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Limit theorems for empirical Fréchet means of independent and nonidentically distributed manifoldvalued random variables
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Kendall, Wilfrid S. and Le, Huiling. (2011) Limit theorems for empirical Fréchet means of independent and nonidentically distributed manifoldvalued random variables. Brazilian Journal of Probability and Statistics, Vol.25 (No.3). pp. 323352. ISSN 01030752

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Official URL: http://dx.doi.org/10.1214/11BJPS141
Abstract
We prove weak laws of large numbers and central limit theorems of Lindeberg type for empirical centres of mass (empirical Fréchet means) of independent nonidentically distributed random variables taking values in Riemannian manifolds. In order to prove these theorems we describe and prove a simple kind of Lindeberg–Feller central approximation theorem for vectorvalued random variables, which may be of independent interest and is therefore the subject of a selfcontained section. This vectorvalued result allows us to clarify the number of conditions required for the central limit theorem for empirical Fréchet means, while extending its scope.
Item Type:  Journal Article 

Subjects:  Q Science > QA Mathematics 
Divisions:  Faculty of Science > Statistics 
Library of Congress Subject Headings (LCSH):  Geometry, Riemannian, Limit theorems (Probability theory), Random variables 
Journal or Publication Title:  Brazilian Journal of Probability and Statistics 
Publisher:  Duke University Press 
ISSN:  01030752 
Date:  2011 
Volume:  Vol.25 
Number:  No.3 
Page Range:  pp. 323352 
Identification Number:  10.1214/11BJPS141 
Status:  Peer Reviewed 
Publication Status:  Published 
Access rights to Published version:  Restricted or Subscription Access 
References:  Afsari, B. (2011). Riemannian Lp center of mass: Existence, uniqueness, and convexity. Proc. Amer. Math. Soc. 139, 655–655. MR2736346 Barbour, A. D. and Gnedin, A. V. (2009). Small counts in the infinite occupancy scheme. Electron. J. Probab. 14, 365–384. MR2480545 Bhattacharya, A. and Bhattacharya, R. (2008). Statistics on Riemannian manifolds: Asymptotic distribution and curvature. Proc. Amer. Math. Soc. 136, 2959–2967. MR2399064 Bhattacharya, R. and Patrangenaru, V. (2003). Large sample theory of intrinsic and extrinsic sample means on manifolds—I. Ann. Statist. 31, 1–29. MR1962498 Bhattacharya, R. and Patrangenaru, V. (2005). Large sample theory of intrinsic and extrinsic sample means on manifolds—II. Ann. Statist. 33, 1225–1259. MR2195634 Bhattacharya, R. and Rao, R. R. (1976). Normal Approximation and Asymptotic Expansions. New York–London–Sydney: Wiley. MR0436272 Billingsley, P. (1986). Probability and Measure. New York: Wiley. MR0830424 Chatterjee, S. (2008). A new method of normal approximation. Ann. Probab. 36, 1584–1610. MR2435859 Chow, Y. S. and Teicher, H. (2003). Probability Theory: Independence, Interchangeability, Martingales. New York–Heidelberg–Berlin: SpringerVerlag. Corcuera, J.M. and Kendall, W. S. (1999). Riemannian barycentres and geodesic convexity. Math. Proc. Cambridge Philos. Soc. 127, 253–269. MR1705458 Feller, W. (1966). An Introduction to Probability Theory and Its Applications, Volume 2. New York: Wiley. MR0210154 Fréchet,M. (1948). Les éléments aléatoires de nature quelconque dans un espace distancié. Ann. Inst. Henri Poincaré 10, 215–310. MR0027464 Karcher, H. (1977). Riemannian center of mass and mollifier smoothing. Comm. Pure Appl. Math. 30, 509–541. MR0442975 Kendall, W. S. (1990). Probability, convexity, and harmonic maps with small image I: Uniqueness and fine existence. Proc. Lond. Math. Soc. (3) 61, 371–406. MR1063050 Kendall, W. S. (1991a). Convex geometry and nonconfluent Γmartingales I: Tightness and strict convexity. In Stochastic Analysis, Proceedings, LMS Durham Symposium, 11th–21st July 1990 (M. T. Barlow and N. H. Bingham, eds.) 163–178. Cambridge: Cambridge Univ. Press. MR1166410 Kendall, W. S. (1991b). Convexity and the hemisphere. J. Lond. Math. Soc. (2) 43, 567–576. MR1113394 Kendall, W. S. (1992a). Convex geometry and nonconfluent Γmartingales II: Wellposedness and Γmartingale convergence. Stochastics and Stochastic Reports 38, 135–147. MR1274899 Kendall, W. S. (1992b). The Propeller: A counterexample to a conjectured criterion for the existence of certain convex functions. J. Lond. Math. Soc. (2) 46, 364–374. MR1182490 Kendall, W. S. and Le, H. (2010). Statistical shape theory. In New Perspectives in Stochastic Geometry (W. S. Kendall and I. S. Molchanov, eds.) 10 348–373. Oxford: Clarendon Press, Oxford Univ. Press. MR2654683 Le, H. (2001). Locating Fréchet means with application to shape spaces. Adv. Appl. Probab. 33, 324–338. MR1842295 Le, H. (2004). Estimation of Riemannian barycentres. LMS J. Comput. Math 7, 193–200. MR2085875 Picard, J. (1994). Barycentres et martingales sur une variété. Ann. Inst. H. Poincaré Probab. Stat. 30, 647–702. MR1302764 Röllin, A. (2011). Stein’s method in high dimensions with applications. Available at arXiv: 1101.4454. Villani, C. (2003). Topics in optimal transportation. Graduate Studies in Mathematics 58. Providence, RI: Amer. Math. Soc. MR1964483 Ziezold, H. (1977). On expected figures and a strong law of large numbers for random elements in quasimetric spaces. Conf. Inf. Theory, Statist. Decision Functions, Random Processes A, 591– 602. MR0501230 Ziezold, H. (1989). On expected figures in the plane. In Geobild ’89 (Georgenthal, 1989). Math. Res. 51, 105–110. Berlin: AkademieVerlag. MR1003326 Ziezold, H. (1994).Mean figures and mean shapes applied to biological figure and shape distributions in the plane. Biom. J. 36, 491–510. MR1292314 
URI:  http://wrap.warwick.ac.uk/id/eprint/37637 
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