2015
DOI: 10.1214/ecp.v20-4341
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Central limit theorem under variance uncertainty

Abstract: We prove the central limit theorem (CLT) for a sequence of independent zero-mean random variables ξ j , perturbed by predictable multiplicative factors λ j with values in intervals [λ j , λ j ]. It is assumed that the sequences λ j , λ j are bounded and satisfy some stabilization condition. Under the classical Lindeberg condition we show that the CLT limit, corresponding to a "worst" sequence λ j , is described by the solution v of one-dimensional G-heat equation. The main part of the proof follows Peng's appr… Show more

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Cited by 4 publications
(3 citation statements)
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“…The reason for the appearance of this assumption we see in the lack of dominated convergence theorem in a general sublinear expectation space. We also refer to [17], where Peng's approach was applied to the one-dimensional problem with variance uncertainty. It was shown that, written in the classical terms, this approach allows to prove the CLT without unnatural assumptions, even in the case of non-identically distributed independent random variables.…”
Section: On the Relationship With The Sublinear Expectations Frameworkmentioning
confidence: 99%
“…The reason for the appearance of this assumption we see in the lack of dominated convergence theorem in a general sublinear expectation space. We also refer to [17], where Peng's approach was applied to the one-dimensional problem with variance uncertainty. It was shown that, written in the classical terms, this approach allows to prove the CLT without unnatural assumptions, even in the case of non-identically distributed independent random variables.…”
Section: On the Relationship With The Sublinear Expectations Frameworkmentioning
confidence: 99%
“…Li [23] proved a CLT for sub-linear expectation for a sequence of m-dependent random variables. Rokhlin [3] gave a CLT under the Lindeberg condition under classical probability with variance uncertainty. Zhang [10] gained a CLT for sub-linear expectation under a moment condition weaker than (2+α)-moments.…”
Section: §1 Introductionmentioning
confidence: 99%
“…Li [ 14 ] proved a CLT for sub-linear expectation for a sequence of m-dependent random variables. Rokhlin [ 19 ] gave a CLT under the Lindeberg condition under classical probability with variance uncertainty. Zhang [ 22 ] gained a CLT for sub-linear expectation under a moment condition weaker than -moments.…”
Section: Introductionmentioning
confidence: 99%