Author:
Andrei N. Frolov
Andrei.Frolov@pobox.spbu.ru
Andrei N. Frolov
Dept. of Mathematics and Mechanics, St. Petersburg State University, St. Petersburg, Russia
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Fifty years ago P. Erdős and A. Rényi published their famous paper on the new law of large numbers. In this survey, we describe numerous results and achievements which are related with this paper or motivated by it during these years.
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[1]
J.-N. Bacro and M. Brito. On Mason’s extension of the Erdős–Rényi law of large numbers.Statist. Probab. Lett., 11:43–47, 1991.
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[2]
J.-N. Bacro, P. Deheuvels, and J. Steinebach. Exact convergence rates in Erdős–Rényi type theorems for renewal processes. Ann. Inst. Henri Poincaré, 23:195–207, 1987.
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[3]
S. A. Book. A version of the Erdős–Rényi law of large numbers for independent random variables. Bull. Inst. Math. Acad. Sinica, 3(2):199–211, 1975.
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[4]
S. A. Book. An extension of the Erdős–Rényi new law of large numbers. Proc. Amer. Math. Soc., 48(2):438–446, 1975.
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[5]
S. A. Book and T.R. Shore. On large intervals in the Csörgő–Révész theorem on increments of a Wiener process. Z. Wahrsch. Verw. Geb., 46:1–11, 1978.
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[6]
K. A. Borovkov. On functional Erdős–Rényi law of large numbers. Teor. veroyatn. i ee primen., 35(4):758–762, 1990. English transl.: Theor. Probab. Appl., 35, no. 4, 762–766, (1990).
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[7]
Z. Cai. Strong approximation and improved Erdős–Rényi laws for sums of independent non-identically distributed random variables. J. Hangzhou Univ., 19(3):240–246, 1992.
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[8]
A. H.C. Chan. Erdős–Rényi type modulus of continuity theorems for Brownian sheets. Studia Sci. Math. Hungar., 11: 59–68, 1976.
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[9]
Y. K. Choi and N. Kôno. How big are increments of a two-parameter Gaussian processes. J. Theoret. Probab., 12:105–129, 1999.
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[10]
E. Csáki and P. Révész. How big must be the increments of a Wiener process? Acta Math. Acad. Sci. Hungar., 33:37–49, 1979.
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[11]
M. Csörgő and P. Révész. How big are the increments of a multiparameter Wiener process? Z. Wahrsch. verw. Geb., 42:1–12, 1978.
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[12]
M. Csörgő and P. Révész. How big are the increments of a Wiener process? Ann. Probab., 7:731–737, 1979.
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[13]
M. Csörgő and P. Révész. Strong approximations in probability and statistics. Akadémiai Kiadó, Budapest, 1981.
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[14]
M. Csörgő and J. Steinebach. Improved Erdős–Rényi and strong approximation laws for increments of partial sums. Ann. Probab., 9:988–996, 1981.
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[15]
S. Csörgő. Erdős–Rényi laws. Ann. Statist., 7: 772–787, 1979.
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[16]
P. Deheuvels. On the Erdős–Rényi theorem for random fields and sequences and its rela-tionships with the theory of runs and spacings. Z. Wahrsch. verw. Geb., 70:91–115, 1985.
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[17]
P. Deheuvels. Functional Erdős–Rényi laws. Studia Sci. Math. Hungar, 26:261–295, 1991.
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[18]
P. Deheuvels. Functional law of the iterated logarithm for small increments of empirical processes. Statist. Neerlandica, 50:261–280, 1996.
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[19]
P. Deheuvels and L. Devroye. Limit laws of Erdős–Rényi–Shepp type. Ann. Probab., 15:1363–1386, 1987.
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[20]
P. Deheuvels, L. Devroye, and J. Lynch. Exact convergence rate in the limit theorems of Erdős–Rényi and Shepp. Ann. Probab., 14:209–223, 1986.
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[21]
P. Deheuvels and J. H.J. Einmahl. Functional limit laws for the increments of Kaplan–Meier product-limit processes and applications. Ann. Probab, 28:1301–1335, 1990.
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[22]
P. Deheuvels and D.M. Mason. Functional law of the iterated logarithm for the increments of empirical and quantile processes. Ann. Probab, 20:1248–1287, 1992.
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[23]
P. Deheuvels and J. Steinebach. Exact convergence rates in strong approximation laws for large increments of partial sums. Probab. Th. Rel. Fields, 76:369–393, 1987.
