Abstract
Let (X k) k≥1 be a sequence of random variables with common distribution function F(x) = P(X 1 ≤ x). Define the empirical distribution function
and the empirical process by \( \sqrt {n} ({{F}_{n}}(x) - F(x)) \) In this chapter we provide a survey of classical as well as modern techniques in the study of empirical processes of dependent data. We begin with a sketch of the early roots of the field in the theory of uniform distribution mod 1, of sequences defined by X k = {n k ω}, ω ∈ [0, 1], dating back to Weyl’s celebrated 1916 paper. In the second section we provide the essential tools of empirical process theory, and we prove Donsker’s classical empirical process invariance principle for i.i.d. processes. The third section provides an introduction to the subject of weakly dependent random variables. We introduce a variety of mixing concepts, provide necessary technical tools like correlation and moment inequalities, and prove central limit theorems for partial sums. The empirical process of weakly dependent data is investigated in the fourth section, where we put special emphasis on almost sure approximation techniques. The fifth section is devoted to the empirical distribution of U-statistics, defined as
for some symmetric kernel h. We give some applications, e.g., to dimension estimation in the analysis of time series, and prove weak convergence of the corresponding empirical process. Empirical processes of long-range dependent data are the topic of the sixth section. We give an introduction to the area of long-range dependent processes, provide important technical tools for the study of their partial sums and investigate the limit behavior of the empirical process. It turns out that the limit process is of a completely different type as in the case of independent or weakly dependent data, and that this has important consequences for various functionals of the empirical process. The final section is devoted to pair correlations, i.e., U-statistics empirical processes over short intervals associated with the kernel h(x, y) = |x − y|
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References
J. Aaronson, R.M. Burton, H.G. Dehling, D. Gilat, T. Hill and B. Weiss: Strong laws for L- and U-statistics. Transactions of the American Mathematical Society 348 (1996), 2845–2865.
M.A. Arcones and B. Yu: Central limit theorems for empirical processes and U-processes of stationary mixing sequences. Journal of Theoretical Probability 7 (1994), 47–71.
F. Avram and M.S. Taqqu: Noncentral limit theorems and Appell polynomials. Annals of Probability 15 (1987), 767–775.
R.C. Baker: Metric number theory and the large sieve. Journal of the London Mathematical Society 24 (1981), 34–40.
J. Beran: Statistical methods for data with long-range dependence (with discussions). Statistical Science 7 (1992), 404–427.
H.C.R Berbee: Random Walks with Stationary Increments and Renewal Theory. Mathematical Centre Tracts 112, Mathematisch Centrum, Amsterdam, 1979.
R.H. Berk: Limit behavior of posterior distributions when the model is incorrect. Annals of Mathematical Statistics 37 (1966), 51–58.
I. Berkes and W. Philipp: An almost sure invariance principle for the empirical distribution function of mixing random variables. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 41 (1977), 115–137.
I. Berkes and W. Philipp: Approximation theorems for independent and weakly dependent random vectors. Annals of Probability 7 (1979), 29–54.
I. Berkes and W. Philipp: The size of trigonometric and Walsh series and uniform distribution mod 1. Journal of the London Mathematical Society 50 (1994), 454–464.
I. Berkes, W. Philipp and R. Tichy: The pair correlation for independent and weakly depedent random variables. Illinois Journal of Mathematics 45 (2001), 559–580.
P. Billingsley: Convergence of Probability Measures. John Wiley & Sons, New York, 1968 (2nd edition: J. Wiley, New York, 1999).
P. Billingsley: Probability and Measure. 3rd edition, John Wiley & Sons, New York, 1995.
N.H. Bingham, C.M. Goldie and J.L. Teugels: Regular Variation. Encyclopedia of Mathematics and Its Applications 27. Cambridge University Press, Cambridge, 1987.
J.R. Blum, D.L. Hanson and L.H. Koopmans: On the strong law of large numbers for a class of stochastic processes. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 2 (1963), 1–11.
S.A. Borovkova: Weak convergence of the empirical process of U-statistics structure for dependent observations. Theory of Stochastic Processes 18 (1995), 115–124.
