Limit theorems for empirical Fr\'echet means of independent and non-identically distributed manifold-valued random variables

We prove weak laws of large numbers and central limit theorems of Lindeberg type for empirical centres of mass (empirical Fr\'echet means) of independent non-identically 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 vector-valued random variables, which may be of independent interest and is therefore the subject of a self-contained section. This vector-valued result allows us to clarify the number of conditions required for the central limit theorem for empirical Fr\'echet means, while extending its scope.

[1]  M. Fréchet.  Les éléments aléatoires de nature quelconque dans un espace distancié , 1948 .

[2]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .

[3]  R. Rao,et al.  Normal Approximation and Asymptotic Expansions , 1976 .

[4]  H. Karcher.  Riemannian center of mass and mollifier smoothing , 1977 .

[5]  H. Ziezold.  On Expected Figures and a Strong Law of Large Numbers for Random Elements in Quasi-Metric Spaces , 1977 .

[6]  H. Teicher,et al.  Probability theory: Independence, interchangeability, martingales , 1978 .

[7]  Patrick Billingsley,et al.  Probability and Measure. , 1986 .

[8]  W. Kendall.  Probability, Convexity, and Harmonic Maps with Small Image I: Uniqueness and Fine Existence , 1990 .

[9]  Stochastic Analysis: Convex geometry and nonconfluent Γ-martingales I: tightness and strict convexity , 1991 .

[10]  W. Kendall.  Convexity and the Hemisphere , 1991 .

[11]  Torsten Bohlin,et al.  Large-sample theory , 1991 .

[12]  W. Kendall.  The Propeller: A Counterexample to a Conjectured Criterion for the Existence of Certain Convex Functions , 1992 .

[13]  W. Kendall.  Convex geometry and nonconfluent γ-martingales II: well-posedness and γ-martingale convergence , 1992 .

[14]  H. Ziezold,et al.  Mean Figures and Mean Shapes Applied to Biological Figure and Shape Distributions in the Plane , 1994 .

[15]  J. Picard.  Barycentres et martingales sur une variété , 1994 .

[16]  Wilfrid S. Kendall,et al.  Riemannian barycentres and geodesic convexity , 1999, Mathematical Proceedings of the Cambridge Philosophical Society.

[17]  H. Le,et al.  Locating Fréchet means with application to shape spaces , 2001, Advances in Applied Probability.

[18]  C. Villani.  Topics in Optimal Transportation , 2003 .

[19]  R. Bhattacharya,et al.  LARGE SAMPLE THEORY OF INTRINSIC AND EXTRINSIC SAMPLE MEANS ON MANIFOLDS—II , 2003 .

[20]  H. Le,et al.  Estimation of Riemannian Barycentres , 2004 .

[21]  R. Bhattacharya,et al.  Large sample theory of intrinsic and extrinsic sample means on manifolds--II , 2005, math/0507423.

[22]  S. Chatterjee.  A NEW METHOD OF NORMAL APPROXIMATION , 2006, math/0611213.

[23]  A. D. Barbour,et al.  Small counts in the infinite occupancy scheme , 2008, 0809.4387.

[24]  R. Bhattacharya,et al.  Statistics on Riemannian manifolds: asymptotic distribution and curvature , 2008 .

[25]  Wilfrid S. Kendall.  Statistical shape theory , 2010 .

[26]  B. Afsari.  Riemannian Lp center of mass: existence, uniqueness, and convexity , 2011 .

[27]  J. Norris.  Appendix: probability and measure , 1997 .

[28]  A. Röllin,et al.  Stein’s method in high dimensions with applications , 2011, 1101.4454.