Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-14T06:47:38.350Z Has data issue: false hasContentIssue false

References

Published online by Cambridge University Press:  16 October 2020

A. A. Borovkov
Translated by
V. V. Ulyanov  and
Mikhail Zhitlukhin
A. A. Borovkov
Affiliation:
Sobolev Institute of Mathematics, Russia
V. V. Ulyanov
Affiliation:
Lomonosov Moscow State University and National Research University Higher School of Economics, Moscow
Mikhail Zhitlukhin
Affiliation:
Steklov Institute of Mathematics, Moscow
Get access

Summary

A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
Image of the first page of this content. For PDF version, please use the ‘Save PDF’ preceeding this image.'
Type
Chapter
Information
Asymptotic Analysis of Random Walks , pp. 410 - 418
Publisher: Cambridge University Press
Print publication year: 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] de Acosta, A. On large deviations of sums of independent random vectors. Probability in Banach spaces, V (Medford, Mass., 1984), 1–14, Lecture Notes in Math., 1153 (Springer, 1985).Google Scholar
[2] Arndt, K. Asymptotic properties of the distribution of the supremum of a random walk on a Markov chain. Theory Probab. Appl., 25:2 (1981), 309–324.CrossRefGoogle Scholar
[3] Arndt, K. On finding the distribution of the supremum of a random walk on a Markov chain in an explicit form. Trudy Instituta Matematiki SO AN SSSR, 1 (1982), 139–146. (In Russian.)Google Scholar
[4] Asmussen, S. Subexponential asymptotics for stochastic processes: extremal behaviour, stationary distributions and first passage probabilities. Ann. Appl. Probab., 8:2 (1998), 354–374.CrossRefGoogle Scholar
[5] Bahadur, R.R., Ranga Rao, R. On deviations of the sample mean. Ann. Math. Statist., 31:4 (1960), 1015–1027.Google Scholar
[6] Bahadur, R.R., Zabell, S.L. Large deviations of the sample mean in general vector spaces. Ann. Probab., 7:4 (1979), 587–621.Google Scholar
[7] Baron, M. Nonparametric adaptive change-point estimation and on-line detection. Sequential Anal., 19:1–2 (2000), 1–23.Google Scholar
[8] Bertoin, J., Doney, R.A. Some asymptotic results for transient random walks. Adv. Appl. Probab., 28 (1996), 207–227.CrossRefGoogle Scholar
[9] Bhattacharya, R.N., Rao, R. Normal Approximation and Asymptotic Expansions (Robert E. Krieger Publishing Company, 1986).Google Scholar
[10] Bickel, P.J., Yahav, J.A. Renewal theory in the plane. Ann. Math. Statist., 36 (1965), 946–955.Google Scholar
[11] Billingsley, P. Convergence of Probability Measures (Wiley, 1968).Google Scholar
[12] Bingham, N.H., Goldie, C.H., Teugels, J.L. Regular Variations (Cambrige University Press, Cambrige, 1987).Google Scholar
[13] Bolthausen, E. On the probability of large deviations in Banach spaces. Ann. Probab., 2:2 (1984) 427–435.Google Scholar
[14] Borisov, I.S. On the rate of convergence in the ‘conditional’ invariance principle. Theory Probab. Appl., 23:1 (1978), 63–76.Google Scholar
[15] Borovkov, A.A. Limit theorems on the distributions of maxima of sums of bounded lattice random variables. I, II. Theory Probab. Appl., 5:2 (1960), 125–155; 5:4 (1960), 341–355.Google Scholar
[16] Borovkov, A.A. New limit theorems for boundary crossing problems for sums of independent summands. Sibirskij Mat. J., 3:5 (1962), 645–694. (In Russian.)Google Scholar
[17] Borovkov, A.A. Analysis of large deviations in boundary crossing problems with arbitrary boundaries. I. Sibirskij Mat. J., 5:2 (1964), 253–289. (In Russian.)Google Scholar
