Probabilistic and deterministic averaging

Bingham, N. H.; Goldie, Charles M.

doi:10.1090/S0002-9947-1982-0637702-1

Probabilistic and deterministic averaging
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Trans. Amer. Math. Soc. 269 (1982), 453-480 Request permission

Abstract:

Let $\{ {S_n}\}$ be a random walk whose step distribution has positive mean $\mu$ and an absolutely continuous component. For any bounded measurable function $f$, a Marcinkiewicz-Zygmund strong law in an $r$-quick version (a ’Lai strong law’) is proved for $f({S_n})$, assuming existence of a suitable higher moment of the step distribution. This is extended to show ${n^{ - \alpha }}\{ \sum \nolimits _1^n {f({S_k})} - \int _0^n {f(\mu t)dt\} \to 0}$ ($r$-quickly). These results remain true when the step distribution is lattice, provided $f$ is constant between lattice points. Certain intermediate results on renewal theory, mixing, local limit theory, ladder height, and a strong law of Lai for mixing random variables are of independent interest.

References

Leonard E. Baum and Melvin Katz, Convergence rates in the law of large numbers, Trans. Amer. Math. Soc. 120 (1965), 108–123. MR 198524, DOI 10.1090/S0002-9947-1965-0198524-1
J. R. Baxter and G. A. Brosamler, Energy and the law of the iterated logarithm, Math. Scand. 38 (1976), no. 1, 115–136. MR 426178, DOI 10.7146/math.scand.a-11622

The equilibrium random walk

Patrick Billingsley, Convergence of probability measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0233396
N. H. Bingham, Tauberian theorems and the central limit theorem, Ann. Probab. 9 (1981), no. 2, 221–231. MR 606985

Tauberian theorems for Borel and related summability methods

N. H. Bingham and Charles M. Goldie, Extensions of regular variation. I. Uniformity and quantifiers, Proc. London Math. Soc. (3) 44 (1982), no. 3, 473–496. MR 656246, DOI 10.1112/plms/s3-44.3.473
K. Chandrasekharan and S. Minakshisundaram, Typical means, Oxford University Press, 1952. MR 0055458
Y. S. Chow and T. L. Lai, Some one-sided theorems on the tail distribution of sample sums with applications to the last time and largest excess of boundary crossings, Trans. Amer. Math. Soc. 208 (1975), 51–72. MR 380973, DOI 10.1090/S0002-9947-1975-0380973-6
Yuan Shih Chow and Henry Teicher, Probability theory, Springer-Verlag, New York-Heidelberg, 1978. Independence, interchangeability, martingales. MR 513230
Kai Lai Chung, Markov chains with stationary transition probabilities, 2nd ed., Die Grundlehren der mathematischen Wissenschaften, Band 104, Springer-Verlag New York, Inc., New York, 1967. MR 0217872

Limit theorems for functionals of processes with independent increments

18

Iu. A. Davydov, Sur une classe des fonctionnelles des processus stables et des marches aléatoires, Ann. Inst. H. Poincaré Sect. B (N.S.) 10 (1974), 1–29 (French, with English summary). MR 0368109

On asymptotic behaviour of some functionals of processes with independent increments

16

J. L. Doob, Stochastic processes, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1953. MR 0058896
William Feller, An introduction to probability theory and its applications. Vol. I, 3rd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0228020
Allan Gut, On the moments and limit distributions of some first passage times, Ann. Probability 2 (1974), 277–308. MR 394857, DOI 10.1214/aop/1176996709
G. H. Hardy, Divergent Series, Oxford, at the Clarendon Press, 1949. MR 0030620
I. A. Ibragimov, Some limit theorems for stationary processes, Teor. Verojatnost. i Primenen. 7 (1962), 361–392 (Russian, with English summary). MR 0148125
I. A. Ibragimov and Yu. V. Linnik, Independent and stationary sequences of random variables, Wolters-Noordhoff Publishing, Groningen, 1971. With a supplementary chapter by I. A. Ibragimov and V. V. Petrov; Translation from the Russian edited by J. F. C. Kingman. MR 0322926

Distributions in statistics: continuous univariate distributions

G. Kallianpur and H. Robbins, The sequence of sums of independent random variables, Duke Math. J. 21 (1954), 285–307. MR 62976
Samuel Karlin, On the renewal equation, Pacific J. Math. 5 (1955), 229–257. MR 70877
J. F. C. Kingman, Regenerative phenomena, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Ltd., London-New York-Sydney, 1972. MR 0350861

