Abstract
Let X, {X n}n≥1 be independent, identically distributed random variables with distribution function F and let {S n}n≥1 be the sequence of their partial sums. Let w and φ be two positive functions. Put w k = w(k) and φ k = φ(k). We study the convergence and the asymptotic behavior with respect to small parameters ε of the series
Main result of this chapter is Theorem 11.1, together with some corollaries, which exhibit possible asymptotics for simple choices of the functions w(⋅) and φ(⋅) in (11.1).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
S. Aljančić, R. Bojanić, and M. Tomić, Sur la valeur asymptotique d’une classe des intégrales définies, Acad. Serbe Sci. Publ. Inst. Math. 7 (1954), 81–94.
L.E. Baum and M. Katz, Convergence rates in the law of large numbers, Trans. Amer. Math. Soc. 120 (1965), no. 1, 108–123.
V.V. Buldygin, Strong laws of large numbers and convergence to zero of Gaussian sequences, Teor. Imovirnost. Matem. Statist. 19 (1978), 33–41; English transl. in Theory Probab. Math. Statist. 19 (1978), 33–41.
V.V. Buldygin, O.I. Klesov, and J.G. Steinebach, Precise asymptotics over a small parameter for a series of large deviation probabilities, Theory Stoch. Process. 13 (29)(2007), no. 1–2, 44–56.
R. Chen, A remark on the tail probability of a distribution, J. Multivariate Anal. 8(1978), no. 2, 328–333.
Y.S. Chow and T.L. Lai, Some one-sided theorems on the tail distribution of samples with applications to the last time and largest excess of boundary crossings, Trans. Amer. Math. Soc. 208(1975), 51–72.
Y.S. Chow and T.L. Lai, Paley-type inequalities and convergence rates related to the law of large numbers and extended renewal theory, Z. Wahrscheinlichkeitstheorie verw. Gebiete 45(1978), no. 1, 1–19.
P. Erdös, On a theorem of Hsu and Robbins, Ann. Math. Statist. 20 (1949), no. 2, 286–291.
P. Erdös, Remark on my paper “On a theorem of Hsu and Robbins”, Ann. Math. Statist. 21 (1950), no. 1, 138.
E. Fisher, M. Berman, N. Vowels, and C. Wilson, Excursions of a normal random walk above a boundary, Statist. Probab. Lett. 48 (2000), 141–151.
M.U. Gafurov and A.D. Slastnikov, Some problems of the exit of a random walk beyond a curvilinear boundary and large deviations, Teor. Veroyatnost. i Primenen. 32 (1987), no. 2, 327–348; English transl. in Theory Probab. Appl. 32 (1987), no. 2, 299–321.
G.N. Griffiths and F.T. Wright, Moments of the number of deviations of sums of independent identically distributed random variables, Sankhyā, Series A 37 (1975), no. 3, 452–455.
A. Gut and A. Spătaru, Precise asymptotics in the Baum-Katz and Davis laws of large Numbers, J. Math. Anal. Appl. 248 (2000), no. 1, 233–246.
A. Gut and J. Steinebach, Convergence rates and precise asymptotics for renewal counting processes and some first passage times, Fields Inst. Comm. 44 (2004), 205–227.
A. Gut and J. Steinebach, Convergence rates in precise asymptotics, J. Math. Anal. Appl. 390 (2012), no. 1, 1–14.
C.C. Heyde, On large deviation probabilities in the case of attraction to a non-normal stable Law, Sankhyā A30 (1968), no. 3, 253–258.
C.C. Heyde, A supplement to the strong law of large numbers, J. Appl. Probab. 12 (1975), no. 1, 173–175.
C.C. Heyde and V.K. Rohatgi, A pair of complementary theorems on convergence rates in the law of large numbers, Proc. Camb. Phil. Soc. 63 (1967), no. 1, 73–82.
P.L. Hsu and H. Robbins, Complete convergence and the law of large numbers, Proc. Nat. Acad. Sci U.S.A. 33 (1947), no. 2, 25–31.
I.A. Ibragimov and Yu.V. Linnik, Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff, Groningen, 1971.
K.-H. Indlekofer and O.I. Klesov, The complete convergence in the strong law of large numbers for double sums indexed by a sector with function boundaries, Teor. Imovir. Mat. Stat. 68 (2003), 44–48 (Ukrainian); English transl. in Theory Probab. Math. Statist. 68 (2004), 49–53.