S.A. Borovkova, R.M. Burton and H.G. Dehling: Consistency of the Takens estimator for the correlation dimension. Annals of Applied Probability 9 (1999), 376–390.
S.A. Borovkova, R.M. Burton and H.G. Dehling: Limit theorems for functionals of mixing processes with applications to U-statistics and dimension estimation. Transactions of the American Mathematical Society 353 (2001), 4261–4318.
S.A. Borovkova, R.M. Burton and H.G. Dehling: From Dimension Estimation to Asymptotics of Dependent U-Statistics. Bolyai Society Mathematical Studies X, Limit Theorems, Balatonlelle, Budapest 2001, 1–34.
R.C. Bradley: Introduction to Strong Mixing Conditions, vol. I. Technical Report, Department of Mathematics, Indiana University, Bloomington, 2001.
L. Breiman: Probability. Addison Wesley, Reading, MA, 1968.
D. L. Burkholder: Distribution function inequalities for martingales. Annals of Probability 1 (1973), 19–42.
L. Carleson: On convergence and growth of partial sums of Fourier series. Acta Mathematica 116 (1966), 135–157.
J.W.S. Cassels: Some metrical theorems of Diophantine approximation III. Proceedings of the Cambridge Philosophical Society 46 (1950), 219–225.
J. W. S. Cassels: An extension of the law of the iterated logarithm. Proceedings of the Cambridge Philosophical Society 47 (1951), 55–64.
K. L. Chung: An estimate concerning the Kolmogorov limit distribution. Transactions of the American Mathematical Society 67 (1949), 36–50.
M. Csörgő and P. Revész: A new method to prove Strassen type laws of invariance principle II. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 31 (1975), 261–269.
A.R. Dabrowski and H.G. Dehling: Estimating conditional occupation time distributions for dependent sequences. The Canadian Journal of Statistics 24 (1996), 55–65.
A.R. Dabrowski, H.G. Dehling, T. Mikosch and O. Sharipov: Poisson limits for U-statistics. Stochastic Processes and Their Applications 99 (2002), 137–157.
Yu.A. Davydov: The invariance principle for stationary processes. Theory of Probability and Its Applications 15 (1970), 487–498.
H.G. Dehling: Grenzwertsätze für Summen schwach abhängiger vektor-wertiger Zufallsvariablen. Dissertation, Georg-August-Universität Göttingen, 1981.
H.G. Dehling: Limit theorems for sums of weakly dependent Banach space valued random variables. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 63 (1983), 393–432.
H.G. Dehling: The functional law of the iterated logarithm for von-Mises functionals and multiple Wiener integrals. Journal of Multivariate Analysis 28 (1989), 177–189.
H.G. Dehling: Complete convergence of triangular arrays and the law of the iterated logarithm for U-statistics. Statistics and Probability Letters 7 (1989), 319–321.
H.G. Dehling and W. Philipp: Almost sure invariance principles for weakly dependent vector-valued random variables. Annals of Probability 10 (1982), 689–701.
H.G. Dehling, M. Denker and W. Philipp: Invariance principles for von Mises and U-statistics. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 67 (1984), 139–167.
H.G. Dehling, M. Denker and W. Philipp: A bounded law of the iterated logarithm for Hilbert space valued martingales and its application to U-statistics. Probability Theory and Related Fields 72 (1986), 111–131.
H.G. Dehling, M. Denker and W. Philipp: The almost sure invariance principle for the empirical process of U-statistic structure. Annales de l’Institut Henri Poincare 23 (1987), 121–134.
H.G. Dehling and M.S. Taqqu: The limit behavior of empirical processes and symmetric statistics. Bulletin of the International Statistical Institute 52 (1987), volume 4, 217–234.
H.G. Dehling and M.S. Taqqu: The empirical process of some long-range dependent sequences with an application to U-statistics. Annals of Statistics 17 (1989), 1767–1783.
H.G. Dehling and M.S. Taqqu: Bivariate symmetric statistics of long-range dependent observations. Journal of Statistical Planning and Inference 28 (1991), 153–165.