[18] Borovkov, A.A. Analysis of large deviations in boundary crossing problems with arbitrary boundaries. II. Sibirskij Mat. J., 5:4 (1964), 750–767. (In Russian.)Google Scholar
[19] Borovkov, A.A., Boundary-value problems for random walks and large deviations in function spaces. Theory Probab. Appl., 12:4 (1967), 575–595.CrossRefGoogle Scholar
[20] Borovkov, A.A. Some inequalities for sums of multidimensional random variables. Theory Probab. Appl., 13:1 (1968), 156–160.Google Scholar
[21] Borovkov, A.A. The convergence of distributions of functionals on stochastic processes. Russian Math. Surveys, 27:1 (1972), 1–42.Google Scholar
[22] Borovkov, A.A. Stochastic Processes in Queueing Theory (Springer, 1976).Google Scholar
[23] Borovkov, A.A. Convergence of measures and random processes. Russian Math. Surveys, 31:2 (1976), 1–69.Google Scholar
[24] Borovkov, A.A. Asymptotic Methods in Queueing Theory (Wiley, 1984).Google Scholar
[25] Borovkov, A.A. On the Cramér transform, large deviations in boundary value problems, and the conditional invariance principle. Siberian Math. J., 36:3 (1995), 417–434.Google Scholar
[26] Borovkov, A.A. On the limit conditional distributions connected with large deviations. Siberian Math. J., 37:4 (1996), 635–646.Google Scholar
[27] Borovkov, A.A. Limit theorems for time and place of the first boundary passage. Doklady Mathematics 55:2 (1997), 254–256.Google Scholar
[28] Borovkov, A.A. An asymptotic exit problem for multidimensional Markov chains. Markov Proc. Rel. Fields, 3:4 (1997), 547–564.Google Scholar
[29] Borovkov, A.A. Ergodicity and Stability of Stochastic Processes (Wiley, 1998).Google Scholar
[30] Borovkov, A.A. Mathematical Statistics (Gordon & Breach, 1998).Google Scholar
[31] Borovkov, A.A. Estimates for the distribution of sums and maxima of sums of random variables without the Cramér condition. Siberian Math. J., 41:5 (2000), 811–848.Google Scholar
[32] Borovkov, A.A. Large deviation probabilities for random walks with semiexponential distributions. Siberian Math. J., 41:6 (2000), 1061–1093.Google Scholar
[33] Borovkov, A.A. On subexponential distributions and asymptotics of the distribution of the maximum of sequential sums. Siberian Math. J., 43:6 (2002), 995–1022.Google Scholar
[34] Borovkov, A.A. On the asymptotic behavior of the distributions of first-passage times, I. Math. Notes, 75:1 (2004), 23–37.Google Scholar
[35] Borovkov, A.A. On the asymptotic behavior of distributions of first-passage times, II. Math. Notes, 75:3 (2004), 322–330.Google Scholar
[36] Borovkov, A.A. Large sample change-point estimation when distributions are unknown. Theory Probab. Appl., 53:3 (2009), 402–418.CrossRefGoogle Scholar
[37] Borovkov, A.A. Integro-local and local theorems on normal and large deviations of the sums of nonidentically distributed random variables in the triangular array scheme. Theory Probab. Appl., 54:4 (2010), 571–587.CrossRefGoogle Scholar
[38] Borovkov, A.A. Large deviation principles for random walks with regularly varying distributions of jumps. Siberian Math. J., 52:3 (2011), 402–410.Google Scholar
[39] Borovkov, A.A. Probability Theory (Springer, 2013).Google Scholar
[40] Borovkov, A.A., Borovkov, K.A. On probabilities of large deviations for random walks. I. Regularly varying distribution tails. Theory Probab. Appl., 46:2 (2002), 193–213.Google Scholar
[41] Borovkov, A.A., Borovkov, K.A. On probabilities of large deviations for random walks. II. Regular exponentially decaying distributions. Theory Probab. Appl., 49:2 (2005), 189–206.Google Scholar
