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Functional Central Limit Theorems
2008Central limit theorems guarantee that the distributions of properly normalized sums of certain random variables are approximately normal. In many cases, however, a more detailed analysis is necessary. When testing for structural constancy in models, we might be interested in the temporal evolution of our sums.
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A Counterexample in the Central Limit Theorem
Bulletin of the London Mathematical Society, 1999We construct a \(c_0\)-valued random variable \(X\) such that \((S_n/\sqrt{n})_{n\in N}\) has a convergent subsequence, but \(X\) does not satisfy the central limit theorem (CLT), thus give a counterexample against the subsequence rule in CLT.
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1981
Here we present some non-trivial limit theorems where the limit is a non-Gaussian self-similar field. The results of the previous chapters may explain at a heuristic level why such results should hold. But a rigorous proof demands much extra work whose consequences may be interesting in themselves.
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Here we present some non-trivial limit theorems where the limit is a non-Gaussian self-similar field. The results of the previous chapters may explain at a heuristic level why such results should hold. But a rigorous proof demands much extra work whose consequences may be interesting in themselves.
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Central Limit Theorems for Aggregate Efficiency
Operations Research, 2018Applied researchers in the field of efficiency and productivity analysis often need to estimate and make inference about aggregate efficiency, such as industry efficiency or aggregate efficiency of a group of distinct firms within an industry (e.g., public versus private firms, regulated versus unregulated firms, etc.).
Léopold Simar, Valentin Zelenyuk
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On the universal A.S. central limit theorem
Acta Mathematica Hungarica, 2007Let \(X_1 ,X_2 ,\dots\) be independent random variables such that for some measurable functions \(g_l \) the weak limit theorem \(g_l (X_1 ,\dots,X_l ) \Rightarrow G\) holds with some distribution function \(G\). The paper gives conditions for the validity of relation \(D_N^{ - 1} \sum\limits_{k = 1}^N {d_k f(g_k (X_1 ,\dots,X_k ))} =\int_{ - \infty }^\
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Scandinavian Actuarial Journal, 1944
If X and Y are mutually independent random variables whith the d. f. 1 F 1(χ) and F 2(χ), it is known 2 that the sum X + Y has the d. f. F 2(χ), defined as the convolution where the integrals are Lebesgue-Stiltjes integrals. One uses the abbreviation More generally the sum X 1 + X 2 + … + X n of n mutually independent random variables with the d. f.
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If X and Y are mutually independent random variables whith the d. f. 1 F 1(χ) and F 2(χ), it is known 2 that the sum X + Y has the d. f. F 2(χ), defined as the convolution where the integrals are Lebesgue-Stiltjes integrals. One uses the abbreviation More generally the sum X 1 + X 2 + … + X n of n mutually independent random variables with the d. f.
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2012
The law of large numbers states that the arithmetic mean of independent, identically distributed random variables converges to the expected value. One interpretation of the central limit theorem is as a (distributional) rate result.
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The law of large numbers states that the arithmetic mean of independent, identically distributed random variables converges to the expected value. One interpretation of the central limit theorem is as a (distributional) rate result.
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On the central limit theorems.
1971This item was digitized as part of a project to share McGill's intellectual legacy with the public. If you are the copyright holder or a relative of the copyright holder who is deceased, you may request withdrawal by emailing escholarship.library@mcgill.ca.
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