H.G. Dehling and M.S. Taqqu: Continuous functions whose level sets are orthogonal to all polynomials of a given degree. Acta Mathematica Hungarica 60 (1992), 217–224.
M. Denker: Asymptotic Distribution Theory in Nonparametric Statistics. Vieweg Verlag, Braunschweig, 1985.
M. Denker and G. Keller: On U-statistics and von Mises statistics for weakly dependent processes. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 64 (1983), 505–522.
M. Denker and G. Keller: Rigorous statistical procedures for data from dynamical systems. Journal of Statistical Physics 44 (1986), 67–93.
M. Denker, Chr. Grillenberger and K. Siegmund: Ergodic Theory on Compact spaces. Lecture Notes in Mathematics 527, Springer Verlag, Berlin, 1976.
M. Denker, C. Grillenberger and G. Keller: A note on invariance principles for von Mises statistics. Metrika 32 (1985), 197–214.
C.M. Deo: A note on empirical processes of strong-mixing sequences. Annals of Probability 1 (1973), 870–875.
R.L. Dobrushin and P. Major: Non-central limit theorems for non-linear functionals of Gaussian fields. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 50 (1979), 27–52.
M.D. Donsker: An invariance principle for certain probability limit theorems. Memoirs of the American Mathematical Society 6 (1951).
M.D. Donsker: Justification and extension of Doob’s heuristic approach to the Kolmogorov-Smirnov theorems. Annals of Mathematical Statistics 23 (1952), 277–281.
J.L. Doob: Heuristic approach to the Kolmogorov-Smirnov theorems. Annals of Mathematical Statistics 20 (1949), 393–403.
J.L. Doob: Stochastic Processes. J. Wiley & Sons, New York, 1953.
P. Doukhan: Mixing: Properties and Examples. Lecture Notes in Statistics 85, Springer Verlag, 1994.
P. Doukhan and F. Portal: Principe d’invariance faible avec vitesse pour un processus empirique dans un cadre multidimensionnel et fortement melangeant. Comptes Rendus Academie des Sciences Paris, Série I 297 (1983), 505–508.
P. Doukhan and F. Portal: Principe d’invariance faible pour la fonction de repartition empirique dans un cadre multidimensionnel et melangeant. Probability and Mathematical Statistics 8 (1987), 117–132.
M. Drmota and R.F. Tichy: Sequences, Discrepancies and Applications. Springer Lecture Notes in Mathematics 1651, 1997.
R.M. Dudley Real Analysis and Probability. Wadsworth, Belmont, California, 1989.
R.M. Dudley and W. Philipp: Invariance principles for sums of Banach space valued random elements and empirical processes. Zeitschrift für Wahrscheinlichkeitstheorie verwandte Gebiete 62 (1983), 509–552.
A. Dvoretzky: Asymptotic normality for sums of dependent random variables. In: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability Theory (1970), vol. II, 513–535.
V. Eastwood and L. Horváth: Limit theorems for short distances in ℝm. Statistics and Probability Letters 45 (1999), 261–268.
P. Erdős: Problems and results in diophantine approximations. Compositio Mathematica 16 (1964), 52–65.
P. Erdős and LS. Gál: On the law of the iterated logarithm, I-H. Indagationes Mathematicae 17 (1955), 65–76, 77–84.
P. Erdős and M. Kac: On certain limit theorems in the theory of probability. Bulletin of the American Mathematical Society 52 (1946), 292–302.
P. Erdős and J.F. Koksma: On the uniform distribution modulo 1 of sequences (f (n, θ)). Indagationes Mathematicae 11 (1949), 299–302.
P. Erdős and P. Turán: On a problem in the theory of uniform distribution I. Indagationes Mathematicae 10 (1948), 370–378.
J. Esary, F. Proschan and D. Walkup: Association of random variables with applications. Annals of Mathematical Statistics 38 (1967), 1466–1474.
H. Fiedler, W. Jurkat and O. Körner: Asymptotic expansions of certain theta series. Acta Arithmetica 32 (1977), 129–146.
A.A. Filippova: Mises’s theorem on the asymptotic behavior of functionals of empirical distribution functions and its statistical applications. Theory of Probability and Its Applications 7 (1961), 24–57.