[42] Borovkov, A.A., Borovkov, K.A. Asymptotic Analysis of Random Walks. Heavy Tailed Distributions (Cambridge University Press, 2008).Google Scholar
[43] Borovkov, A.A., Korshunov, D.A. Large-deviation probabilities for one-dimensional Markov chains. Part 2: Prestationary distributions in the exponential case. Theory Probab. Appl., 45:3 (2001), 379–405.Google Scholar
[44] Borovkov, A.A., Linke, Yu.Yu. Asymptotically optimal estimates in the smooth change-point problem. Math. Methods Stat., 13:1 (2004), 1–24.Google Scholar
[45] Borovkov, A.A., Linke, Yu.Yu. Change-point problem for large samples and incomplete information on distribution. Math. Methods Stat., 14:4 (2006), 404–430.Google Scholar
[46] Borovkov, A.A., Mogul’skii, A.A. Probabilities of large deviations in topological spaces, I. Siberian Math. J., 19:5 (1978), 697–709.Google Scholar
[47] Borovkov, A.A., Mogul’skii, A.A. Probabilities of large deviations in topological spaces, II. Siberian Math. J., 21:5 (1980), 653–664.CrossRefGoogle Scholar
[48] Borovkov, A.A., Mogul’skii, A.A. Large Deviations and Testing Statistical Hypotheses (Nauka, 1992; in Russian).Google Scholar
[49] Borovkov, A.A., Mogul’skii, A.A. Large deviations and testing statistical hypothesis. Siberian Adv. Math., 2 (1993), 3 (1993).Google Scholar
[50] Borovkov, A.A., Mogul’skii, A.A. The second rate function and the asymptotic problems of renewal and hitting the boundary for multidimensional random walks. Siberian Math. J., 37:4 (1996), 647–682.Google Scholar
[51] Borovkov, A.A., Mogulskii, A.A. Itegro-local limit theorems including large deviations for sums of random vectors, I. Theory Probab. Appl., 43:1 (1999), 1–12.Google Scholar
[52] Borovkov, A.A., Mogulskii, A.A. Integro-local limit theorems including large deviations for sums of random vectors, II. Theory Probab. Appl., 45:1 (2001), 3–22.Google Scholar
[53] Borovkov, A.A., Mogul’skii, A.A. Limit theorems in the boundary hitting problem for a multidimensional random walk. Siberian Math. J., 42:2 (2001), 245–270.Google Scholar
[54] Borovkov, A.A., Mogul’skii, A.A. Integro-local theorems for sums of independent random vectors in the series scheme. Math. Notes, 79:4 (2006), 468–482.Google Scholar
[55] Borovkov, A.A., Mogul’skii, A.A. Integro-local and integral theorems for sums of random variables with semiexponential distributions. Siberian Math. J., 47:6 (2006), 990–1026.Google Scholar
[56] Borovkov, A.A., Mogul’skii, A.A. On large and superlarge deviations for sums of independent random vectors under Cramér’s condition, I. Theory Probab. Appl., 51:2 (2007), 227–255.Google Scholar
[57] Borovkov, A.A., Mogul’skii, A.A. On large and superlarge deviations of sums of independent random vectors under Cramér’s condition, II. Theory Probab. Appl., 51:4 (2007), 567–594.Google Scholar
[58] Borovkov, A.A., Mogul’skii, A.A. On large deviations of sums of independent random vectors on the boundary and outside of the Cramér zone, I. Theory Probab. Appl., 53:2 (2009), 301–311.Google Scholar
[59] Borovkov, A.A., Mogul’skii, A.A. On large deviations of sums of independent random vectors on the boundary and outside of the Cramér zone, II. Theory Probab. Appl., 53:4 (2009), 573–593.Google Scholar
[60] Borovkov, A.A., Mogul’skii, A.A. On large deviation principles in metric spaces. Siberian Math. J., 51:6 (2010), 989–1003.Google Scholar
