Deriving the asymptotic distribution of U- and V-statistics of dependent data using weighted empirical processes

[1] Aaronson, J., Burton, R., Dehling, H., Gilat, D., Hill, T. and Weiss, B. (1996). Strong laws for <i>L</i>- and <i>U</i>-statistics. <i>Trans. Amer. Math. Soc.</i> <b>348</b> 2845–2866.

[2] Beutner, E. and Zähle, H. (2010). A modified functional delta method and its application to the estimation of risk functionals. <i>J. Multivariate Anal.</i> <b>101</b> 2452–2463.

[3] Bradley, R.C. (2005). Basic properties of strong mixing conditions. A survey and some open questions. <i>Probab. Surv.</i> <b>2</b> 107–144 (electronic).

[4] Chen, X. and Fan, Y. (2006). Estimation of copula-based semiparametric time series models. <i>J. Econometrics</i> <b>130</b> 307–335.

[5] Dehling, H. (2006). Limit theorems for dependent <i>U</i>-statistics. In <i>Dependence in Probability and Statistics. Lecture Notes in Statist.</i> <b>187</b> 65–86. New York: Springer.

[6] Dehling, H. and Wendler, M. (2010). Central limit theorem and the bootstrap for <i>U</i>-statistics of strongly mixing data. <i>J. Multivariate Anal.</i> <b>101</b> 126–137.

[8] Denker, M. and Keller, G. (1986). Rigorous statistical procedures for data from dynamical systems. <i>J. Statist. Phys.</i> <b>44</b> 67–93.

[9] Dewan, I. and Prakasa Rao, B.L.S. (2001). Asymptotic normality for <i>U</i>-statistics of associated random variables. <i>J. Statist. Plann. Inference</i> <b>97</b> 201–225.

[10] Dewan, I. and Prakasa Rao, B.L.S. (2002). Central limit theorem for <i>U</i>-statistics of associated random variables. <i>Statist. Probab. Lett.</i> <b>57</b> 9–15.

[11] Doukhan, P. (1994). <i>Mixing</i>: <i>Properties and Examples. Lecture Notes in Statistics</i> <b>85</b>. New York: Springer.

[12] Fernholz, L.T. (1983). <i>Von Mises Calculus for Statistical Functionals. Lecture Notes in Statistics</i> <b>19</b>. New York: Springer.

[13] Gill, R.D. (1989). Non- and semi-parametric maximum likelihood estimators and the von Mises method. I. <i>Scand. J. Statist.</i> <b>16</b> 97–128.

[14] Hoeffding, W. (1948). A class of statistics with asymptotically normal distribution. <i>Ann. Math. Statistics</i> <b>19</b> 293–325.

[16] Lee, A.J. (1990). <i>U-statistics</i>: <i>Theory and Practice. Statistics</i>: <i>Textbooks and Monographs</i> <b>110</b>. New York: Marcel Dekker Inc.

[19] Sen, P.K. (1972/73). Limiting behavior of regular functionals of empirical distributions for stationary mixing processes. <i>Z. Wahrsch. Verw. Gebiete</i> <b>25</b> 71–82.

[24] Shao, Q.M. and Yu, H. (1996). Weak convergence for weighted empirical processes of dependent sequences. <i>Ann. Probab.</i> <b>24</b> 2098–2127.

[25] Shiryaev, A.N. (1996). <i>Probability</i>, 2nd ed. <i>Graduate Texts in Mathematics</i> <b>95</b>. New York: Springer.

[27] von Mises, R. (1947). On the asymptotic distribution of differentiable statistical functions. <i>Ann. Math. Statistics</i> <b>18</b> 309–348.

[28] van der Vaart, A.W. (1998). <i>Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics</i> <b>3</b>. Cambridge: Cambridge Univ. Press.

[30] Wu, W.B. (2008). Empirical processes of stationary sequences. <i>Statist. Sinica</i> <b>18</b> 313–333.

[31] Yoshihara, K.i. (1976). Limiting behavior of <i>U</i>-statistics for stationary, absolutely regular processes. <i>Z. Wahrsch. Verw. Gebiete</i> <b>35</b> 237–252.

[32] Zähle, H. (2011). Rates of almost sure convergence of plug-in estimates for distortion risk measures. <i>Metrika</i> <b>74</b> 267–285.

[7] Denker, M. (1985). <i>Asymptotic Distribution Theory in Nonparametric Statistics. Advanced Lectures in Mathematics</i>. Braunschweig: Friedr. Vieweg &amp; Sohn.

[15] Kallenberg, O. (2002). <i>Foundations of Modern Probability</i>, 2nd ed. <i>Probability and Its Applications</i> (<i>New York</i>). New York: Springer.

[17] Lindner, A.M. (2008). Stationarity, Mixing, Distributional Properties and Moments of GARCH(<i>p</i>, <i>q</i>) Processes. In <i>Handbook of Financial Time Series</i> (T.G. Andersen, R.A. Davis, J.-P. Kreiß, T. Mikosch, eds.) 41–69. Berlin: Springer.

[18] Reeds, J.A. (1976). On the definition of von Mises functionals. Research Report S-44. Dept. Statistics, Harvard Univ.

[20] Sen, P.K. (1981). <i>Sequential Nonparametrics</i>: <i>Invariance principles and statistical inference. Wiley Series in Probability and Mathematical Statistics</i>. New York: Wiley.

[21] Sen, P.K. (1991). Introduction to Hoeffding (1948): A class of statistics with asymptotically normal distribution. In <i>Breakthroughs in Statistics</i> (S. Kotz and N.L. Johnson, eds.) <b>I</b> 299–307. Springer, New York.

[22] Sen, P.K. (1996). Statistical functionals, Hadamard differentiability and martingales. In <i>Probability Models and Statistics</i> 29–47. New Delhi: New Age.

[23] Serfling, R.J. (1980). <i>Approximation Theorems of Mathematical Statistics. Wiley Series in Probability and Mathematical Statistics</i>. New York: Wiley.

[26] Shorack, G.R. and Wellner, J.A. (1986). <i>Empirical Processes with Applications to Statistics. Wiley Series in Probability and Mathematical Statistics</i>: <i>Probability and Mathematical Statistics</i>. New York: Wiley.

[29] van der Vaart, A.W. and Wellner, J.A. (1996). <i>Weak Convergence and Empirical Processes</i>: <i>With Applications to Statistics. Springer Series in Statistics</i>. New York: Springer.