H. Finkelstein: The law of the iterated logarithm for empirical distributions. Annals of Mathematical Statistics 42 (1971), 607–615.
I. S. Gál and J.F. Koksma: Sur l’ordre de grandeur des fonctions sommables. Proceedings Koninklijke Nederlandse Akademie van Wetenschappen 53 (1950), 638–653.
L. Giraitis and D. Surgailis: Central limit theorem for the empirical process of a linear sequence with long memory. Journal of Statistical Planning and Inference 80 (1999), 81–93.
P. Grassberger and I. Procaccia: Characterization of strange attractors. Physical Review Letters 50 (1983), 346–349.
Y. Guivarc’h, M. Keane and B. Roynette: Marches aléatoires sur les groupes de Lie. Lecture Notes in Mathematics 624, Springer Verlag, Berlin, 1977.
P. Halmos: The theory of unbiased estimation. Annals of Mathematical Statistics 17 (1946), 34–43.
R. Helmers, P. Janssen and R. Serfling: Glivenko-Cantelli properties of some generalized empirical DFs and strong convergence of generalized L-statistics. Probability Theory and Related Fields 79 (1988), 75–93.
H.-C. Ho and T. Hsing: On the asymptotic expansion of the empirical process of long-memory moving averages. Annals of Statistics 24 (1996), 992–1024.
W. Hoeffding: A class of statistics with asymptotically normal distribution. Annals of Mathematical Statistics 19 (1948), 293–325.
W. Hoeffding: The strong law of large numbers for U-statistics. University of North Carolina Mimeo Report No. 302, 1961.
J. Hoffmann-Jørgensen: Stochastic Processes on Polish Spaces. Various Publication Series 39, Aarhus Universitet, Aarhus, Denmark, 1991.
L. Horváth: Short distances on the line. Stochastic Processes and Their Applications 39 (1991), 65–80.
R.A. Hunt: On the convergence of Fourier series, orthogonal expansions and their continuous analogues. Southern Illinois University Press (1968), 235–255.
H.E. Hurst: Long-term storage capacity of reservoirs. Transactions of the American Society of Civil Engineers 116 (1951), 770–808.
I.A. Ibragimov: Some limit theorems for stochastic processes stationary in the strict sense. Doklady Akademii Nauk SSSR 125 (1959), 711–714.
I.A. Ibragimov: Some limit theorems for stationary processes. Theory of Probability and Its Applications 7 (1962), 349–382.
I.A. Ibragimov and Yu.V. Linnik: Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff Publishing, Groningen, 1971.
M. Iosifescu: A very simple proof of a generalization of the Gauss-Kuzmin-Lévy theorem on continued fractions and related questions. Revue Roumaine Mathematiques Pures et Appliques 37 (1992), 901–914.
V. Isham: Statistical aspects of chaos: a review. In: Networks and Chaos — Statistical and Probabilistic Aspects (O.E. Barndorff-Nielsen, J.L. Jensen and W.S. Kendall, eds.). Chapman & Hall, London, 1993.
J. Kiefer: Skorohod embedding of multivariate rv’s and the sample df. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 24 (1972), 1–35.
A. Kolmogoroff and G. Seliverstoff: Sur la convergence de series de Fourier. Atti Accad. Naz. Lincei Ser.6 3 (1926), 307–310.
A.N. Kolmogorov and Yu.A. Rozanov: On strong mixing conditions for stationary Gaussian processes. Theory of Probability and Its Applications 5 (1960), 204–208.
J. Komlós, P. Major and G. Tusnady: An approximation of partial sums of independent RV’s and the sample DF. I. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 32 (1975), 111–131.
J. Komlós, P. Major and G. Tusnady: An approximation of partial sums of independent RV’s and the sample DF. II. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 34 (1975), 33–58.
H. Koul and D. Surgailis: Asymptotic expansion of the empirical process of long memory moving averages. In: Empirical Processes for Dependent Data, (H.G. Dehling, T. Mikosch and M. Sørensen, eds.). Birkhäuser, Boston (2002).
