A Functional Law of the Iterated Logarithm for Weighted Empirical Distributions
نویسندگان
چکیده
منابع مشابه
On the law of the iterated logarithm.
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...
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This was first proved for Bernoulli random variables by Khintchine. Salem and Zygmund [SZ2] considered the case when the Xk are replaced by functions ak cosnkx on [−π, π] and gave an upper bound ( ≤ 1) result; this was extended to the full upper and lower bound by Erdös and Gál [EG]. Takahashi [T1] extends the result of Salem and Zygmund: Consider a real measurable function f satisfying f(x + 1...
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We find conditions which are sufficient and nearly necessary for the compact and bounded law of the iterated logarithm for Markov chains with a countable state space.
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Let X,X1, X2, . . . be a sequence of centered iid random variables. Let f(n) be a strongly additive arithmetic function such that ∑ p<n f2(p) p → ∞ and put An = ∑ p<n f(p) p . If EX2 < ∞ and f satisfies a Lindeberg-type condition, we prove the following law of the iterated logarithm: lim sup N→∞ ∑N n=1 f(n)Xn AN √ 2N log logN a.s. = ‖X‖2. We also prove the validity of the corresponding weighted...
متن کاملA law of the iterated logarithm for Grenander's estimator.
In this note we prove the following law of the iterated logarithm for the Grenander estimator of a monotone decreasing density: If f(t0) > 0, f'(t0) < 0, and f' is continuous in a neighborhood of t0, then [Formula: see text]almost surely where [Formula: see text]here [Formula: see text] is the two-sided Strassen limit set on [Formula: see text]. The proof relies on laws of the iterated logarith...
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ژورنال
عنوان ژورنال: The Annals of Probability
سال: 1975
ISSN: 0091-1798
DOI: 10.1214/aop/1176996263