Abstract
Infinitely divisible stochastic processes form a broad family whose structure is reasonably well understood. A stochastic process \({\bigl (X(t),\,t \in T\bigr )}\) is said to be infinitely divisible if for every n = 1, 2, …, there is a stochastic process \(\bigl( Y(t),\, t\in T\bigr)\) such that
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Bibliography
P. Billingsley, Probability and Measure, 3rd edn. (Wiley, New York, 1995)
S. Cohen, G. Samorodnitsky, Random rewards, Fractional Brownian local times and stable self-similar processes. Ann. Appl. Probab. 16, 1432–1461 (2006)
L. Decreusefond, A. Üstünel, Stochastic analysis of the Fractional Brownian motion. Potential Anal. 10, 177–214 (1999)
C. Dombry, N. Guillotin-Plantard, Discrete approximation of a stable self-similar stationary increments process. Bernoulli 15, 195–222 (2009)
T. Ferguson, M. Klass, A representation theorem of independent increment processes without Gaussian component. Ann. Math. Stat. 43, 1634–1643 (1972)
Z. Kabluchko, S. Stoev, Stochastic integral representations and classification of sum- and max-infinitely divisible processes. Bernoulli 22, 107–142 (2016)
A. Kolmogorov, Wienersche Spiralen und einige andere interessante kurven in Hilbertschen raum. C.R. (Doklady) Acad. Sci. USSR (N.S.) 26, 115–118 (1940)
S. Kwapień, W. Woyczyński, Random Series and Stochastic Integrals: Single and Multiple (Birkhäuser, Boston, 1992)
R. LePage, M. Woodroofe, J. Zinn, Convergence of a stable distribution via order statistics. Ann. Prob. 9, 624–632 (1981)
B. Mandelbrot, J. Van Ness, Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10, 422–437 (1968)
B. Rajput, J. Rosiński, Spectral representations of infinitely divisible processes. Probab. Theory Relat. Fields 82, 451–488 (1989)
J. Rosiński, On series representation of infinitely divisible random vectors. Ann. Probab. 18, 405–430 (1990)
J. Rosiński, On the structure of stationary stable processes. Ann. Probab. 23, 1163–1187 (1995)
J. Rosiński, Lévy and Related Jump-type Infinitely Divisible Processes. Lecture Notes, Cornell University (2007)
J. Rosiński, G. Samorodnitsky, Infinitely Divisible Stochastic Processes (2016). In preparation
J. Rosiński, T. Żak, The equivalence of ergodicity and weak mixing for infinitely divisible processes. J. Theor. Probab. 10, 73–86 (1997)
E. Roy, Ergodic properties of Poissonian ID processes. Ann. Probab. 35, 551–576 (2007)
G. Samorodnitsky, Null flows, positive flows and the structure of stationary symmetric stable processes. Ann. Probab. 33, 1782–1803 (2005)
G. Samorodnitsky, M. Taqqu, Stable Non-Gaussian Random Processes (Chapman and Hall, New York, 1994)
K. Sato, Lévy Processes and Infinitely Divisible Distributions (Cambridge University Press, Cambridge, 1999)
W. Vervaat, On a stochastic difference equation and a representation of non-negative infinitely divisible random variables. Adv. Appl. Probab. 11, 750–783 (1979)
A. Yaglom, Correlation theory of processes with stationary random increments of order n. Mat. Sb. 37, 141–196 (1955); English translation in Am. Math. Soc. Translations Ser. 2 8, 87–141 (1958)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Samorodnitsky, G. (2016). Infinitely Divisible Processes. In: Stochastic Processes and Long Range Dependence. Springer Series in Operations Research and Financial Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-45575-4_3
Download citation
DOI: https://doi.org/10.1007/978-3-319-45575-4_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-45574-7
Online ISBN: 978-3-319-45575-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)