J. Kuelbs: Kolmogorov’s law of the iterated logarithm for Banach space valued random variables. Illinois Journal of Mathematics 21 (1977), 784–800.
J. Kuelbs and W. Philipp: Almost sure invariance principles for partial sums of mixing B-valued random variables. The Annals of Probability 8 (1980), 1003–1036.
L. Kuipers and H. Niederreiter: Uniform Distribution of Sequences. John Wiley, New York, 1974.
T.L. Lai: Reproducing kernel Hilbert spaces and the law of the iterated logarithm for Gaussian processes. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 29 (1974), 7–19.
P. Major: Multiple Wiener-Itô Integrals. Lecture Notes in Mathematics 849, Springer Verlag, Berlin, 1981.
A. Mandelbaum and M.S. Taqqu: Invariance principle for symmetric statistics. Annals of Statistics 12 (1984), 483–496.
B.B. Mandelbrot: The Fractal Geometry of Nature. W.H.Freeman and Company, New York, 1982.
B.B. Mandelbrot and J.W. van Ness: Fractional Brownian motion, fractional noises and applications. SIAM Review 10 (1968), 909–918.
D. Monrad and W. Philipp: Nearby variables with nearby conditional laws and a strong approximation theorem for Hilbert space valued martingales. Probability Theory and Related Fields 88 (1991), 381–404.
D. Nolan and D. Pollard: Functional limit theorems for U-processes. Annals of Probability 16 (1988), 1291–1298.
W. Philipp: The central limit problem for mixing sequences of random variables. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 12 (1969), 155–171.
W. Philipp: Mixing sequences of random variables and probabilistic number theory. Memoirs of the American Mathematical Society 114 (1971).
W. Philipp: Empirical distribution functions and uniform distribution mod 1. In: Diophantine Approximation and Its Applications. (C. Osgood, ed.), Academic Press, New York (1973), 211–234.
W. Philipp: Limit theorems for lacunary series and uniform distribution mod 1. Acta Arithmetica 26 (1974), 241–251.
W. Philipp: A functional law of the iterated logarithm for empirical distribution functions of weakly dependent random variables. Annals of Probability 5 (1977), 319–350.
W. Philipp: Invariance principles for sums of mixing random elements and the multivariate empirical process. In: Colloquium Mathematicae Societatis Janos Bolyái: Limit Theorems in Probability and Statistics, Veszprem (1982).
W. Philipp: Invariance principles for independent and weakly dependent random variables. In: Dependence in Probability and Statistics. (Eberlein, E., Taqqu, M.S., eds.) Progress in Probability and Statistics 11 (1986), Birkhäuser, Boston, 225–269.
W. Philipp: Empirical distribution functions and strong approximation theorems for dependent random variables. A problem of Baker in probabilistic number theory. Transactions of the American Mathematical Society 345 (1994), 705–727.
W. Philipp and W.F. Stout: Almost sure invariance principles for partial sums of weakly dependent random variables. Memoirs of the American Mathematical Society 161 (1915).
A. Plessner: Über Konvergenz von trigonometrischen Reihen. Journal für reine und angewandte Mathematik 155 (1926), 15–25.
D. Pollard: Convergence of Stochastic Processes. Springer Verlag, New York, 1984.
P. Revész: The Laws of Large Numbers. Academic Press, New York, London, 1968.
E. Rio: Covariance inequalities for strongly mixing processes. Annales de l’Institut Henri Poincaré Probability and Statistics 29 (1993), 587–597.
E. Rio: About the Lindeberg method for strongly mixing sequences. ESAIM: Probaba-bility and Statistics 1 (1995), 35–61.
E. Rio: Théorie asymptotique des processus aléatoires faiblement dépendants. Mathé-matiques et applications 31, Springer Verlag, 2000.
M. Römersperger: A note on nearby variables with nearby conditional laws. Probability Theory and Related Fields 106 (1996), 371–377.
M. Rosenblatt: A central limit theorem and a strong mixing condition. Proceedings of the National Academy of Sciences USA 42 (1956), 43–47.