[61] Borovkov, A.A., Mogul’skii, A.A. Properties of a functional of trajectories which arises in studying the probabilities of large deviations of random walks. Siberian Math. J., 52:4 (2011), 612–627.Google Scholar
[62] Borovkov, A.A., Mogul’skii, A.A. Chebyshev type exponential inequalities for sums of random vectors and random walk trajectories. Theory Probab. Appl., 56:1 (2012), 21–43.Google Scholar
[63] Borovkov, A.A., Mogulskii, A.A. On large deviation principles for random walk trajectories, I. Theory Probab. Appl., 56:4 (2012), 538–561.Google Scholar
[64] Borovkov, A.A., Mogul’skii, A.A. On large deviation principles for random walk trajectories, II. Theory Probab. Appl., 57:1 (2013), 1–27.Google Scholar
[65] Borovkov, A.A., Mogul’skii, A.A. Inequalities and principles of large deviations for the trajectories of processes with independent increments. Siberian Math. J., 54:2 (2013), 217–226.Google Scholar
[66] Borovkov, A.A., Mogulskii, A.A. Large deviation principles for random walk trajectories, III. Theory Probab. Appl., 58:1 (2014), 25–37.Google Scholar
[67] Borovkov, A.A., Mogul’skii, A.A. Moderately large deviation principles for trajectories of random walks and processes with independent increments. Theory Probab. Appl., 58:4 (2014), 562–581.Google Scholar
[68] Borovkov, A.A., Mogul’skii, A.A. Conditional moderately large deviation principles for the trajectories of random walks and processes with independent increments. Siberian Adv. Math., 25:1 (2015), 39–55.Google Scholar
[69] Borovkov, A.A., Foss, S.G. Estimates for overshooting an arbitrary boundary by a random walk and their applications. Theory Probab. Appl., 44:2 (2000), 231–253.Google Scholar
[70] Borovkov, A.A., Rogozin, B.A. Boundary value problems for some two-dimensional random walks. Theory Probab. Appl., 9:3 (1964), 361–388.Google Scholar
[71] Borovkov, A.A., Sycheva, N.M. On asymptotically optimal non-parametric criteria. Theory Probab. Appl., 13:3 (1968), 359–393.Google Scholar
[72] Borovkov, A.A., Mogul’skii, A.A., Sakhanenko, A.A. Limit Theorems for Random Processes (VINITI, Moscow, 1995; in Russian).Google Scholar
[73] Borovkov, K.A. Stability theorems and estimates of the rate of convergence of the components of factorizations for walks defined on Markov chains. Theory Probab. Appl., 25:2 (1980), 325–334.Google Scholar
[74] Boukai, B., Zhou, H. Nonparametric estimation in a two change-point model. J. Nonparametric Stat., 3 (1997), 275–292.Google Scholar
[75] Brodskij, B., Darkhovskij, B. Nonparametric methods in change-point problems (Kluwer, 1993).Google Scholar
[76] Carlstein, E. Nonparametric change-point estimation. Ann. Stat., 16:1 (1988), 188– 197.Google Scholar
[77] Carlstein, E., Muller, H.-G., Sigmund, D., eds. Change-Point Problems. Institute of Math. Statist. Lecture Notes, vol. 23 (1994).Google Scholar
[78] Chover, J., Ney, P., Wainger, S. Function of probability measures. J. Anal. Math., 26 (1973), 255–302.Google Scholar
[79] Cohen, J.W. Analysis of Random Walks (IOS Press, 1992).Google Scholar
[80] Cramér, H. Sur un nouveau théorème-limite de la théorie des probabilités. In: Actualités scientifiques et industrielles (Hermann, 1938; in French).Google Scholar
[81] Darkhovshkii, B.S. A non-parametric method for the a posteriori detection of the disorder time of a sequence of independent random variables. Theory Probab Appl., 21 (1976), 178–183.Google Scholar
[82] Darkhovskii, B.S., Brodskii, B.E. A nonparametric method for fastest detection of a change in the mean of a random sequence. Theory Probab. Appl., 32:4 (1987), 640–648.Google Scholar
