## Electronic Journal of Probability

### Renewal theory for asymmetric $U$-statistics

Svante Janson

#### Abstract

We extend a functional limit theorem for symmetric $U$-statistics [Miller and Sen, 1972] to asymmetric $U$-statistics, and use this to show some renewal theory results for asymmetric $U$-statistics.

Some applications are given.

#### Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 129, 27 pp.

Dates
Accepted: 27 November 2018
First available in Project Euclid: 21 December 2018

https://projecteuclid.org/euclid.ejp/1545361593

Digital Object Identifier
doi:10.1214/18-EJP252

Mathematical Reviews number (MathSciNet)
MR3896866

Zentralblatt MATH identifier
07021685

#### Citation

Janson, Svante. Renewal theory for asymmetric $U$-statistics. Electron. J. Probab. 23 (2018), paper no. 129, 27 pp. doi:10.1214/18-EJP252. https://projecteuclid.org/euclid.ejp/1545361593

#### References

• [1] Robert H. Berk, Limiting behavior of posterior distributions when the model is incorrect. Ann. Math. Statist. 37 (1966), 51–58.
• [2] Patrick Billingsley, Convergence of Probability Measures. Wiley, New York, 1968.
• [3] Miklós Bóna, The copies of any permutation pattern are asymptotically normal. Preprint, 2007. arXiv:0712.2792
• [4] Miklós Bóna, On three different notions of monotone subsequences. Permutation Patterns, 89–114, London Math. Soc. Lecture Note Ser., 376, Cambridge Univ. Press, Cambridge, 2010.
• [5] Herold Dehling, Manfred Denker & Walter Philipp, Invariance principles for von Mises and $U$-statistics. Z. Wahrsch. Verw. Gebiete 67 (1984), no. 2, 139–167.
• [6] Víctor H. de la Peña and Evarist Giné, Decoupling. Springer-Verlag, New York, 1999.
• [7] M. Denker, Ch. Grillenberger & G. Keller, A note on invariance principles for v. Mises’ statistics. Metrika 32 (1985), no. 3-4, 197–214.
• [8] William Feller, An Introduction to Probability Theory and Its Application, volume I, third edition, Wiley, New York, 1968.
• [9] A. A. Filippova, The theorem of von Mises on limiting behaviour of functionals of empirical distribution functions and its statistical applications. (Russian.) Teor. Verojatnost. i Primenen. 7 (1962), 26–60.
• [10] Philippe Flajolet, Wojciech Szpankowski and Brigitte Vallée. Hidden word statistics. J. ACM 53 (2006), no. 1, 147–183.
• [11] Gavin G. Gregory, Large sample theory for $U$-statistics and tests of fit. Ann. Statist. 5 (1977), no. 1, 110–123.
• [12] Allan Gut, Stopped Random Walks 2nd ed. Springer, New York, 2009.
• [13] Allan Gut, Probability: A Graduate Course, 2nd ed. Springer, New York, 2013.
• [14] Allan Gut & Svante Janson, Converse results for existence of moments and uniform integrability for stopped random walks. Ann. Probab. 14 (1986), 1296–1317.
• [15] Peter Hall, On the invariance principle for $U$-statistics. Stochastic Process. Appl. 9 (1979), no. 2, 163–174.
• [16] Wassily Hoeffding, A class of statistics with asymptotically normal distribution. Ann. Math. Statistics 19 (1948), 293–325.
• [17] Wassily Hoeffding, The strong law of large numbers for $U$-statistics. Institute of Statistics, Univ. of North Carolina, Mimeograph series 302 (1961). https://repository.lib.ncsu.edu/handle/1840.4/2128
• [18] Wassily Hoeffding, Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 (1963), 13–30.
• [19] Svante Janson, Moments for first passage and last exit times, the minimum, and related quantities for random walks with positive drift. Adv. Appl. Probab. 18 (1986), 865–879.
• [20] Svante Janson, Gaussian Hilbert Spaces, Cambridge Univ. Press, Cambridge, UK, 1997.
• [21] Svante Janson, Large deviations for sums of partly dependent random variables. Random Structures Algorithms 24 (2004), no. 3, 234–248.
• [22] Svante Janson, Patterns in random permutations avoiding some sets of multiple patterns. Preprint, 2018. arXiv:1804.06071
• [23] Svante Janson, Brian Nakamura & Doron Zeilberger, On the asymptotic statistics of the number of occurrences of multiple permutation patterns. Journal of Combinatorics 6 (2015), no. 1-2, 117–143.
• [24] Svante Janson & Michael J. Wichura, Invariance principles for stochastic area and related stochastic integrals. Stochastic Process. Appl. 16 (1984), no. 1, 71–84.
• [25] Olav Kallenberg, Foundations of Modern Probability. 2nd ed., Springer, New York, 2002.
• [26] R. G. Miller, Jr. & Pranab Kumar Sen, Weak convergence of $U$-statistics and von Mises’ differentiable statistical functions. Ann. Math. Statist. 43 (1972), 31–41.
• [27] Georg Neuhaus, Functional limit theorems for $U$-statistics in the degenerate case. J. Multivariate Anal. 7 (1977), no. 3, 424–439.
• [28] A. F. Ronzhin, A functional limit theorem for homogeneous $U$-statistics with degenerate kernel. (Russian) Teor. Veroyatnost. i Primenen. 30 (1985), no. 4, 759–762. English transl.: Theory Probab. Appl. 30 (1985), no. 4, 806–809.
• [29] H. Rubin & R.A. Vitale, Asymptotic distribution of symmetric statistics. Ann. Statist. 8 (1980), 165–170.
• [30] Pranab Kumar Sen, Weak convergence of generalized $U$-statistics Ann. Probability 2 (1974), 90–102.
• [31] Rodica Simion and Frank W. Schmidt, Restricted permutations. European J. Combin. 6 (1985), no. 4, 383–406.
• [32] Raymond N. Sproule, Asymptotic properties of $U$-statistics. Trans. Amer. Math. Soc. 199 (1974), 55–64.