The principle of large deviations for the almost everywhere central limit theorem

AbstractWe study atypical behavior in bootstrap percolation on the Erdős–Rényi random graph. Initially a set S is infected. Other vertices are infected once at least r of their neighbors become infected. Janson et al. (Ann Appl Probab 22(5):1989–2047, 2012) locates the critical size of S, above which it is likely that the infection will spread almost everywhere. Below this threshold, a central limit theorem is proved for the size of the eventually infected set. In this work, we calculate the rate function for the event that a small set S eventually infects an unexpected number of vertices, and identify the least-cost trajectory realizing such a large deviation.

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An almost everywhere central limit theorem

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100065750 ◽

1988 ◽

Vol 104 (3) ◽

pp. 561-574 ◽

Cited By ~ 159

Author(s):

Gunnar A. Brosamler

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Weak Convergence ◽

Central Limit ◽

Finite Measure ◽

Standard Normal Distribution ◽

Almost Everywhere ◽

Real Functions ◽

Standard Normal ◽

Image Position

The purpose of this paper is the proof of an almost everywhere version of the classical central limit theorem (CLT). As is well known, the latter states that for IID random variables Y1, Y2, … on a probability space (Ω, , P) with we have weak convergence of the distributions of to the standard normal distribution on ℝ. We recall that weak convergence of finite measures μn on a metric space S to a finite measure μ on S is defined to mean thatfor all bounded, continuous real functions on S. Equivalently, one may require the validity of (1·1) only for bounded, uniformly continuous real functions, or even for all bounded measurable real functions which are μ-a.e. continuous.

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On the behaviour of a long cascade of linear reservoirs

Journal of Applied Probability ◽

10.1017/s002190020001562x ◽

2000 ◽

Vol 37 (02) ◽

pp. 417-428

Author(s):

John E. Glynn ◽

Peter W. Glynn

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Large Deviations ◽

Asymptotic Behaviour ◽

Central Limit ◽

Tail Probabilities ◽

Logarithmic Asymptotics ◽

Random Fluctuations

This paper describes the limiting asymptotic behaviour of a long cascade of linear reservoirs fed by stationary inflows into the first reservoir. We show that the storage in the nth reservoir becomes asymptotically deterministic as n → ∞, and establish a central limit theorem for the random fluctuations about the deterministic approximation. In addition, we prove a large deviations theorem that provides precise logarithmic asymptotics for the tail probabilities associated with the storage in the nth reservoir when n is large.

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On the behaviour of a long cascade of linear reservoirs

Journal of Applied Probability ◽

10.1239/jap/1014842547 ◽

2000 ◽

Vol 37 (2) ◽

pp. 417-428

Author(s):

John E. Glynn ◽

Peter W. Glynn

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Large Deviations ◽

Asymptotic Behaviour ◽

Central Limit ◽

Tail Probabilities ◽

Logarithmic Asymptotics ◽

Random Fluctuations

This paper describes the limiting asymptotic behaviour of a long cascade of linear reservoirs fed by stationary inflows into the first reservoir. We show that the storage in the nth reservoir becomes asymptotically deterministic as n → ∞, and establish a central limit theorem for the random fluctuations about the deterministic approximation. In addition, we prove a large deviations theorem that provides precise logarithmic asymptotics for the tail probabilities associated with the storage in the nth reservoir when n is large.

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