[83] Dembo, A., Zeitouni, O. Large Deviations Techniques and Applications, 2nd edn. (Springer, 1998).Google Scholar
[84] Dinwoodie, I.H. A note on the upper bound for i.i.d. large deviations. Ann. Probab., 18:4 (1999), 1732–1736.Google Scholar
[85] Dobrushin, R.L., Pecherskii, E.A. Large deviations for random processes with independent increments on an infinite interval. Probl. Inform. Transmiss., 34:4 (1998), 354–382.Google Scholar
[86] Doney, R.A. An analog of the renewal theorem in higher dimensions. Proc. London Math. Soc., 16 (1966), 669–684.Google Scholar
[87] Doney, R.A. On the asymptotic behaviour of first passage times for transient random walk. Probab. Theory Rel. Fields 18 (1989), 239–246.Google Scholar
[88] Dumbgen, L. The asymptotic behavior of some nonparametric change-point estimation. Ann. Stat., 19:3 (1991), 1471–1495.Google Scholar
[89] Ellis, R.S. Entropy, Large Deviations, and Statistical Mechanics (Springer, 2006).Google Scholar
[90] Emery, D.J. Limiting behaviour of the distribution of the maxima of partial sums of certain random walks. J. Appl. Probab., 9 (1972), 572–579.Google Scholar
[91] Essen, M. Banach algebra methods in renewal theory. J. Anal. Math., 26 (1973) 303–335.Google Scholar
[92] Feller, W. An Introduction to Probability Theory and its Applications II, 2nd edn (Wiley, 1968).Google Scholar
[93] Feng, J., Kurtz, T.G. Large Deviations for Stochastic Processes (Amer. Math. Soc., Providence, RI, 2006).Google Scholar
[94] Gikhman, I.I., Skorokhod, A.V. Introduction to the Theory of Random Processes (Dover, 1996).Google Scholar
[95] Gnedenko, B.V. On a local limit theorem of the theory of probability. Uspekhi Mat. Nauk, 3:3 (1948), 187–194. (In Russian.)Google Scholar
[96] Gnedenko, B.V., Kolmogorov, A.N. Limit Distributions for Sums of Independent Random Variables (Addison-Wesley, 1954).Google Scholar
[97] Gut, A. Stopped Random Walks: Limit Theorems and Applications (Springer, 1988).Google Scholar
[98] Hawkins, D.L. A simple least squares method for estimating a change. Commun. Statist. Simula., 15:3 (1986), 655–679.Google Scholar
[99] Heyde, C.C. Asymptotic renewal results for a natural generalization of classical renewal theory. J. Roy. Statist. Soc. Ser. B., 29 (1967), 141–150.Google Scholar
[100] Huang, W.-T., Chang, Y.-P. Nonparametric estimation in change-point models. J. Statis. Plan. Inference, 35 (1993), 335–347.Google Scholar
[101] Hunter, J.J. Renewal theory in two dimensions: asymptotic results. Adv. Appl. Probab., 6 (1974), 546–562.Google Scholar
[102] Ibragimov, I.A., Linnik, Yu.V. Independent and Stationary Sequences of Random Variables (Wolters-Noordhoff, 1971).Google Scholar
[103] Ikeda, N., Watanabe, S. Stochastic Differential Equations and Diffusion Processes (North-Holland, 1981).Google Scholar
[104] Janson, S. Moments for first-passage and last-exit times. The minimum, and related quantities for random walks with positive drift. Adv. Appl. Probab., 18 (1986), 865–879.Google Scholar
[105] Kantorovich, L.V., Akilov, G.P. Functional Analysis (Pergamon Press, 1982).Google Scholar
[106] Keilson, J., Wishart, D.M.G. A central limit theorem for processes defined on a finite Markov chain. Proc. Camb. Phil. Soc., 60 (1964), 547.Google Scholar
[107] Kesten, H., Maller, R.A. Two renewal theorems for general random walks tending to infinity. Probab. Theory Rel. Fields, 106:1 (1996), 1–38.Google Scholar