W. Rudin: Principles of Mathematical Analysis. McGraw-Hill, 1976.
Z. Rudnick and P. Sarnak: The pair correlation function of fractional parts of polynomials. Communications in Mathematical Physics 194 (1998), 61–70.
Z. Rudnick and A. Zaharescu: A metric result on the pair correlation of fractional parts of sequences. Acta Arithmetica 89 (1999), 283–293.
R. Salem and A. Zygmund: On lacunary trigonometric series. Proceedings of the National Academy of Sciences USA 33 (1947), 333–338.
R. Salem and A. Zygmund: La loi du logarithme itéré pour les séries trigonométriques lacunaires. Bulletin des Sciences Mathématiques (2) 74 (1950), 220–224.
P.K. Sen: Anote on weak convergence of empirical processes for sequences of ∅-mixing random variables. Annals of Mathematical Statistics 42 (1971), 2131–2133.
P.K. Sen: Almost sure behavior of U-statistics and von Mises’ differentiable statistical functionals. Annals of Statistics 2 (1974), 387–395.
R.J. Serfling: Approximation Theorems of Mathematical Statistics. John Wiley & Sons, New York, 1980.
R. Serfling: Generalized L-, M- and R-statistics. Annals of Statistics 12 (1984), 76–86.
B.W. Silverman: Limit theorems for dissociated random variables. Advances in Applied Probability 8 (1976), 806–819.
B.W. Silverman: Convergence of a class of empirical distribution functions of dependent random variables. Annals of Probability 11 (1983), 745–751.
B.W. Silverman and T. Brown: Short distances, flat triangles and Poisson limits. Journal of Applied Probability 15 (1978), 815–825.
A.V. Skorohod: On a representation of random variables. Theory of Probability and Its Applications 21 (1976), 628–632.
D.A. Sotres and M. Ghosh: Strong convergence of linear rank statistics for mixing processes. Sankya, Series B 39 (1977), 1–11.
W.F. Stout: Almost Sure Convergence. Academic Press, New York-London, 1974.
V. Strassen and R.M. Dudley: The central limit theorem and ɛ-entropy. In: Lecture Notes in Mathematics 89, Springer-Verlag, 1969, 224–231.
D. Surgailis: Zones of attraction of self-similar multiple integrals. Lithuanian Mathematical Journal 22 (1983), 327–340.
F. Takens: Detecting strange attractors in turbulence. In: Dynamical Systems and Turbulence. Lecture Notes in Mathematics 898, Springer-Verlag, 1981, 336–381.
F. Takens: On the numerical determination of the dimension of the attractor. In: Dynamical Systems and Bifurcations. Lecture Notes in Mathematics 1125, Springer-Verlag, 1985, 99–106.
M.S. Taqqu: Weak convergence to fractional Brownian motion and to the Rosenblatt process. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 31 (1975), 287–302.
M.S. Taqqu: Convergence of integrated processes of arbitrary Hermite rank. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 50 (1979), 53–83.
VA. Volkonskii and Yu.A. Rozanov: Some limit theorems for random functions I. Theory of Probability and Its Applications 4 (1959), 178–197.
A. Walfisz: Ein metrischer Satz über Diophantische Approximationen. Fundamenta Mathematicae 16 (1930), 361–385.
H. Weyl: Über die Gleichverteilung von Zahlen mod eins. Mathematische Annalen 77 (1916), 313–352.
CS. Withers: Convergence of empirical processes of mixing random variables. Annals of Statistics 3 (1975), 1101–1108.
K. Yoshihara: Limiting behaviour of U-statistics for stationary, absolutely regular processes. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 35 (1976), 237–252.
H. Yu: A Glivenko-Cantelli lemma and weak convergence for empirical processes of associated sequences. Probability Theory and Related Fields 95 (1993), 357–370.
V.V. Yurinskii: On the error of the Gaussian approximation for convolutions. Theory of Probability and Its Applications 22 (1977), 236–247.
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Dehling, H., Philipp, W. (2002). Empirical Process Techniques for Dependent Data. In: Dehling, H., Mikosch, T., Sørensen, M. (eds) Empirical Process Techniques for Dependent Data. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0099-4_1
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