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2015 Central limit theorem under variance uncertainty
Dmitry Rokhlin
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Electron. Commun. Probab. 20: 1-10 (2015). DOI: 10.1214/ECP.v20-4341


We prove the central limit theorem (CLT) for a sequence of independent zero-mean random variables $\xi_j$, perturbed by predictable multiplicative factors $\lambda_j$ with values in intervals $[\underline\lambda_j,\overline\lambda_j]$. It is assumed that the sequences $\underline\lambda_j$, $\overline\lambda_j$ are bounded and satisfy some stabilization condition. Under the classical Lindeberg condition we show that the CLT limit, corresponding to a "worst'' sequence $\lambda_j$, is described by the solution $v$ of one-dimensional $G$-heat equation. The main part of the proof follows Peng's approach to the CLT under sublinear expectations, and utilizes Hölder regularity properties of $v$. Under the lack of such properties, we use the technique of half-relaxed limits from the theory of viscosity solutions.


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Dmitry Rokhlin. "Central limit theorem under variance uncertainty." Electron. Commun. Probab. 20 1 - 10, 2015.


Accepted: 26 September 2015; Published: 2015
First available in Project Euclid: 7 June 2016

zbMATH: 1329.60042
MathSciNet: MR3407210
Digital Object Identifier: 10.1214/ECP.v20-4341

Primary: 60F05
Secondary: 35D40

Keywords: $G$-heat equation , central limit theorem , Lindeberg condition , Variance uncertainty

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