[108] Khodjibaev, V.R. Asymptotic analysis of distributions in two-boundaries problems for random walks with continuous time. Limit theorems for sums of random variables. Trudy Instituta Matematiki SO AN SSSR, 3 (1984), 77–93. (In Russian.)Google Scholar
[109] Kligene, N., Tel’ksnis, L. Methods of detecting instants of change of random process properties. Automation and Remote Control, 44:10 (1983), 1241–1283.Google Scholar
[110] Kolmogorov, A.N., Fomin, S.V. Elements of the Theory of Functions and Functional Analysis (Dover, 1968).Google Scholar
[111] Korolyuk, V.S., Borovskikh, Yu.V. Analytical Problems for Asymptotics of Probability Distributions (Naukova Dumka, 1981; in Russian).Google Scholar
[112] Krein, S.G., ed. Functional Analysis (Nauka, 1964; in Russian).Google Scholar
[113] Kushner, H.J., Clark, D.S. Stochastic Approximation Methods for Constrained and Unconstrained Systems. Applied Mathematical Sciences, vol. 26 (Springer, 1978).Google Scholar
[114] Kushner, H.J., Dupuis, P.,G. Numerical methods for stochastic control problems in continuous time (Springer, 1992).Google Scholar
[115] Liggett, T.M. An invariance principle for conditioned sums of independent random variables. J. Math. Mech., 18:6 (1968), 559–570.Google Scholar
[116] Liptser, R.Sh., Shiryaev, A.N. Statistics of Random Processes (Springer, 2001).Google Scholar
[117] Lorden, G. Procedures for reacting to a change in distribution. Ann. Math. Statist., 42:6 (1971), 1897–1908.Google Scholar
[118] Lotov, V.I. Asymptotic analysis of distributions in problems with two boundaries, I. Theory Probab. Appl., 24:3 (1979), 480–491.Google Scholar
[119] Lotov, V.I. Asymptotic analysis of distributions in problems with two boundaries, II. Theory Probab. Appl., 24:4 (1979), 869–876.Google Scholar
[120] Lotov, V.I. On the asymptotics of distributions in two-boundaries problems for random walks defined on a Markov chain. Asymptotic analysis of distributions of random processes. Trudy Instituta Matematiki SO AN SSSR, 13 (1989), 116–136. (In Russian.)Google Scholar
[121] Lotov, V.I., Khodzhibaev, V.R. Asymptotic expansions in a boundary problem. Siberian Math. J., 25:5 (1984), 758–764.Google Scholar
[122] Lotov, V.I., Khodzhibaev, V.R. On limit theorems for the first exit time from a strip for stochastic processes, I. Siberian Adv. Math., 8:3 (1998), 90–113.Google Scholar
[123] Lotov, V.I., Khodzhibaev, V.R. On limit theorems for the first exit time from a strip for stochastic processes, II. Siberian Adv. Math., 8:4 (1998), 41–59.Google Scholar
[124] Lynch, J., Sethuraman, J. Large deviations for processes with independent increments. Ann. Prob., 15:2 (1987), 610–627.Google Scholar
[125] Malyshev, V.A. Random Walks. The Wiener–Hopf Equation in a Quadrant of the Plane. Galois Automorphisms (Moscow University Publishing House, 1972; in Russian).Google Scholar
[126] Markushevich, A.I. The Theory of Analytical Functions: A Brief Course (Mir, 1983).Google Scholar
[127] Mikosh, T., Nagaev, A.V. Large deviations of heavy-tailed sums with applications in insurance. Extremes, 1:1 (1998), 81–110.Google Scholar
[128] Miller, H.D. A matrix factorization problem in the theory of random variables defined on a finite Markov chain. Proc. Camb. Phil. Soc., 58:2 (1962), 265–285.Google Scholar
[129] Mogul’skii, A.A. Absolute estimates for moments of certain boundary functionals. Theory Probab. Appl., 18:2 (1973), 340–347.Google Scholar
[130] Mogul’skii, A.A. Large deviations in the space C(0, 1) for sums given on a finite Markov chain. Siberian Math. J., 15:1 (1974), 43–53.Google Scholar
