A Robust $$\alpha $$ α -Stable Central Limit Theorem Under Sublinear Expectation without Integrability Condition

Lianzi Jiang () and Gechun Liang ()
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Lianzi Jiang: Shandong University of Science and Technology
Gechun Liang: The University of Warwick

Journal of Theoretical Probability, 2024, vol. 37, issue 3, 2394-2424

Abstract: Abstract This article fills a gap in the literature by relaxing the integrability condition for the robust $$\alpha $$ α -stable central limit theorem under sublinear expectation. Specifically, for $$\alpha \in (0,1]$$ α ∈ ( 0 , 1 ] , we prove that the normalized sums of i.i.d. non-integrable random variables $$\big \{n^{-\frac{1}{\alpha }}\sum _{i=1}^{n}Z_{i}\big \}_{n=1}^{\infty }$$ { n - 1 α ∑ i = 1 n Z i } n = 1 ∞ converge in law to $${\tilde{\zeta }}_{1}$$ ζ ~ 1 , where $$({\tilde{\zeta }}_{t})_{t\in [0,1]}$$ ( ζ ~ t ) t ∈ [ 0 , 1 ] is a multidimensional nonlinear symmetric $$\alpha $$ α -stable process with jump uncertainty set $${\mathcal {L}}$$ L . The limiting $$\alpha $$ α -stable process is further characterized by a fully nonlinear partial integro-differential equation (PIDE): $$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \partial _{t}u(t,x)-\sup \limits _{F_{\mu }\in {\mathcal {L}}}\left\{ \int _{{\mathbb {R}}^{d}}\delta _{\lambda }^{\alpha }u(t,x)F_{\mu }(d\lambda )\right\} =0,\\ \displaystyle u(0,x)=\phi (x),\quad \forall (t,x)\in [0,1]\times {\mathbb {R}}^{d}, \end{array} \right. \end{aligned}$$ ∂ t u ( t , x ) - sup F μ ∈ L ∫ R d δ λ α u ( t , x ) F μ ( d λ ) = 0 , u ( 0 , x ) = ϕ ( x ) , ∀ ( t , x ) ∈ [ 0 , 1 ] × R d , where $$\begin{aligned} \delta _{\lambda }^{\alpha }u(t,x):=\left\{ \begin{array}{ll} u(t,x+\lambda )-u(t,x)-\langle D_{x}u(t,x),\lambda \mathbbm {1}_{\{|\lambda |\le 1\}}\rangle , &{}\quad \alpha =1,\\ u(t,x+\lambda )-u(t,x), &{}\quad \alpha \in (0,1). \end{array} \right. \end{aligned}$$ δ λ α u ( t , x ) : = u ( t , x + λ ) - u ( t , x ) - ⟨ D x u ( t , x ) , λ 1 { | λ | ≤ 1 } ⟩ , α = 1 , u ( t , x + λ ) - u ( t , x ) , α ∈ ( 0 , 1 ) . The approach used in this study involves the utilization of several tools, including a weak convergence approach to obtain the limiting process, a Lévy–Khintchine representation of the nonlinear $$\alpha $$ α -stable process and a truncation technique to estimate the corresponding $$\alpha $$ α -stable Lévy measures. In addition, the article presents a probabilistic method for proving the existence of a solution to the above fully nonlinear PIDE.

Keywords: Robust stable central limit theorem; Partial integro-differential equation; $$\alpha $$ α -stable distribution; Sublinear expectation; 60F05; 60G65; 45K05 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10959-023-01298-x

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