Long memory and self-similar processes
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 15 (2006) no. 1, pp. 107-123.

Cet article est une synthèse de résultats et idées classiques ou nouveaux sur la longue mémoire, les changements d’échelles et l’autosimilarité, à la fois dans le cas de queues de distributions lourdes ou légères.

This paper is a survey of both classical and new results and ideas on long memory, scaling and self-similarity, both in the light-tailed and heavy-tailed cases.

DOI : 10.5802/afst.1115
Samorodnitsky, Gennady 1

1 School of Operations Research and Industrial Engineering, and Department of Statistical Science, Cornell University, Ithaca, NY 14853.
@article{AFST_2006_6_15_1_107_0,
     author = {Samorodnitsky, Gennady},
     title = {Long memory and self-similar processes},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {107--123},
     publisher = {Universit\'e Paul Sabatier, Institut de math\'ematiques},
     address = {Toulouse},
     volume = {Ser. 6, 15},
     number = {1},
     year = {2006},
     doi = {10.5802/afst.1115},
     mrnumber = {2225749},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/afst.1115/}
}
TY  - JOUR
AU  - Samorodnitsky, Gennady
TI  - Long memory and self-similar processes
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2006
SP  - 107
EP  - 123
VL  - 15
IS  - 1
PB  - Université Paul Sabatier, Institut de mathématiques
PP  - Toulouse
UR  - http://www.numdam.org/articles/10.5802/afst.1115/
DO  - 10.5802/afst.1115
LA  - en
ID  - AFST_2006_6_15_1_107_0
ER  - 
%0 Journal Article
%A Samorodnitsky, Gennady
%T Long memory and self-similar processes
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2006
%P 107-123
%V 15
%N 1
%I Université Paul Sabatier, Institut de mathématiques
%C Toulouse
%U http://www.numdam.org/articles/10.5802/afst.1115/
%R 10.5802/afst.1115
%G en
%F AFST_2006_6_15_1_107_0
Samorodnitsky, Gennady. Long memory and self-similar processes. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 15 (2006) no. 1, pp. 107-123. doi : 10.5802/afst.1115. http://www.numdam.org/articles/10.5802/afst.1115/

[Astrauskas et al. (1991)] Astrauskas, A.; Levy, J.; Taqqu, M. S. The asymptotic dependence structure of the linear fractional Lévy motion, Lietuvos Matematikos Rinkinys (Lithuanian Mathematical Journal), Volume 31 (1991), pp. 1-28 | MR | Zbl

[Beran (1994)] Beran, J. Statistics for Long-Memory Processes, Chapman and Hall, New York, 1994 | MR | Zbl

[Cohen and Samorodnitsky (2005)] Cohen, S.; Samorodnitsky, G. Random rewards, Fractional Brownian local times and stable self-similar processes (2005) (Preprint)

[Embrechts and Maejima (2002)] Embrechts, P.; Maejima, M. Selfsimilar Processes, Princeton University Press, Princeton and Oxford, 2002 | MR | Zbl

[Hurst (1951)] Hurst, H. Long-term storage capacity of reservoirs, Transactions of the American Society of Civil Engineers, Volume 116 (1951), pp. 770-808

[Krengel (1985)] Krengel, U. Ergodic Theorems, de Gruyter Studies in Mathematics, 6, Walter de Gruyter & Co., Berlin, New York, 1985 | MR | Zbl

[Mandelbrot (1965)] Mandelbrot, B. Une classe de processus stochastiques homothétiques à soi; application à loi climatologique de H.E. Hurst, Comptes Rendus Acad. Sci. Paris, Volume 240 (1965), pp. 3274-3277 | MR | Zbl

[Mandelbrot (1983)] Mandelbrot, B. The Fractal Geometry of Nature, W.H. Freeman and Co., San Francisco, 1983 | MR | Zbl

[Mandelbrot and Van Ness (1968)] Mandelbrot, B.; Van Ness, J. Fractional Brownian motions, fractional noises and applications, SIAM Review, Volume 10 (1968), pp. 422-437 | MR | Zbl