[131] Mogul’skii, A.A. Large deviations for trajectories of multidimensional random walks. Theory Probab. Appl., 21:2 (1977), 300–315.Google Scholar
[132] Mogul’skii, A.A. Large deviation probabilities of random walks. Trudy Instituta Matematiki SO AN SSSR, 3 (1983), 93–124. (In Russian.)Google Scholar
[133] Mogul’skii, A.A. An integro-local theorem applicable on the whole half-axis to the sums of random variables with regularly varying distributions. Siberian Math. J., 49:4 (2008), 669–683.Google Scholar
[134] Mogul’skii, A.A. Local limit theorem for the first crossing time of a fixed level by a random walk. Siberian Adv. Math., 20:3 (2010), 191–200.Google Scholar
[135] Mogul’skii, A.A. The expansion theorem for the deviation integral. Siberian Adv. Math., 23:4 (2010), 250–262.Google Scholar
[136] Mogul’skii, A.A. On the upper bound in the large deviation principle for sums of random vectors. Siberian Adv. Math., 24:2 (2014), 140–152.Google Scholar
[137] Mogul’skii, A.A., Pagma, Ch. Superlarge deviations for sums of random variables with arithmetical super-exponential distributions. Siberian Adv. Math., 18:3 (2008), 185–208.Google Scholar
[138] Mogul’skii, A.A., Rogozin, B.A. Random walks in the positive quadrant, I–III. Siberian Adv. Math., 10:1 (2000), 34–72; 10:2 (2000), 35–103; 11:2 (2001), 35–59.Google Scholar
[139] Moustakides, G.V. Optimal stopping times for detecting changes in distributions. Ann. Statist., 14:4 (1986), 1379–1387.Google Scholar
[140] Nagaev, A.V. Local theorems with an allowance of large deviations. In Limit Theorems and Random Processes, 71–88 (Fan, 1967; in Russian).Google Scholar
[141] Nagaev, A.V. Limit theorems for a scheme of series. In Limit Theorems and Random Processes, pp. 43–70 (Fan Publishers, 1967; in Russian).Google Scholar
[142] Nagaev, A.V. Integral limit theorems taking into account large deviations when Cramér’s condition does not hold, I–II. Theory Probab. Appl., 14:1 (1969), 51–64; 14:2 (1969), 193–208.Google Scholar
[143] Nagaev, A.V. Renewal theorems in Rd. Theory Probab. Appl., 24:3 (1980), 572–581.Google Scholar
[144] Nagaev, S.V. Some limit theorems for large deviations. Theory Probab. Appl., 10:2 (1965), 214–235.Google Scholar
[145] Nagaev, S.V. On the asymptotic behaviour of one-sided large deviation probabilities. Theory Probab. Appl., 26:2 (1982), 362–366.Google Scholar
[146] Page, E.S. Continuous inspection schemes. Biometrika, 41:1 (1954), 100–115.Google Scholar
[147] Paulauskas, V.I. Estimates of the remainder term in limit theorems in the case of stable limit law. Lithuanian Math. J., 14:1 (1974), 127–146.Google Scholar
[148] Paulauskas, V.I. Uniform and nonuniform estimates of the remainder term in a limit theorem with a stable limit law. Lithuanian Math. J., 14:4 (1974), 661–672.Google Scholar
[149] Petrov, V.V. A generalization of Cramér’s limit theorem. Russian Math. Surveys, 9:4 (1954), 195–202. (In Russian.)Google Scholar
[150] Petrov, V.V. Limit theorems for large deviations violating Cramér’s condition. Vestnik Leningrad. Univ. Ser. Mat. Meh. Astronom., 19 (1963), 49–68. (In Russian.)Google Scholar
[151] Petrov, V.V. Sums of Independent Random Variables (Springer, 1975).Google Scholar
[152] Pollak, M. Optimal detection of a change in distribution. Ann. Statist, 13:1 (1985), 206–227.Google Scholar
[153] Pollak, M. Average run length of optimal method of detecting a change in distribution. Ann. Statist., 15:2 (1987), 749–779.Google Scholar
[154] Presman, E.L. Factorization methods and boundary problems for sums of random variables given on Markov chains. Math. USSR Izvestiya, 3:4 (1969), 815–852.Google Scholar
