A central limit theorem for the isotropic random sphere

Let be a real-valued, homogeneous, and isotropic random field indexed in . When restricted to those indices with , the Euclidean length of , equal to r (a positive constant), then the random field resides on the surface of a sphere of radius r. Using a modified stratified spherical sampling plan (Brown (1993)) on the sphere, define to be a realization of the random process and to be the cardinality of . Without specifying the dependence structure of nor the marginal distribution of the , conditions for asymptotic normality of the standardized sample mean, , are given. The conditions on and are motivated by the ideas and results for dependent stationary sequences.

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On the Zagreb Index of Random Recursive Trees

Journal of Applied Probability ◽

10.1017/s0021900200008706 ◽

2011 ◽

Vol 48 (04) ◽

pp. 1189-1196 ◽

Cited By ~ 4

Author(s):

Qunqiang Feng ◽

Zhishui Hu

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Asymptotic Normality ◽

Random Process ◽

Central Limit ◽

Topological Indices ◽

Zagreb Index ◽

Martingale Central Limit Theorem ◽

Recursive Tree ◽

Recursive Trees

We investigate the Zagreb index, one of the topological indices, of random recursive trees in this paper. Through a recurrence equation, the first two moments ofZn, the Zagreb index of a random recursive tree of sizen, are obtained. We also show that the random process {Zn− E[Zn],n≥ 1} is a martingale. Then the asymptotic normality of the Zagreb index of a random recursive tree is given by an application of the martingale central limit theorem. Finally, two other topological indices are also discussed in passing.

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Counterexamples to the classical central limit theorem for triplewise independent random variables having a common arbitrary margin

Dependence Modeling ◽

10.1515/demo-2021-0120 ◽

2021 ◽

Vol 9 (1) ◽

pp. 424-438

Author(s):

Guillaume Boglioni Beaulieu ◽

Pierre Lafaye de Micheaux ◽

Frédéric Ouimet

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Central Limit ◽

Marginal Distribution ◽

Random Variables ◽

Independent Random Variables ◽

Sample Mean ◽

Non Gaussian ◽

Mutually Independent ◽

Specific Sequences

Abstract We present a general methodology to construct triplewise independent sequences of random variables having a common but arbitrary marginal distribution F (satisfying very mild conditions). For two specific sequences, we obtain in closed form the asymptotic distribution of the sample mean. It is non-Gaussian (and depends on the specific choice of F). This allows us to illustrate the extent of the ‘failure’ of the classical central limit theorem (CLT) under triplewise independence. Our methodology is simple and can also be used to create, for any integer K, new K-tuplewise independent sequences that are not mutually independent. For K [four.tf], it appears that the sequences created using our methodology do verify a CLT, and we explain heuristically why this is the case.

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On the Zagreb Index of Random Recursive Trees

Journal of Applied Probability ◽

10.1239/jap/1324046027 ◽

2011 ◽

Vol 48 (4) ◽

pp. 1189-1196 ◽

Cited By ~ 4

Author(s):

Qunqiang Feng ◽

Zhishui Hu

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Asymptotic Normality ◽

Random Process ◽

Central Limit ◽

Topological Indices ◽

Zagreb Index ◽

Martingale Central Limit Theorem ◽

Recursive Tree ◽

Recursive Trees

We investigate the Zagreb index, one of the topological indices, of random recursive trees in this paper. Through a recurrence equation, the first two moments of Zn, the Zagreb index of a random recursive tree of size n, are obtained. We also show that the random process {Zn − E[Zn], n ≥ 1} is a martingale. Then the asymptotic normality of the Zagreb index of a random recursive tree is given by an application of the martingale central limit theorem. Finally, two other topological indices are also discussed in passing.

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The Transparent Dead Leaves Model

Advances in Applied Probability ◽

10.1017/s0001867800005425 ◽

2012 ◽

Vol 44 (01) ◽

pp. 1-20 ◽

Cited By ~ 1

Author(s):

B. Galerne ◽

Y. Gousseau

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Random Field ◽

Central Limit ◽

Gaussian Random Field ◽

Natural Images ◽

Simulation Algorithm ◽

New Model ◽

Grain Model ◽

Dead Leaves Model

In this paper we introduce the transparent dead leaves (TDL) random field, a new germ-grain model in which the grains are combined according to a transparency principle. Informally, this model may be seen as the superposition of infinitely many semitransparent objects. It is therefore of interest in view of the modeling of natural images. Properties of this new model are established and a simulation algorithm is proposed. The main contribution of the paper is to establish a central limit theorem, showing that, when varying the transparency of the grain from opacity to total transparency, the TDL model ranges from the dead leaves model to a Gaussian random field.

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A Central Limit Theorem and Law of the Iterated Logarithm for a Random Field with Exponential Decay of Correlations

Canadian Journal of Mathematics ◽

10.4153/cjm-2004-010-6 ◽

2004 ◽

Vol 56 (1) ◽

pp. 209-224 ◽

Cited By ~ 3

Author(s):

Byron Schmuland ◽

Wei Sun

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Random Field ◽

Central Limit ◽

Exponential Decay ◽

Gibbs Measures ◽

Strong Mixing ◽

Iterated Logarithm ◽

Exponential Decay Of Correlations ◽

The Law

AbstractIn [6], Walter Philipp wrote that “… the law of the iterated logarithm holds for any process for which the Borel-Cantelli Lemma, the central limit theorem with a reasonably good remainder and a certain maximal inequality are valid.” Many authors [1], [2], [4], [5], [9] have followed this plan in proving the law of the iterated logarithm for sequences (or fields) of dependent random variables.We carry on this tradition by proving the law of the iterated logarithm for a random field whose correlations satisfy an exponential decay condition like the one obtained by Spohn [8] for certain Gibbs measures. These do not fall into the ϕ-mixing or strong mixing cases established in the literature, but are needed for our investigations [7] into diffusions on configuration space.The proofs are all obtained by patching together standard results from [5], [9] while keeping a careful eye on the correlations.

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Central Limit Theorem for the Free Energy of the Random Field Ising Model

Journal of Statistical Physics ◽

10.1007/s10955-019-02249-9 ◽

2019 ◽

Vol 175 (1) ◽

pp. 185-202

Author(s):

Sourav Chatterjee

Keyword(s):

Free Energy ◽

Central Limit Theorem ◽

Limit Theorem ◽

Ising Model ◽

Random Field ◽

Central Limit ◽

Random Field Ising Model

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A central limit theorem for exchangeable variates with geometric applications

Journal of Applied Probability ◽

10.2307/3212385 ◽

1973 ◽

Vol 10 (4) ◽

pp. 837-846 ◽

Cited By ~ 7

Author(s):

P. A. P. Moran

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Asymptotic Normality ◽

Central Limit ◽

Random Variables ◽

Multinomial Distribution ◽

Sum Of Random Variables ◽

Th Cell ◽

Occupancy Problems

A central limit theorem is proved for the sum of random variables Xr all having the same form of distribution and each of which depends on a parameter which is the number occurring in the rth cell of a multinomial distribution with equal probabilities in N cells and a total n, where nN–1 tends to a non-zero constant. This result is used to prove the asymptotic normality of the distribution of the fractional volume of a large cube which is not covered by N interpenetrating spheres whose centres are at random, and for which NV–1 tends to a non-zero constant. The same theorem can be used to prove asymptotic normality for a large number of occupancy problems.

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