A Law of the Iterated Logarithm for the Sample Covariance Matrix
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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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...
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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...
متن کاملThe law of the iterated logarithm for LNQD sequences
Let [Formula: see text] be a stationary LNQD sequence of random variables with zero means and finite variance. In this paper, by the Kolmogorov type maximal inequality and Stein's method, we establish the result of the law of the iterated logarithm for LNQD sequence with less restriction of moment conditions. We also prove the law of the iterated logarithm for a linear process generated by an L...
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ژورنال
عنوان ژورنال: Electronic Communications in Probability
سال: 2003
ISSN: 1083-589X
DOI: 10.1214/ecp.v8-1070