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Central Limit Theorems revisited
Statistics & Probability Letters, 2000The authors prove four central limit theorems (CLT). The first CLT is for a triangular array of random elements taking values in a real separable Hilbert space, where the components of the array are row-wise independent and have finite second moments, the covariance operator of the row sum satisfies mild convergence requirements, and the array ...
Majumdar, Suman +2 more
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Conditional Central Limit Theorem
Theory of Probability & Its Applications, 2017zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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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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1978
Central limit theorems have played a paramount role in probability theory starting—in the case of independent random variables—with the DeMoivreLaplace version and culminating with that of Lindeberg-Feller. The term “central” refers to the pervasive, although nonunique, role of the normal distribution as a limit of d.f.s of normalized sums of ...
Yuan Shih Chow, Henry Teicher
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Central limit theorems have played a paramount role in probability theory starting—in the case of independent random variables—with the DeMoivreLaplace version and culminating with that of Lindeberg-Feller. The term “central” refers to the pervasive, although nonunique, role of the normal distribution as a limit of d.f.s of normalized sums of ...
Yuan Shih Chow, Henry Teicher
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2008
At the end of the 17th century, the mathematician Abraham de Moivre first used the normal distribution as an approximation for the percentage of successes in a large number of experiments. Later on, Laplace generalized his results, but it took 20th century mathematics to give an exact and complete description of this subject. So let me now describe the
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At the end of the 17th century, the mathematician Abraham de Moivre first used the normal distribution as an approximation for the percentage of successes in a large number of experiments. Later on, Laplace generalized his results, but it took 20th century mathematics to give an exact and complete description of this subject. So let me now describe the
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1997
A key ingredient for the asymptotic normality proof, as outlined in Chapter 8, is that the normalized score vector can be expressed as a linear function of random variables ζ n which converge in distribution, cf. Assumption 8.1(g). In this chapter we present central limit theorems (CLTs) which can be used to imply this distributional convergence of ζ n
Benedikt M. Pötscher, Ingmar R. Prucha
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A key ingredient for the asymptotic normality proof, as outlined in Chapter 8, is that the normalized score vector can be expressed as a linear function of random variables ζ n which converge in distribution, cf. Assumption 8.1(g). In this chapter we present central limit theorems (CLTs) which can be used to imply this distributional convergence of ζ n
Benedikt M. Pötscher, Ingmar R. Prucha
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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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2018
Let P be a Markov kernel on \(\mathsf {X}\times \mathscr {X}\) that admits an invariant probability measure \(\pi \) and let Open image in new window be the canonical Markov chain.
Randal Douc +3 more
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Let P be a Markov kernel on \(\mathsf {X}\times \mathscr {X}\) that admits an invariant probability measure \(\pi \) and let Open image in new window be the canonical Markov chain.
Randal Douc +3 more
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1989
Actually, we had a central limit theorem (CLT) in lesson 14: the sequence X1X2X3… consists of IID Bernoulli RVs with parameter 0; the partial sum \({S_{n}} = \sum\nolimits_{{k = 1}}^{n} {{X_{k}}}\) is standardized to $${V_{n}} = ({S_{n}} - n\theta )/\surd (n\theta (1 - \theta );$$ the limiting distribution of Vnis normal.
Hung T. Nguyen, Gerald S. Rogers
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Actually, we had a central limit theorem (CLT) in lesson 14: the sequence X1X2X3… consists of IID Bernoulli RVs with parameter 0; the partial sum \({S_{n}} = \sum\nolimits_{{k = 1}}^{n} {{X_{k}}}\) is standardized to $${V_{n}} = ({S_{n}} - n\theta )/\surd (n\theta (1 - \theta );$$ the limiting distribution of Vnis normal.
Hung T. Nguyen, Gerald S. Rogers
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2019
In this chapter, we will discuss central limit theorems for some of the functionals introduced in Chapter 2. Further we develop general techniques that will later in Chapters 7–9 be applied to find central limit theorems also for other more specific statistics which are based on functionals from Chapter 2.
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In this chapter, we will discuss central limit theorems for some of the functionals introduced in Chapter 2. Further we develop general techniques that will later in Chapters 7–9 be applied to find central limit theorems also for other more specific statistics which are based on functionals from Chapter 2.
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