On the asymptotic behaviour of the distribution of functionals of the type

of diffusion processes

8

Tze Leung Lai, On $r$-quick convergence and a conjecture of Strassen, Ann. Probability 4 (1976), no. 4, 612–627. MR 431326, DOI 10.1214/aop/1176996031
Tze Leung Lai, Convergence rates and $r$-quick versions of the strong law for stationary mixing sequences, Ann. Probability 5 (1977), no. 5, 693–706. MR 471043, DOI 10.1214/aop/1176995713
Tze Leung Lai and William Stout, Limit theorems for sums of dependent random variables, Z. Wahrsch. Verw. Gebiete 51 (1980), no. 1, 1–14. MR 566103, DOI 10.1007/BF00533812
D. L. McLeish, A maximal inequality and dependent strong laws, Ann. Probability 3 (1975), no. 5, 829–839. MR 400382, DOI 10.1214/aop/1176996269
Isaac Meilijson, The average of the values of a function at random points, Israel J. Math. 15 (1973), 193–203. MR 329010, DOI 10.1007/BF02764606
Harald Niederreiter, Quasi-Monte Carlo methods and pseudo-random numbers, Bull. Amer. Math. Soc. 84 (1978), no. 6, 957–1041. MR 508447, DOI 10.1090/S0002-9904-1978-14532-7
V. V. Petrov, Sums of independent random variables, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 82, Springer-Verlag, New York-Heidelberg, 1975. Translated from the Russian by A. A. Brown. MR 0388499

Almost sure invariance principles for partial sums of weakly dependent random variables

N. I. Portenko and L. N. Prokopenko, A certain method of proving limit theorems for functionals of random walks, Theory of random processes, No. 1 (Russian), Izdat. “Naukova Dumka”, Kiev, 1973, pp. 107–119, 140 (Russian). MR 0375476
N. U. Prabhu, Stochastic processes. Basic theory and its applications, The Macmillan Company, New York; Collier Macmillan Ltd., London, 1965. MR 0192522
Pál Révész, Some remarks on the random ergodic theorem. I, Magyar Tud. Akad. Mat. Kutató Int. Közl. 5 (1960), 375–381 (English, with Russian summary). MR 125593
M. Rosenblatt, A central limit theorem and a strong mixing condition, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 43–47. MR 74711, DOI 10.1073/pnas.42.1.43
R. J. Serfling, Moment inequalities for the maximum cumulative sum, Ann. Math. Statist. 41 (1970), 1227–1234. MR 268938, DOI 10.1214/aoms/1177696898
S. H. Siraždinov and M. Mamatov, On mean convergence for densities, Teor. Verojatnost. i Primenen. 7 (1962), 433–437 (Russian, with English summary). MR 0143237
A. V. Skorohod and N. P. Slobodenjuk, Predel′nye teoremy dlya sluchaĭ nykh bluzhdaniĭ, Izdat. “Naukova Dumka”, Kiev, 1970 (Russian). MR 0282419
Charles Stone, On characteristic functions and renewal theory, Trans. Amer. Math. Soc. 120 (1965), 327–342. MR 189151, DOI 10.1090/S0002-9947-1965-0189151-0
Charles Stone, On absolutely continuous components and renewal theory, Ann. Math. Statist. 37 (1966), 271–275. MR 196795, DOI 10.1214/aoms/1177699617
William F. Stout, Almost sure convergence, Probability and Mathematical Statistics, Vol. 24, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 0455094
Volker Strassen, Almost sure behavior of sums of independent random variables and martingales, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66) Univ. California Press, Berkeley, Calif., 1967, pp. 315–343. MR 0214118
Murad S. Taqqu, Law of the iterated logarithm for sums of non-linear functions of Gaussian variables that exhibit a long range dependence, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 40 (1977), no. 3, 203–238. MR 471045, DOI 10.1007/BF00736047
A. F. Taraskin, Certain limit theorems for stochastic integrals, Theory of random processes, No. 1, (Russian), Izdat. “Naukova Dumka”, Kiev, 1973, pp. 119–133, 140 (Russian). MR 0368140