A Robust α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-Stable Central Limit Theorem Under Sublinear Expectation without Integrability Condition

Abstract

This article fills a gap in the literature by relaxing the integrability condition for the robust α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-stable central limit theorem under sublinear expectation. Specifically, for α∈(0,1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\in (0,1]$$\\end{document}, we prove that the normalized sums of i.i.d. non-integrable random variables {n-1α∑i=1nZi}n=1∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\big \\{n^{-\\frac{1}{\\alpha }}\\sum _{i=1}^{n}Z_{i}\\big \\}_{n=1}^{\\infty }$$\\end{document} converge in law to ζ~1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ilde{\\zeta }}_{1}$$\\end{document}, where (ζ~t)t∈[0,1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({\ ilde{\\zeta }}_{t})_{t\\in [0,1]}$$\\end{document} is a multidimensional nonlinear symmetric α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-stable process with jump uncertainty set L\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {L}}$$\\end{document}. The limiting α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-stable process is further characterized by a fully nonlinear partial integro-differential equation (PIDE):∂tu(t,x)-supFμ∈L∫Rdδλαu(t,x)Fμ(dλ)=0,u(0,x)=ϕ(x),∀(t,x)∈[0,1]×Rd,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{l} \\displaystyle \\partial _{t}u(t,x)-\\sup \\limits _{F_{\\mu }\\in {\\mathcal {L}}}\\left\\{ \\int _{{\\mathbb {R}}^{d}}\\delta _{\\lambda }^{\\alpha }u(t,x)F_{\\mu }(d\\lambda )\\right\\} =0,\\\\ \\displaystyle u(0,x)=\\phi (x),\\quad \\forall (t,x)\\in [0,1]\ imes {\\mathbb {R}}^{d}, \\end{array} \\right. \\end{aligned}$$\\end{document}where δλαu(t,x):=u(t,x+λ)-u(t,x)-⟨Dxu(t,x),λ1{|λ|≤1}⟩,α=1,u(t,x+λ)-u(t,x),α∈(0,1).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\delta _{\\lambda }^{\\alpha }u(t,x):=\\left\\{ \\begin{array}{ll} u(t,x+\\lambda )-u(t,x)-\\langle D_{x}u(t,x),\\lambda \\mathbbm {1}_{\\{|\\lambda |\\le 1\\}}\\rangle , &{}\\quad \\alpha =1,\\\\ u(t,x+\\lambda )-u(t,x), &{}\\quad \\alpha \\in (0,1). \\end{array} \\right. \\end{aligned}$$\\end{document}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 α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-stable process and a truncation technique to estimate the corresponding α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-stable Lévy measures. In addition, the article presents a probabilistic method for proving the existence of a solution to the above fully nonlinear PIDE.