.: On subexponential tails for the maxima of negatively driven compound renewal and Lévy processes (2018), Stochastic Processes and their Applications, 128(4), 1316–1332.CrossRefGoogle Scholar
Korshunov, D.A.: The critical case of the Cramer–Lundberg theorem on the asymptotic tail behavior of the maximum of a negative drift random walk (2005), Siberian Math. J., 46(6), 1077–1081.Google Scholar
Loéve, M.: Probability Theory (1960), Van Nostrand Co.Google Scholar
Macci, C.: Large deviations for compound Markov renewal processes with dependent jumps sizes and jumps waiting times (2007), Bull. Belg. Math. Soc. Simon Stevin, 14(2), 213–228.Google Scholar
Meyn, S.P., Tweedie, R.L.: Markov Chains and Stochastic Stability (1993), Springer.Google Scholar
Miller, H.D.: A convexity property in the theory of random variables defined on a finite Markov chain (1961), Annals Math. Statist., (32), 1260–1270.CrossRefGoogle Scholar
Mitalauskas, A.: On a multidimensional limit theorem for lattice distributions (1960), Trudy Lit. SSR Ser. B, 2 3–14.Google Scholar
Mogul’skii, A.A.: Small deviations in the sample function space (1975), Theory Probab. Appl., 19(1), 726–736.Google Scholar
Mogul’skii, A.A.: The Fourier method for finding the asymptotic behavior of small deviations of a Wiener process (1982), Siberian Math. J., 23(3), 420–431.Google Scholar
Mogul’skii, A.A.: The large deviation principle for a compound Poisson process (2017), Siberian Adv. Math., 2(3), 160–186.Google Scholar
Mogul’skii, A.A.: On a property of the Legendre transform (2018), Siberian Adv. Math., 28(1), 65–73.Google Scholar
Mogul’skii, A.A.: The extended large deviation principle for a process with independent increments (2017), Siberian Math. J., 58(3), 515–524.Google Scholar
Mogul’skii, A.A.: Local theorems for arithmetic compound renewal processes when Cramer’s condition holds (2019), Sib. Èlektron. Mat. Izv., 16, 21–41.Google Scholar
Mogul’skii, A.A., Prokopenko, E. I.: Integro-local theorems for multidimensional compound renewal processes, when Cramer’s condition holds I, II, III (2018), Sib. Èlektron. Mat. Izv., 15, 475–502; Sib. Èlektron. Mat. Izv., 15, 503–527; Sib. Èlektron. Mat. Izv., 15, 528–553.Google Scholar
Mogul’skii, A.A., Prokopenko, E.I.: Local theorems for multivariatearithmetic compound renewal processes under the Cramér condition (2019), Mat. Tr., 22(2), 106–133.Google Scholar
Nagaev, A.V., Cramer large deviations when the extreme conjugate distribution is heavytailed (1999), Theory Probab. Appl., 43(3), 405–421.Google Scholar
Ney, P., Nummelin, T.: Markov additive processes. I. Eigenvalue properties and limit theorems (1987), Annals of Probability, 15(2), 561–592.Google Scholar
Ney, P., Nummelin, T.: Markov additive processes. II. Large deviations (1987), Annals of Probability, 15(2), 593–609.CrossRefGoogle Scholar
Nummelin, E.: General Irreducible Markov Chains and Nonnegative Operators (1984), Cambridge University Press.Google Scholar
Presman, E.L.: Factorization methods and boundary problems for sums of random variables given on Markov chains (1969), Math. USSR-Izv., 3(4), 815–852.Google Scholar
Prokhorov, Yu. V.: On a local limit theorem for lattice distributions (1954), Dokl. Akad. Nauk SSSR, n. Ser., 98, 535–538.Google Scholar
Prokhorov, Yu. V.: Transient phenomena in processes of mass service, Lit. Matem. Sb., 3, 199–206.Google Scholar
Raudelyunas, A.K.: On a multidimensional local limit theorem (1964), Lit. Matem. Sb., 4, 141–145.Google Scholar
Renyi, A.: On the asymptotic distribution of the sum of a random number of independent random variables (1957), Acta Math. Acad. Sci. Hungar., 8, 193–199.Google Scholar
