On the law of the iterated logarithm for permuted lacunary sequences
نویسندگان
چکیده
منابع مشابه
On the law of the iterated logarithm for the discrepancy of lacunary sequences
A classical result of Philipp (1975) states that for any sequence (nk)k≥1 of integers satisfying the Hadamard gap condition nk+1/nk ≥ q > 1 (k = 1, 2, . . .), the discrepancy DN of the sequence (nkx)k≥1 mod 1 satisfies the law of the iterated logarithm (LIL), i.e. 1/4 ≤ lim supN→∞NDN (nkx)(N log logN)−1/2 ≤ Cq a.e. The value of the limsup is a long standing open problem. Recently Fukuyama expli...
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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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متن کاملOn 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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ژورنال
عنوان ژورنال: Proceedings of the Steklov Institute of Mathematics
سال: 2012
ISSN: 0081-5438,1531-8605
DOI: 10.1134/s0081543812010026