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[24]
P. Deheuvels and J. Steinebach. Sharp rates for increments of renewal processes. Ann. Probab., 17:700–722, 1989.
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[25]
U. Einmahl and D.M. Mason. Some universal results on the behaviour of increments of partial sums. Ann. Probab., 24: 1388–1407, 1996.
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[26]
P. Erdős and A. Rényi. On a new law of large numbers. J. Analyse Math., 23:103–111, 1970.
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[27]
A. N. Frolov. On one-sided strong laws for large increments of sums. Statist. Probab. Lett., 37:155–165, 1998.
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[28]
A. N. Frolov. On the asymptotic behaviour of increments of sums of independent random variables. Doklady Akademii nauk, 372(5):596–599, 2000. English transl.: Doklady Math., 61, no. 3, 409–412, (2000).
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[29]
A. N. Frolov. On asymptotic behaviour of large increments of sums of independent random variables. Theor. Probab. Appl., 47(2):366–374, 2002. English transl.: Theor. Probab. Appl., 47, no. 2, 315–323, (2003).
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[30]
A. N. Frolov. One-sided strong laws for increments of sums of i.i.d. random variables. Studia Sci. Math. Hungar., 39: 333–359, 2002.
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[31]
A. N. Frolov. Limit theorems for increments of processes with independent increments. Doklady Akademii nauk, 393(2):165–169, 2003. English transl.: Doklady Math., 68, no. 3, 345–349, (2003).
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[32]
A. N. Frolov. Limit theorems for increments of sums independent random variables. Theor. Probab. Appl., 48(1):104–121, 2003. English transl.: Theor. Probab. Appl., 48, no. 1, 93–107, (2004).
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[33]
A. N. Frolov. On asymptotic behaviour of increments of random fields. Zap. Nauchn. Semin. POMI, 298:191–207, 2003. English transl.: J. Math. Sci., 128, no. 1, 2604–2613, (2005).
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[34]
A. N. Frolov. Strong limit theorems for increments of renewal processes. Zap. Nauchn. Semin. POMI, 298:208–225, 2003. English transl.: J. Math. Sci., 128, no. 1, 2614–2624, (2005).
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[35]
A. N. Frolov. On the law of the iterated logarithm for increments of sums of independent random variables. Zap. Nauchn. Semin. POMI, 320:174–186, 2004. English transl.: J. Math. Sci., 137, no. 1, 4575–4582, (2006).
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[36]
A. N. Frolov. Strong limit theorems for increments of sums of independent random variables. Zap. Nauchn. Semin. POMI, 311:260–285, 2004. English transl.: J. Math. Sci., 133, no. 3, 1356–1370, (2006).
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[37]
A. N. Frolov. Universal strong laws for increments of processes with independent increments. Theor. Probab. Appl., 49(3):601–609, 2004. English transl.: Theor. Probab. Appl., 49, no. 3, 531–540, (2005).
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[38]
A. N. Frolov. Converses to the Csörgő–Révész laws. Statist. Probab. Lett., 72:113–123, 2005.
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[39]
A. N. Frolov. Limit theorems for increments of compound renewal processes. Zap. Nauchn. Semin. POMI, 351:259–283, 2007. English transl.: J. Math. Sci., 152, no. 6, 944–957, (2008).
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[40]
A. N. Frolov. Universal theory for strong limit theorems of probability. World Scientific, Singapore, 2019.
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[41]
A. N. Frolov, A. I. Martikainen, and J. Steinebach. Erdős–Rényi–Shepp type laws in non-i.i.d. case. Studia Sci. Math. Hungar., 33:127–151, 1997.
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[42]
N. Gantert. Functional Erdős–Rényi laws for semiexponential random variables. Ann. Probab, 26:1356–1369, 1998.
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[43]
D. L. Hanson and R.P. Russo. Some results on increments of Wiener process with applica-tions to lag sums of independent identically distributed random variables. Ann. Probab., 11:609–623, 1983.
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[44]
D. L. Hanson and R.P. Russo. Some limit results for lag sums of independent, non-i.i.d. random variables. Z. Wahrsch. verw. Geb., 68:425–445, 1985.
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[45]
V. R. Huse and J. Steinebach. On an improved Erdős–Rényi type law for increments of partial sums. Canad. J. Statist, 13:311–315, 1985.
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[46]
J. Komlós, P. Major, and G. Tusnády. An approximation of partial sums of independent rv’s, and the sample df. I. Z. Wahrsch. Verw. Geb., 32:111–131, 1975.
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[47]
J. Komlós, P. Major, and G. Tusnády. An approximation of partial sums of independent rv’s, and the sample df. II. Z. Wahrsch. Verw. Geb., 34:33–58, 1976.
