A central limit theorem for exchangeable variates with geometric applications

1973 ◽  
Vol 10 (4) ◽  
pp. 837-846 ◽  
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.

A central limit theorem for exchangeable variates with geometric applications

1973 ◽  
Vol 10 (04) ◽  
pp. 837-846 ◽  
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.

Waitingtimes between record observations

1967 ◽  
Vol 4 (01) ◽  
pp. 206-208 ◽  
Author(s):  
Marcel F. Neuts
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Asymptotic Normality ◽  
Central Limit ◽  
Law Of Large Numbers ◽  
Continuous Distribution ◽  
Random Variables ◽  
Large Numbers ◽  
Upper Record ◽  
Independent Identically Distributed

If Δ r denotes the waitingtime between the (r − 1)st and the rth upper record in a sequence of independent, identically distributed random variables with a continuous distribution, then it is shown that Δ r satisfies the weak law of large numbers and a central limit theorem. This theorem supplements those of Foster and Stuart and Rényi, who investigated the index Vr of the rth upper record. Qualitatively the theorems establish the intuitive fact that for higher records, the waitingtime between the last two records outweighs even the total waitingtime for previous records. This explains also why the asymptotic normality of logVr is very inadequate for approximation purposes—Barton and Mallows.

Waitingtimes between record observations

1967 ◽  
Vol 4 (1) ◽  
pp. 206-208 ◽  
Author(s):  
Marcel F. Neuts
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Asymptotic Normality ◽  
Central Limit ◽  
Law Of Large Numbers ◽  
Continuous Distribution ◽  
Random Variables ◽  
Large Numbers ◽  
Upper Record ◽  
Independent Identically Distributed

If Δr denotes the waitingtime between the (r − 1)st and the rth upper record in a sequence of independent, identically distributed random variables with a continuous distribution, then it is shown that Δr satisfies the weak law of large numbers and a central limit theorem.This theorem supplements those of Foster and Stuart and Rényi, who investigated the index Vr of the rth upper record.Qualitatively the theorems establish the intuitive fact that for higher records, the waitingtime between the last two records outweighs even the total waitingtime for previous records. This explains also why the asymptotic normality of logVr is very inadequate for approximation purposes—Barton and Mallows.

Asymptotic normality of some Graph-Related statistics

1989 ◽  
Vol 26 (1) ◽  
pp. 171-175 ◽  
Author(s):  
Pierre Baldi ◽  
Yosef Rinott
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Asymptotic Normality ◽  
Central Limit ◽  
Simple Proof ◽  
Random Function ◽  
Random Variables ◽  
Local Maxima ◽  
Colored Graphs ◽  
U Statistics

Petrovskaya and Leontovich (1982) proved a central limit theorem for sums of dependent random variables indexed by a graph. We apply this theorem to obtain asymptotic normality for the number of local maxima of a random function on certain graphs and for the number of edges having the same color at both endpoints in randomly colored graphs. We briefly motivate these problems, and conclude with a simple proof of the asymptotic normality of certain U-statistics.

Asymptotic normality of some Graph-Related statistics

1989 ◽  
Vol 26 (01) ◽  
pp. 171-175 ◽  
Author(s):  
Pierre Baldi ◽  
Yosef Rinott
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Asymptotic Normality ◽  
Central Limit ◽  
Simple Proof ◽  
Random Function ◽  
Random Variables ◽  
Local Maxima ◽  
Colored Graphs ◽  
U Statistics

Petrovskaya and Leontovich (1982) proved a central limit theorem for sums of dependent random variables indexed by a graph. We apply this theorem to obtain asymptotic normality for the number of local maxima of a random function on certain graphs and for the number of edges having the same color at both endpoints in randomly colored graphs. We briefly motivate these problems, and conclude with a simple proof of the asymptotic normality of certain U-statistics.

scholarly journals Central limit theorem for linear processes generated by IID random variables under the sub-linear expectation

2021 ◽  
Vol 36 (2) ◽  
pp. 243-255
Author(s):  
Wei Liu ◽  
Yong Zhang
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Invariance Principle ◽  
Random Variables ◽  
Probability Measures ◽  
Linear Processes ◽  
The Central Limit Theorem

AbstractIn this paper, we investigate the central limit theorem and the invariance principle for linear processes generated by a new notion of independently and identically distributed (IID) random variables for sub-linear expectations initiated by Peng [19]. It turns out that these theorems are natural and fairly neat extensions of the classical Kolmogorov’s central limit theorem and invariance principle to the case where probability measures are no longer additive.

scholarly journals A counterexample to the existence of a general central limit theorem for pairwise independent identically distributed random variables

2021 ◽  
Vol 499 (1) ◽  
pp. 124982
Author(s):  
Benjamin Avanzi ◽  
Guillaume Boglioni Beaulieu ◽  
Pierre Lafaye de Micheaux ◽  
Frédéric Ouimet ◽  
Bernard Wong
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Random Variables ◽  
Independent Identically Distributed
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