[155] Pukhalskii, A.A. On the theory of large deviations. Theory Probab. Appl., 38:3 (1993), 490–497.Google Scholar
[156] Puhalskii, A.A. Large Deviations and Idempotent Probability (Chapman & Hall/CRC, 2001).Google Scholar
[157] Riesz, F., Szökefalvi-Nagy, B. Functional Analysis (Blackie & Son, London, 1956).Google Scholar
[158] Roberts, S.W. A comparison of some control chart procedures. Technometrics, 8 (1966), 411–430.Google Scholar
[159] Rockefeller, R.T. Convex Analysis (Princeton University Press, 1970).Google Scholar
[160] Rogozin, B.A. On distributions of functionals related to boundary problems for processes with independent increments. Theory Probab. Appl., 11:4 (1966), 580– 591.CrossRefGoogle Scholar
[161] Rogozin, B.A. Distribution of the maximum of a process with independent increments. Siberian Math. J., 10:6 (1969), 989–1010.Google Scholar
[162] Rozovskii, L.V. Probabilities of large deviations on the whole axis. Theory Probab. Appl., 38:1 (1993), 53–79.Google Scholar
[163] Rvacheva, E.L. On domains of attraction of multi-dimensional distributions. Select. Transl. Math. Statist. Probab., 2 (1962), 183–205.Google Scholar
[164] Sanov, I.N. On the probability of large deviations of random magnitudes. Sbornik: Mathematics, 42:1 (1957), 11–44. (In Russian.)Google Scholar
[165] Saulis, L., Statulevicius, V.A. Limit Theorems for Large Deviations (Kluwer, Dordrecht, 1991).Google Scholar
[166] Shepp, L.A. A local limit theorem. Ann. Math. Stat., 35 (1964), 419–423.Google Scholar
[167] Shiryaev, A.N. Minimax optimality of the method of cumulative sums (CUSUM) in the case of continuous time. Russian Math. Surveys, 51:4 (1996), 750–751.Google Scholar
[168] Shiryaev, A.N. Optimal Stopping Rules (Springer, 2008). (Translated from the 1976 Russian original.)Google Scholar
[169] Skorokhod, A.V. Limit theorems for stochastic processes. Theory Probab. Appl., 1:3 (1956), 261–290.Google Scholar
[170] Slaby, M. On the upper bound for large deviations of sums of i.i.d. random vectors. Ann. Probab., 16:3 (1988), 978–990.Google Scholar
[171] Smith, W.L. Renewal theory and its ramifications. J. Roy. Statist. Soc. B., 20:2 (1961), 95–150.Google Scholar
[172] Stam, A.J. Renewal theory in r-dimensions, I, II. Composito Math. 21 (1969), 383–399; 23, 1–13.Google Scholar
[173] Stone, C. On characteristic function and renewal theory. Trans. Amer. Math. Soc., 20:2 (1965), 327–342.Google Scholar
[174] Stone, C. A local limit theorems for nonlattice multidimensional distribution functions. Ann. Math. Statistics, 36 (1965), 546–551.Google Scholar
[175] Stone, C. On local and ratio limit theorems. In: Proc. Fifth Berkeley Symp. Math. Stat. Probab., vol. II, part II, 217–224 (University of California, 1966).Google Scholar
[176] Stroock, D.W., Varadhan, S.R.S. Multidimensional Diffusion Processes (Springer, 1979).Google Scholar
[177] Varadhan, S.R.S. Asymptotic probabilities and differential equations. Comm. Pure Appl. Math. 19:3 (1966), 261–286.Google Scholar
[178] Varadhan, S.R.S. Large Deviations and Applications (SIAM, 1984).Google Scholar
[179] Varadhan, S.R.S. Large deviations. Ann. Probab. 36:2 (2008), 397–419.Google Scholar
[180] Wald, A. Sequential Analysis (Wiley, 1947).Google Scholar
[181] Zaigraev, O. Large-Deviation Theorems for Sums of Independent and Identically Distributed Random Vectors (Universytet Mikolaja Kopernika, 2005).Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×