K.-H. Indlekofer and O.I. Klesov, The asymptotic behavior over a small parameter of a series of large deviation probabilities weighted with the Dirichlet divisors function, Funct. Approx. Comment. Math. 35 (2006), no. 1, 117–131.
K.-H. Indlekofer and O.I. Klesov, Strong law of large numbers for multiple sums whose indices belong to a sector with function boundaries, Teor. Veroyatnost. i Primenen. 52 (2007), no. 4, 803–810 (Russian); English transl. in Theory Probab. Appl. 52 (2008), no. 4, 711–719.
Chung-siung Kao, On the time and the excess of linear boundary crossings of sample sums, Ann. Statist. 6 (1978), no. 1, 191–199.
J. Karamata, Sur un mode de croissance régulière. Theéorèmes fondamentaux, Bull. Soc. Math. France 61 (1933), 55–62.
M. Katz, The probability in the tail of a distribution, Ann. Math. Statist. 34 (1963), no. 1, 312–318.
F.C. Klebaner, Expected number of excursions above curved boundarie by a random walk, Bull. Austral. Math. Soc. 41 (1990), 207–213.
O. Klesov and U. Stadtmüler, Existence of moments of a counting process and convergence in multidimensional time, Adv. in Appl. Probab. 48 (2016), no. A, 181–201.
A.I. Martikainen and V.V. Petrov, On necessary and sufficient conditions for the law of the iterated logarithm, Teor. Veroyatnost. i Primenen. 22 (1977), no. 1, 18–26 (Russian); English transl. in Theory Probab. Appl. 22 (1977), no. 1, 16–23.
S. Parameswaran, Partition functions whose logarithms are slowly oscillating, Trans. Amer. Math. Soc. 100 (1961), no. 2, 217–240.
Yu.V. Prokhorov, On the strong law of large numbers, Izv. Akad. Nauk SSSR, Ser. Mat. 14 (1950), no. 6, 523–536. (Russian)
H. Robbins, D. Siegmund, and J. Wendell, The limiting distribution of the last time S n ≥ εn, Proc. Nat. Acad Sci. U.S.A. 61 (1968), 1228–1230.
L.V. Rozovskiı̆, On exact asymptotics in the weak law of large numbers for sums of independent random variables with a common distribution function from the domain of attraction of a stable law, Teor. Veroyatnost. i Primenen. 48 (2003), no. 3, 589–596 (Russian); English transl. in Theory Probab. Appl. 48 (2004), no. 3, 561–568.
L.V. Rozovskiı̆, On exact asymptotics in the weak law of large numbers for sums of independent random variables with a common distribution function from the domain of attraction of a stable law. II, Teor. Veroyatnost. i Primenen. 49 (2004), no. 4, 803–813 (Russian); English transl. in Theory Probab. Appl. 49 (2005), no. 4, 724–734.
H.-P. Scheffler, Precise asymptotics in Spitzer and Baum–Katz’s law of large numbers: the semistable case, J. Math. Anal. Appl. 288 (2003), no. 1, 285–298.
E. Seneta, Regularly Varying Functions, Springer-Verlag, Berlin–Heidelberg–New York, 1976.
J. Slivka and N.C. Severo, On the strong law of large numbers, Proc. Amer. Math. Soc. 24 (1970), 729–734.
A. Spătaru, Precise asymptotics in Spitzer’s law of large numbers, J. Theor. Probab. 12 (1999), no. 3, 811–819.
F. Spitzer, A combinatorial lemma and its application to probability theory, Trans. Amer. Math. Soc. 82 (1956), no. 2, 323–339.
H.H. Stratton, Moments of oscillations and ruled sums, Ann.Math. Statist. 43 (1972), no. 3, 1012–1016.
N.A. Volodin and S.V. Nagaev, A remark on the strong law of large numbers, Teor. Veroyatnost. i Primenen. 22 (1977), no. 4, 829–831 (Russian); English transl. in Theory Probab. Appl. 22 (1977), no. 4, 810–813.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Buldygin, V.V., Indlekofer, KH., Klesov, O.I., Steinebach, J.G. (2018). Spitzer Series and Regularly Varying Functions. In: Pseudo-Regularly Varying Functions and Generalized Renewal Processes. Probability Theory and Stochastic Modelling, vol 91. Springer, Cham. https://doi.org/10.1007/978-3-319-99537-3_11
Download citation
DOI: https://doi.org/10.1007/978-3-319-99537-3_11
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-99536-6
Online ISBN: 978-3-319-99537-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)