The self-normalized Donsker theorem revisited
Pub. online: 18 September 2017
Type: Research Article
Open Access
Open Access
Received
18 May 2017
18 May 2017
Revised
9 August 2017
9 August 2017
Accepted
9 August 2017
9 August 2017
Published
18 September 2017
18 September 2017
Abstract
We extend the Poincaré–Borel lemma to a weak approximation of a Brownian motion via simple functionals of uniform distributions on n-spheres in the Skorokhod space . This approach is used to simplify the proof of the self-normalized Donsker theorem in Csörgő et al. (2003). Some notes on spheres with respect to -norms are given.
References
Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1968). MR0233396
Csörgő, M., Hu, Z.: Weak convergence of self-normalized partial sums processes. In: Asymptotic Laws and Methods in Stochastics. Fields Inst. Commun., 76, pp. 3–15. Springer (2015). MR3409822. doi:10.1007/978-1-4939-3076-0_1
Csörgő, M., Szyszkowicz, B., Wang, Q.: Donsker’s theorem for self-normalized partial sums processes. Ann. Probab. 31 (3), 1228–1240 (2003). MR1988470. doi:10.1214/aop/1055425777
Cutland, N., Ng, S.-A.: The wiener sphere and wiener measure. Ann. Probab. 21 (1), 1–13 (1993). MR1207212
de laPeña, V.H., Lai, T.L., Shao, Q.-M.: Self-normalized Processes. Springer, Berlin (2009). MR2488094. doi:10.1007/978-3-540-85636-8
Diaconis, P., Freedman, D.: A dozen de finetti-style results in search of a theory. Ann. Inst. Henri Poincaré Probab. Stat. 23 (2), 397–423 (1987). MR0898502
Dryden, I.L.: Statistical analysis on high-dimensional spheres and shape spaces. Ann. Stat. 33 (4), 1643–1665 (2005). MR2166558. doi:10.1214/009053605000000264
Giné, E., Götze, F., Mason, D.M.: When is the student t-statistic asymptotically standard normal? Ann. Probab. 25, 1514–1531 (1997). MR1457629. doi:10.1214/aop/1024404523
Kallenberg, O.: Foundations of Modern Probability. Second Edition. Probability and Its Applications. Springer, New York (2002). MR1876169. doi:10.1007/978-1-4757-4015-8
Lifshits, M.: Lectures on Gaussian Processes. Springer, New York (2012). MR3024389. doi:10.1007/978-3-642-24939-6
McKean, H.P.: Geometry of differential space. Ann. Probab. 1 (2), 197–206 (1973). MR0353471
Mishura, Y.: Stochastic Calculus for Fractional Brownian Motion and Related Processes. Springer, Berlin (2008). MR2378138. doi:10.1007/978-3-540-75873-0
Naor, A., Romik, D.: Projecting the surface measure of the sphere of . Ann. Inst. Henri Poincaré Probab. Stat. 39 (2), 241–261 (2003). MR1962135. doi:10.1016/S0246-0203(02)00008-0
Parczewski, P.: A fractional donsker theorem. Stoch. Anal. Appl. 32 (2), 328–347 (2014). MR3177074. doi:10.1080/07362994.2013.866521
Rachev, S.T., Rüschendorf, L.: Approximate independence of distributions on spheres and their stability properties. Ann. Probab. 19 (3), 1311–1337 (1991). MR1112418
Schechtman, G., Zinn, J.: On the volume of the intersection of two balls. Proc. Am. Math. Soc. 110 (1), 217–224 (1990). MR1015684. doi:10.2307/2048262
Taqqu, M.S.: Weak convergence to fractional brownian motion and to the rosenblatt process. Z. Wahrscheinlichkeitstheor. Verw. Geb. 31, 287–302 (1975). MR0400329. doi:10.1007/BF00532868
Vershik, A.M.: Does a lebesgue measure in an infinite-dimensional space exist? Tr. Mat. Inst. Steklova 259, 256–281 (2007). MR2433687. doi:10.1134/S0081543807040153
