## The Annals of Probability

### Brownian limits, local limits and variance asymptotics for convex hulls in the ball

#### Abstract

Schreiber and Yukich [Ann. Probab. 36 (2008) 363–396] establish an asymptotic representation for random convex polytope geometry in the unit ball $\mathbb{B}^{d}$, $d\geq 2$, in terms of the general theory of stabilizing functionals of Poisson point processes as well as in terms of generalized paraboloid growth processes. This paper further exploits this connection, introducing also a dual object termed the paraboloid hull process. Via these growth processes we establish local functional limit theorems for the properly scaled radius-vector and support functions of convex polytopes generated by high-density Poisson samples. We show that direct methods lead to explicit asymptotic expressions for the fidis of the properly scaled radius-vector and support functions. Generalized paraboloid growth processes, coupled with general techniques of stabilization theory, yield Brownian sheet limits for the defect volume and mean width functionals. Finally we provide explicit variance asymptotics and central limit theorems for the $k$-face and intrinsic volume functionals.

#### Article information

Source
Ann. Probab., Volume 41, Number 1 (2013), 50-108.

Dates
First available in Project Euclid: 23 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.aop/1358951981

Digital Object Identifier
doi:10.1214/11-AOP707

Mathematical Reviews number (MathSciNet)
MR3059193

Zentralblatt MATH identifier
1278.60020

#### Citation

Calka, Pierre; Schreiber, Tomasz; Yukich, J. E. Brownian limits, local limits and variance asymptotics for convex hulls in the ball. Ann. Probab. 41 (2013), no. 1, 50--108. doi:10.1214/11-AOP707. https://projecteuclid.org/euclid.aop/1358951981

#### References

• [1] Affentranger, F. (1992). Random approximation of convex bodies. Publ. Mat. 36 85–109.
• [2] Bárány, I., Fodor, F. and Vígh, V. (2010). Intrinsic volumes of inscribed random polytopes in smooth convex bodies. Adv. in Appl. Probab. 42 605–619.
• [3] Bárány, I. and Reitzner, M. (2010). Poisson polytopes. Ann. Probab. 38 1507–1531.
• [4] Baryshnikov, Y. and Yukich, J. E. (2005). Gaussian limits for random measures in geometric probability. Ann. Appl. Probab. 15 213–253.
• [5] Bickel, P. J. and Wichura, M. J. (1971). Convergence criteria for multiparameter stochastic processes and some applications. Ann. Math. Statist. 42 1656–1670.
• [6] Buchta, C. (1985). Zufällige Polyeder—eine Übersicht. In Zahlentheoretische Analysis (E. Hlawka, ed.). Lecture Notes in Math. 1114 1–13. Springer, Berlin.
• [7] Calka, P. and Yukich, J. E. (2012). Variance asymptotics for random polytopes in smooth convex bodies. Unpublished manuscript.
• [8] Chen, L. H. Y. and Shao, Q.-M. (2004). Normal approximation under local dependence. Ann. Probab. 32 1985–2028.
• [9] Dedecker, J., Doukhan, P., Lang, G., León R., J. R., Louhichi, S. and Prieur, C. (2007). Weak Dependence: With Examples and Applications. Lecture Notes in Statistics 190. Springer, New York.
• [10] Eddy, W. F. (1980). The distribution of the convex hull of a Gaussian sample. J. Appl. Probab. 17 686–695.
• [11] Gruber, P. M. (1997). Comparisons of best and random approximation of convex bodies by polytopes. Rend. Circ. Mat. Palermo (2) Suppl. 50 189–216.
• [12] Hsing, T. (1994). On the asymptotic distribution of the area outside a random convex hull in a disk. Ann. Appl. Probab. 4 478–493.
• [13] Molchanov, I. S. (1996). On the convergence of random processes generated by polyhedral approximations of compact convex sets. Theory Probab. Appl. 40 383–390 (translated from Teor. Veroyatnost. i Primenen. 40 (1995) 438–444).
• [14] Pardon, J. (2011). Central limit theorems for random polygons in an arbitrary convex set. Ann. Probab. 39 881–903.
• [15] Penrose, M. D. (2007). Gaussian limits for random geometric measures. Electron. J. Probab. 12 989–1035 (electronic).
• [16] Penrose, M. D. (2007). Laws of large numbers in stochastic geometry with statistical applications. Bernoulli 13 1124–1150.
• [17] Penrose, M. D. and Yukich, J. E. (2001). Central limit theorems for some graphs in computational geometry. Ann. Appl. Probab. 11 1005–1041.
• [18] Penrose, M. D. and Yukich, J. E. (2003). Weak laws of large numbers in geometric probability. Ann. Appl. Probab. 13 277–303.
• [19] Penrose, M. D. and Yukich, J. E. (2005). Normal approximation in geometric probability. In Stein’s Method and Applications. Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap. 5 37–58. Singapore Univ. Press, Singapore.
• [20] Reiss, R. D. (1993). A Course on Point Processes. Springer, New York.
• [21] Reitzner, M. (2005). Central limit theorems for random polytopes. Probab. Theory Related Fields 133 483–507.
• [22] Reitzner, M. (2010). Random polytopes. In New Perspectives in Stochastic Geometry 45–76. Oxford Univ. Press, Oxford.
• [23] Rényi, A. and Sulanke, R. (1963). Über die konvexe Hülle von $n$ zufällig gewählten Punkten. Z. Wahrsch. Verw. Gebiete 2 75–84 (1963).
• [24] Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Applied Probability. A Series of the Applied Probability Trust 4. Springer, New York.
• [25] Schneider, R. (1988). Random approximation of convex sets. J. Microscopy 151 211–227.
• [26] Schneider, R. (1993). Convex Bodies: The Brunn-Minkowski Theory. Encyclopedia of Mathematics and Its Applications 44. Cambridge Univ. Press, Cambridge.
• [27] Schneider, R. (1997). Discrete aspects of stochastic geometry. In Handbook of Discrete and Computational Geometry (J. E. Goodman and J. O’Rourke, eds.) 167–184. CRC, Boca Raton, FL.
• [28] Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Springer, Berlin.
• [29] Schreiber, T. (2010). Limit theorems in stochastic geometry. In New Perspectives in Stochastic Geometry 111–144. Oxford Univ. Press, Oxford.
• [30] Schreiber, T. and Yukich, J. E. (2008). Variance asymptotics and central limit theorems for generalized growth processes with applications to convex hulls and maximal points. Ann. Probab. 36 363–396.
• [31] Shank, N. B. (2006). Limit theorems for random Euclidean graphs. Ph.D. thesis, Dept. Mathematics.
• [32] Vu, V. (2006). Central limit theorems for random polytopes in a smooth convex set. Adv. Math. 207 221–243.
• [33] Vu, V. H. (2005). Sharp concentration of random polytopes. Geom. Funct. Anal. 15 1284–1318.
• [34] Weil, W. and Wieacker, J. A. (1993). Stochastic geometry. In Handbook of Convex Geometry, Vol. A, B 1391–1438. North-Holland, Amsterdam.