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[48]
H. Lanzinger. A law of the single logarithm for moving averages of random variables under exponential moment conditions. Studia Sci. Math. Hungar., 36:65–91, 2000.
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[49]
H. Lanzinger and U. Stadtmüller. Maxima of increments of partial sums for certain subex-ponential distributions. Stoch. Processes Appl., 86:307–322, 2000.
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[50]
Z. Y. Lin. The Erdős–Rényi laws of large numbers for non-identically distributed random variables. Chin. Ann. Math., 11B:376–383, 1990.
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[51]
Z. Y. Lin, Y. K. Choi, and K. S. Hwang. Some limit theorems on the increments of a multi-parameter fractional Brownian motion. Stoch. Anal. Appl., 19:499–517, 2001.
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[52]
Z. Y. Lin, C. R. Lu, and Q.-M. Shao. Contribution to the limit theorems. Contemporary Math., 118:221–237, 1991.
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[53]
J. Lynch. Some comments on the Erdős–Rényi law and a theorem of Shepp. Ann. Probab., 11:801–802, 1983.
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[54]
D. M. Mason. An extended version of the Erdős–Rényi strong law of large numbers. Ann. Probab., 17:257–265, 1989.
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[55]
J. Ortega and M. Wschebor. On the increments of the Wiener process. Z. Wahrsch. Verw. Geb., 65:329–339, 1984.
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[56]
W. Pfuhl and J. Steinebach. On precise asymtotics for the Erdős–Rényi increments of random fields. Pub. Inst. Stat. Univ. Paris, 33:49–66, 1988.
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[57]
P. Révész. A generalization of Strassen’s functional law of iterated logarithm. Z. Wahrsch. verw. Geb, 50:257–264, 1979.
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[58]
P. Révész. On the increments of Wiener and related processes. Ann. Probab., 10:613–622, 1982.
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[59]
P. Révész. Random walk in random and non-random environment. World Scientific Publ., Singapore, 1990.
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[60]
O. E. Scherbakova. Rate of convergence of increments for random fields. Zap. Nauchn. Semin. POMI, 298:304–315, 2003. English transl.: J. Math. Sci., 128, no. 1, 2669–2676, (2005).
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[61]
O. E. Scherbakova. Convergence rate for large increments of random fields. Zap. Nauchn. Semin. POMI, 320:187–225, 2004. English transl.: J. Math. Sci., 137, no. 1, 4583–4608, (2006).
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[62]
Q.-M. Shao. On a problem of Csörgő and Révész. Ann. Probab., 17:809–812, 1989.
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[63]
Q.-M. Shao. On a conjecture of Révész. Proc. AMS, 123:575–582, 1995.
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[64]
L. A. Shepp. A limit law concerning moving averages. Ann. Math. Statist., 35:424–428, 1964.
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[65]
J. Steinebach. On a necessary condition for the Erdős–Rényi law of large numbers. Proc. Amer. Math. Soc., 68:97–100, 1978.
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[66]
J. Steinebach. On general versions of Erdős–Rényi laws. Z. Wahrsch. Verw. Geb., 56:549–554, 1981.
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[67]
J. Steinebach. Between invariance principles and Erdős–Rényi laws. Coll. Math. Soc. J. Bolyai, 36:981–1005, 1982.
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[68]
J. Steinebach. On the increments of partial sum processes with multidimensional indices. Z. Wahrsch. verw. Geb., 63:59–70, 1983.
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[69]
J. Steinebach. Improved Erdős–Rényi and strong approximation laws for increments of renewal processes. Ann. Probab., 14:547–559, 1986.
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[70]
J. Steinebach. Strong laws for small increments of renewal processes. Ann. Probab., 19:1768–1776, 1991.
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[71]
J. Steinebach. On a conjecture of Révész and its analogue for renewal processes. In: Szyszkowicz B. (Ed.), Asymptotic methods in probability and statistics. ICAMPS 197. Amsterdam, North Holland/Elsevier, pages 311–322, 1998.
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[72]
V. Strassen. An invariance principle for the law of the iterated logarithm. Z. Wahrsch. Verw. Geb., 3:211–226, 1964.
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[73]
N. M. Zinchenko. Asymptotic for increments of stable stochastic processes with jumps of one sign. Theor. Probab. Appl., 32(4):793–796, 1987. English transl.: Theor. Probab. Appl., 32, no. 4, 724–727, (1987).
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[74]
N. M. Zinchenko. On the asymptotic behaviour of increments of certain classes of random fields. Theor. Probab. Math. Statst., 48:7–11, 1994.
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