Roots of Kostlan polynomials: moments, strong Law of Large Numbers and Central Limit Theorem

An array of random variables, indexed by a multidimensional parameter set, is said to be dissociated if the random variables are independent whenever their indexing sets are disjoint. The idea of dissociated random variables, which arises rather naturally in data analysis, was first studied by McGinley and Sibson(7). They proved a Strong Law of Large Numbers for dissociated random variables when their fourth moments are uniformly bounded. Silver man (8) extended the analysis of dissociated random variables by proving a Central Limit Theorem when the variables also satisfy certain symmetry relations. It is the aim of this paper to show that a Strong Law of Large Numbers (under more natural moment conditions), a Central Limit Theorem and in variance principle are consequences of the symmetry relations imposed by Silverman rather than the independence structure. To prove these results, reversed martingale techniques are employed and thus it is shown, in passing, how the well known Central Limit Theorem for U-statistics can be derived from the corresponding theorem for reversed martingales (as was conjectured by Loynes(6)).

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The estimation of frequency

Journal of Applied Probability ◽

10.2307/3212772 ◽

1973 ◽

Vol 10 (3) ◽

pp. 510-519 ◽

Cited By ~ 136

Author(s):

E. J. Hannan

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Central Limit ◽

Law Of Large Numbers ◽

Sinusoidal Oscillation ◽

Unknown Parameters ◽

Strong Law ◽

The Central Limit Theorem ◽

Large Numbers

Very general forms of the strong law of large numbers and the central limit theorem are proved for estimates of the unknown parameters in a sinusoidal oscillation observed subject to error. In particular when the unknown frequency θ0, is in fact 0 or <it is shown that the estimate, , satisfies for N ≧ N0 (ω) where N0 (ω) is an integer, determined by the realisation, ω, of the process, that is almost surely finite.

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The estimation of frequency

Journal of Applied Probability ◽

10.1017/s002190020011839x ◽

1973 ◽

Vol 10 (03) ◽

pp. 510-519 ◽

Cited By ~ 10

Author(s):

E. J. Hannan

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Central Limit ◽

Law Of Large Numbers ◽

Sinusoidal Oscillation ◽

Unknown Parameters ◽

Strong Law ◽

The Central Limit Theorem ◽

Large Numbers

Very general forms of the strong law of large numbers and the central limit theorem are proved for estimates of the unknown parameters in a sinusoidal oscillation observed subject to error. In particular when the unknown frequency θ 0, is in fact 0 or <it is shown that the estimate, , satisfies for N ≧ N 0 (ω) where N 0 (ω) is an integer, determined by the realisation, ω, of the process, that is almost surely finite.

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A Central Limit Theorem for Multiplicative Systems

Canadian Mathematical Bulletin ◽

10.4153/cmb-1973-014-3 ◽

1973 ◽

Vol 16 (1) ◽

pp. 67-73 ◽

Cited By ~ 4

Author(s):

J. Komlós

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Central Limit ◽

Law Of Large Numbers ◽

Law Of Iterated Logarithm ◽

Independent Random Variables ◽

Strong Law ◽

Multiplicative Systems ◽

Large Numbers ◽

Strong Notion

The central limit theorem was originally proved for independent random variables. The independence is a very strong notion and hard to check. There are various efforts to prove different theorems on independent variables (e.g. strong law of large numbers, central limit theorem, the law of iterated logarithm, convergence theorem of Kolmogorov) under weaker conditions, like mixing, martingale-difference, orthogonality. Among these concepts the weakest one is orthogonality, but this ensures only the validity of law of large numbers.

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