On the law of the iterated logarithm.
نویسنده
چکیده
The law of the iterated logarithm provides a family of bounds all of the same order such that with probability one only finitely many partial sums of a sequence of independent and identically distributed random variables exceed some members of the family, while for others infinitely many do so. In the former case, the total number of such excesses has therefore a proper probability distribution, and it is shown here that whenever the law applies this distribution possesses no moments of positive order. This result further elucidates the celebrated precision of this law of probability concerning fluctuations of sums of random variables.
منابع مشابه
Asymptotic Behaviors of the Lorenz Curve for Left Truncated and Dependent Data
The purpose of this paper is to provide some asymptotic results for nonparametric estimator of the Lorenz curve and Lorenz process for the case in which data are assumed to be strong mixing subject to random left truncation. First, we show that nonparametric estimator of the Lorenz curve is uniformly strongly consistent for the associated Lorenz curve. Also, a strong Gaussian approximation for ...
متن کاملSome Probability Inequalities for Quadratic Forms of Negatively Dependent Subgaussian Random Variables
In this paper, we obtain the upper exponential bounds for the tail probabilities of the quadratic forms for negatively dependent subgaussian random variables. In particular the law of iterated logarithm for quadratic forms of independent subgaussian random variables is generalized to the case of negatively dependent subgaussian random variables.
متن کاملLaw of iterated logarithm for NA sequences with non-identical distributions
Based on a law of the iterated logarithm for independent random variables sequences, an iterated logarithm theorem for NA sequences with non-identical distributions is obtained. The proof is based on a Kolmogrov-type exponential inequality.
متن کاملExponential Bounds in the Law of Iterated Logarithm for Martingales
In this paper non-asymptotic exponential estimates are derived for tail of maximum martingale distribution by naturally norm-ing in the spirit of the classical Law of Iterated Logarithm.
متن کاملA Supplement to Precise Asymptotics in the Law of the Iterated Logarithm for Self-normalized Sums
Let X, X1, X2, . . . be i.i.d. random variables with zero means, variance one, and set Sn = ∑n i=1 Xi, n ≥ 1. Gut and Spǎtaru [3] established the precise asymptotics in the law of the iterated logarithm and Li, Nguyen and Rosalsky [7] generalized their result under minimal conditions. If P(|Sn| ≥ ε √ 2n log log n) is replaced by E{|Sn|/√n− ε √ 2 log log n}+ in their results, the new one is call...
متن کاملLaw of Iterated Logarithm and Invariance Principle for One-parameter Families of Interval Maps
We show that for almost every map in a transversal one-parameter family of piecewise expanding unimodal maps the Birkhoff sum of suitable observables along the forward orbit of the turning point satisfies the law of iterated logarithm. This result will follow from an almost sure invariance principle for the Birkhoff sum, as a function on the parameter space. Furthermore, we obtain a similar res...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Proceedings of the National Academy of Sciences of the United States of America
دوره 63 2 شماره
صفحات -
تاریخ انتشار 1969