[Mandelbrot and Wallis (1968)] Mandelbrot, B.; Wallis, J. Noah, Joseph and operational hydrology, Water Resources Research, Volume 4 (1968), pp. 909-918

[Mandelbrot and Wallis (1969)] Mandelbrot, B.; Wallis, J. Robustness of the rescaled range R/S in the measurement of noncyclic long-run statistical dependence, Water Resour. Res., Volume 5 (1969), pp. 967-988

[Mikosch and Samorodnitsky (2000)] Mikosch, T.; Samorodnitsky, G. Ruin probability with claims modeled by a stationary ergodic stable process, Annals of Probability, Volume 28 (2000), pp. 1814-1851 | MR | Zbl

[Nolan (1988)] Nolan, J. Path properties of index-β stable fields, Annals of Probability, Volume 16 (1988), pp. 1596-1607 | MR | Zbl

[Pipiras and Taqqu (2002a)] Pipiras, V.; Taqqu, M. S. Decomposition of self-similar stable mixing moving averages, Probability Theory and Related Fields, Volume 123 (2002), pp. 412-452 | MR | Zbl

[Pipiras and Taqqu (2002b)] Pipiras, V.; Taqqu, M. S. The structure of self-similar stable mixing moving averages, Annals of Probability, Volume 30 (2002), pp. 898-932 | MR | Zbl

[Resnick (1987)] Resnick, S. Extreme values, regular variation and point processes, Springer-Verlag, New York, 1987 | MR | Zbl

[Resnick et al. (1999] Resnick, S.; Samorodnitsky, G.; Xue, F. How misleading can sample ACF’s of stable MA’s be? (Very!), Annals of Applied Probability, Volume 9 (1999), pp. 797-817 | MR | Zbl

[Resnick et al. (2000)] Resnick, S.; Samorodnitsky, G.; Xue, F. Growth rates of sample covariances of stationary symmetric α-stable processes associated with null recurrent Markov chains, Stochastic Processes and Their Applications, Volume 85 (2000), pp. 321-339 | MR | Zbl

[Rosiński (1995)] Rosiński, J. On the structure of stationary stable processes, The Annals of Probability, Volume 23 (1995), pp. 1163-1187 | MR | Zbl

[Rosiński, Samorodnitsky (1996)] Rosiński, J.; Samorodnitsky, G. Classes of mixing stable processes, Bernoulli, Volume 2 (1996), p. 3655-378 | MR | Zbl

[Rosiński, Żak (1996)] Rosiński, J.; Zak, T. Simple conditions for mixing of infinitely divisible processes, Stochastic Processes and Their Applications, Volume 61 (1996), pp. 277-288 | MR | Zbl

[Samorodnitsky (2002)] Samorodnitsky, G. Long range dependence, heavy tails and rare events, MaPhySto (Lecture Notes), Centre for Mathematical Physics and Stochastics, Aarhus, 2002

[Samorodnitsky (2004)] Samorodnitsky, G. Extreme value theory, ergodic theory, and the boundary between short memory and long memory for stationary stable processes, Annals of Probability, Volume 32 (2004), pp. 1438-1468 | MR | Zbl

[Samorodnitsky (2005)] Samorodnitsky, G. Null flows, positive flows and the structure of stationary symmetric stable processes, Annals of Probability, Volume 33 (2005), pp. 1782-1803 | MR | Zbl

[Samorodnitsky and Taqqu (1990)] Samorodnitsky, G.; Taqqu, M. (1/α)-self-similar processes with stationary increments, Journal of Multivariate Analysis, Volume 35 (1990), pp. 308-313 | MR | Zbl

[Samorodnitsky and Taqqu (1994)] Samorodnitsky, G.; Taqqu, M. Stable Non-Gaussian Random Processes, Chapman and Hall, New York, 1994 | MR | Zbl

Cité par Sources :