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Generalized Kronecker’s theorem and strong limit power functions
AIP Conference Proceedings, 2011In this paper Kronecker’s theorem is extended to a quite general setting and the new version of the theorem is applied to investigate strong limit power functions. Three fundamental theorems of Fourier expansion are shown to be equivalent. Some principles for the convergence of Fourier series are given.
Chuanyi Zhang +2 more
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Strong Limit Theorems for Increments of Renewal Processes
Journal of Mathematical Sciences, 2005Let \(X_i\), \(i\geq 1\), be i.i.d. nonnegative nondegenerate random variables with \(\text{ess\,inf\,}X_1= 0\), \(0< EX_1=\mu a_T\). The main theorems show that under moderate conditions \(\limsup W_T/b_T= \limsup(u_T- a_T/\mu)= \limsup u_T/b_T= 1\), a.s. and even lim instead of lim\,sup.
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On Strong Versions of the Central Limit Theorem
Mathematische Nachrichten, 1988AbstractLet Sn be the sum of n i.i.d.r.v. and let 1(‐∞,x)(·) be the indicator function of the interval (‐∞, x). Then the sequence 1(‐∞, x)(Sn/√n) does not converge for any x. Likewise the arithmetic means of this sequence converge only with probability zero. But the logarithmic means converge with probability one to the standard normal distribution Ø(x)
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Majorizing potentials in strong ratio limit theorems
Mathematical Notes, 2008In [1], the strong ratio limit theorems associated with Markov chains were first proved for some “test” functions with specific properties and were then generalized to a wider family of functions. In the present paper, this family is significantly extended by functions that can be majorized in a sense by the potentials of the original functions.
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On the Lin Condition in Strong Ratio Limit Theorems
Mathematical Notes, 2004Let \((X_n,n\geq 0)\) be a Markov chain with state space \((E,{\mathcal B})\) and \(P\) be a transition operator. It is proved that for a wide class of Markov chains and, in particular, for many random walks on groups the formula (Lin condition) \(\liminf (v(P^{n+1}f)/ v(P^nf))=1\) holds, where \(v\) and \(f\) are a probability measure on \({\mathcal B}
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The Work of A. N. Komogorov on Strong Limit Theorems
Theory of Probability & Its Applications, 1990See the review in Zbl 0662.60045.
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Strong limit theorems: the strong law of large numbers
1995Abstract The second part of this lemma was found by Erdös and Renyi. In a traditional formulation of the Borel-Cantelli lemma the condition of pairwise independence is replaced by the stronger condition of the mutual independence of the events A1, ... , An for every n.
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Strong Limit Theorems for Pairwise NQD Random Variables
Communications in Statistics - Theory and Methods, 2013A number of strong laws of large numbers for sequences of pairwise negative quadrant dependent (NQD) random variables have been established by using the generalized three series theorem. In this article, we obtain a strong law of large numbers by using the truncation technique and method of subsequences instead of the generalized three series theorem ...
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On a local limit theorem in strong sense
Statistics & Probability Letters, 1997Let (1) \(\xi_1,\dots,\xi_n,\dots\) be a sequence of independent integer-valued random variables. Denote \(S_{n,k}= \xi_{k+1}+\cdots+ \xi_{k+n}\), \(A_{k,n}= ES_{k,n}\), \(B^2_{k,n}= DS_{k,n}\), \(S_{0,n}= S_n\), \(B^2_{0,n}= B^2_n\), \(A_{0,n}= A_n\). The sequence (1) is said to satisfy the local limit theorem if \[ P(S_n= m)= B^{-1}_n\exp\{(m- A_n)^2/
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