Probability Inequalities for Empirical Processes and a Law of the Iterated Logarithm
نویسندگان
چکیده
منابع مشابه
Law of the Iterated Logarithm for Stationary Processes
There has been recent interest in the conditional central limit question for (strictly) stationary, ergodic processes · · ·X−1, X0, X1, · · · whose partial sums Sn = X1 + · · · + Xn are of the form Sn = Mn+Rn, where Mn is a square integrable martingale with stationary increments and Rn is a remainder term for which E(R 2 n) = o(n). Here we explore the Law of the Iterated Logarithm (LIL) for the...
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...
متن کاملA Law of the Iterated Logarithm for General Lacunary Series
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...
متن کاملEmpirical law of the iterated logarithm for Markov chains with a countable state space
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.
متن کاملA Law of the Iterated Logarithm for Arithmetic Functions
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 کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: The Annals of Probability
سال: 1984
ISSN: 0091-1798
DOI: 10.1214/aop/1176993141