Rogozin, B.A.: The distribution of the maximum of a process with independent increments (1969), Siberian Math. J., 10(6), 989–1010.Google Scholar
Rozanov, Yu.A.: On a local limit theorem for lattice distributions (1957), Theory Probab. Appl., 2(2), 260–265.Google Scholar
Rozovskii, L.V.: A lower bound for large deviation probabilities of sum of independent random variables with finite variances (2002), J. Math. Sci. (New York), 109(6), 2192–2209.Google Scholar
Rockafellar, R.: Convex Analysis (1970), Princeton University Press.Google Scholar
Rvacheva, E. L.: On domain of attraction of multidimensional distributions (1960), Select Transl. Math. Statist. and Probability. vol. 2, pp. 183–205.Google Scholar
Saulis, L., Statulevicius, V.: Limit Theorems for Large Deviations (1991), Kluwer.Google Scholar
Shepp, L.A.: A local limit theorem (1964), Ann. Math. Statist., 35, 419–423.Google Scholar
Skorokhod, A.V.: Limit theorems for stochastic processes (1956), Theory Probab. Appl., 1(3), 261–290.Google Scholar
Smirnov, N.V., Limit distributions for the terms of a variational series (1949), Trudy Mat. Inst. Steklov., 25, Acad. Sci. USSR, Moscow–Leningrad, 25, 3–60.Google Scholar
Steinebach, J.: Invariance principles for renewal processes when only moments of low order exist (1988), J. Multivariate Analysis, 26(2), 166–183.Google Scholar
Steinebach, J.: On the optimality of strong approximation rates for compound renewal processes (1988), Statistics and Probability Letters, 6(4), 263–267.Google Scholar
Stone, C.: A local limit theorems for nonlattice multidimensional distribution functions (1965), Ann. Math. Statist., 36(2), 546–551.Google Scholar
Stone, C.: On local and ratio limit theorems (1966), Proc. Fifth Berkeley Symposium of Math Statistics and Probability. Berkeley and Los Angeles, University of California. Vol. II, Part II, pp. 217–224.Google Scholar
Tkachuk, S.G.: Local limit theorems with admissible large deviations for stable limit laws (1973), Izv. Akad. Nauk UzSSR, Ser. fiz.-Mat. Nauk, 17 (21), 30–33.Google Scholar
Torrang, I.: The law of the iterated logarithm–cluster points of deterministic and random subsequences (1987), Probab. Math. Statist., VIII, 133–141.Google Scholar
Varadhan, S.R.S.: Asymptotic probabilities and differential equations (1966), Comm. Pure Appl. Math., 19(3), 261–286.Google Scholar
Volkov, I.S.: On the distribution of sums of random variables defined on a homogeneous Markov chain with a finite number of states (1958), Theory Probab. Appl., 3(4), 384–399.Google Scholar
Volkov, I.S.: Analysis of some limit theorems for large deviations (1961), Theory Probab. Appl., 6(3), 302–304.Google Scholar
Zachary, S., Foss, S.G.: On the exact distributional asymptotics for the supremum of a random walk with increments in a class of light-tailed distributions (2006), Siberian Math. J., 47(6), 1034–1041.Google Scholar

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  • References
  • A. A. Borovkov
  • Translated by Alexey Alimov, Steklov Institute of Mathematics, Moscow
  • Book: Compound Renewal Processes
  • Online publication: 16 June 2022
  • Chapter DOI: https://doi.org/10.1017/9781009093965.011
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  • References
  • A. A. Borovkov
  • Translated by Alexey Alimov, Steklov Institute of Mathematics, Moscow
  • Book: Compound Renewal Processes
  • Online publication: 16 June 2022
  • Chapter DOI: https://doi.org/10.1017/9781009093965.011
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  • References
  • A. A. Borovkov
  • Translated by Alexey Alimov, Steklov Institute of Mathematics, Moscow
  • Book: Compound Renewal Processes
  • Online publication: 16 June 2022
  • Chapter DOI: https://doi.org/10.1017/9781009093965.011
Available formats
×