Some iterated logarithm results related to the central limit theorem.
Author:
R. J. Tomkins
Journal:
Trans. Amer. Math. Soc. 156 (1971), 185192
MSC:
Primary 60.30
MathSciNet review:
0275503
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: An iterated logarithm theorem is presented for sequences of independent, not necessarily bounded, random variables, the distribution of whose partial sums is related to the standard normal distribution in a particular manner. It is shown that if a sequence of independent random variables satisfies the Central Limit Theorem with a sufficiently rapid rate of convergence, then the law of the iterated logarithm holds. In particular, it is demonstrated that these results imply several known iterated logarithm results, including Kolmogorov's celebrated theorem.
 [1]
Andrew
C. Berry, The accuracy of the Gaussian
approximation to the sum of independent variates, Trans. Amer. Math. Soc. 49 (1941), 122–136. MR 0003498
(2,228i), http://dx.doi.org/10.1090/S00029947194100034983
 [2]
Kai
Lai Chung, A course in probability theory, Harcourt, Brace
& World, Inc., New York, 1968. MR 0229268
(37 #4842)
 [3]
W.
Feller, Generalization of a probability limit
theorem of Cramér, Trans. Amer. Math.
Soc. 54 (1943),
361–372. MR 0009262
(5,125b), http://dx.doi.org/10.1090/S00029947194300092625
 [4]
William
Feller, An introduction to probability theory and its applications.
Vol. I, John Wiley and Sons, Inc., New York; Chapman and Hall, Ltd.,
London, 1957. 2nd ed. MR 0088081
(19,466a)
 [5]
William
Feller, An introduction to probability theory and its applications.
Vol. II, John Wiley & Sons, Inc., New YorkLondonSydney, 1966. MR 0210154
(35 #1048)
 [6]
Philip
Hartman and Aurel
Wintner, On the law of the iterated logarithm, Amer. J. Math.
63 (1941), 169–176. MR 0003497
(2,228h)
 [7]
Philip
Hartman, Normal distributions and the law of the iterated
logarithm, Amer. J. Math. 63 (1941), 584–588.
MR
0004405 (3,2i)
 [8]
A.
Kolmogoroff, Über das Gesetz des iterierten Logarithmus,
Math. Ann. 101 (1929), no. 1, 126–135 (German).
MR
1512520, http://dx.doi.org/10.1007/BF01454828
 [9]
Michel
Loève, Probability theory, Third edition, D. Van
Nostrand Co., Inc., Princeton, N.J.Toronto, Ont.London, 1963. MR 0203748
(34 #3596)
 [10]
J. Marcinkiewicz and A. Zygmund, Remarque sur la loi du logarithme itéré, Fund. Math. 29 (1937), 215222.
 [11]
V.
V. Petrov, A bound for the deviation of the distribution of a sum
of independent random variables from the normal law, Dokl. Akad. Nauk
SSSR 160 (1965), 1013–1015 (Russian). MR 0178497
(31 #2754)
 [12]
V.
V. Petrov, On a relation between an estimate of the remainder in
the central limit theorem and the law of iterated logarithm, Teor.
Verojatnost. i Primenen 11 (1966), 514–518 (Russian,
with English summary). MR 0212855
(35 #3720)
 [13]
Valentin
V. Petrov, On the law of the iterated logarithm without assumptions
about the existence of moments, Proc. Nat. Acad. Sci. U.S.A.
59 (1968), 1068–1072. MR 0228052
(37 #3636)
 [14]
V.
Strassen, An invariance principle for the law of the iterated
logarithm, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete
3 (1964), 211–226 (1964). MR 0175194
(30 #5379)
 [15]
Mary
Weiss, On the law of the iterated logarithm, J. Math. Mech.
8 (1959), 121–132. MR 0102853
(21 #1639)
 [1]
 A. C. Berry, The accuracy of the Gaussian approximation to the sum of independent variates, Trans. Amer. Math. Soc. 49 (1941), 122136. MR 2, 228. MR 0003498 (2:228i)
 [2]
 K. L. Chung, A course in probability theory, Harcourt, Brace and World, New York, 1968. MR 0229268 (37:4842)
 [3]
 W. K. Feller, Generalization of a probability limit theorem of Cramér, Trans. Amer. Math. Soc. 54 (1943), 361372. MR 5, 125. MR 0009262 (5:125b)
 [4]
 , An introduction to probability theory and its applications. Vol. I, 2nd ed., Wiley, New York, 1962. MR 0088081 (19:466a)
 [5]
 W. K. Feller, An introduction to probability theory and its applications. Vol. II, Wiley, New York, 1966. MR 35 #1048. MR 0210154 (35:1048)
 [6]
 P. Hartman and A. Wintner, On the law of the iterated logarithm, Amer. J. Math. 63 (1941), 169176. MR 2, 228. MR 0003497 (2:228h)
 [7]
 P. Hartman, Normal distributions and the law of the iterated logarithm, Amer. J. Math. 63 (1941), 584588. MR 3, 2. MR 0004405 (3:2i)
 [8]
 A. Kolmogorov, Über das Gesetz des Iterierten Logarithmus, Math. Ann. 101 (1929), 126135. MR 1512520
 [9]
 M. Loève, Probability theory. Foundations. Random sequences, 3rd ed., Van Nostrand, Princeton, N. J., 1963. MR 34 #3596. MR 0203748 (34:3596)
 [10]
 J. Marcinkiewicz and A. Zygmund, Remarque sur la loi du logarithme itéré, Fund. Math. 29 (1937), 215222.
 [11]
 V. V. Petrov, A bound for the deviation of the distribution of a sum of independent random variables from the normal law, Dokl. Akad. Nauk SSSR 160 (1965), 10131015 = Soviet Math. Dokl. 6 (1965), 242244. MR 31 #2754. MR 0178497 (31:2754)
 [12]
 , On a relation between an estimate of the remainder in the central limit theorem and the law of iterated logarithm, Teor. Verojatnost. i Primenen. 11 (1966), 514518 = Theor. Probability Appl. 11 (1966), 454458. MR 35 #3720. MR 0212855 (35:3720)
 [13]
 , On the law of the iterated logarithm without assumptions about the existence of moments, Proc. Nat. Acad. Sci. U.S.A. 59 (1968), 10681072. MR 37 #3636. MR 0228052 (37:3636)
 [14]
 V. Strassen, An invariance principle for the law of the iterated logarithm, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 3 (1964), 211226. MR 30 #5379. MR 0175194 (30:5379)
 [15]
 M. Weiss, On the law of the iterated logarithm, J. Math. Mech. 8 (1959), 121132. MR 21 #1639. MR 0102853 (21:1639)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC:
60.30
Retrieve articles in all journals
with MSC:
60.30
Additional Information
DOI:
http://dx.doi.org/10.1090/S0002994719710275503X
PII:
S 00029947(1971)0275503X
Keywords:
Independent random variables,
law of the iterated logarithm,
Central Limit Theorem,
normal distribution,
BerryEsseen bounds,
Lindeberg's criterion,
Kolmogorov's theorem
Article copyright:
© Copyright 1971
American Mathematical Society
