Random change of time

Stochastic processes derived by varying the rate of time’s passage, which is more convenient than I imagined

🏗 Various notes on a.e. continuous monotonic random changes of index in order build new processes.

In Warping and registration problems you try to align two or more processes; this can sometimes be an alignment problem, but not necessarily.

To explore

Lamperti representation for continuous state branching processes,

Ogata’s time rescaling: Intensity estimation for point processes uses this as a statistical test.

Relation to e.g. martingale transforms.


A subordinator is a non-decreasing Lévy Process taking values on the reals. AFAICS this is precisely a Gamma process. Curiously, upon giving that definition, many proceed to immediately assert that such a process is a model for a random change of time. This is not insane per se, but doesn’t have much narrative momentum, as a Gamma process can model a bunch of other things than time, and it is a weird specialisation in a field that normally tends to excessive generality.

🏗 I should explain why one would bother doing such an arbitrary thing as changing time though; Basically it is because a time-changed Lévy process is still a Lévy process.

Point process rate transform

As used in point process residual goodness of fit tests.

A summary in (Vere-Jones and Schoenberg 2004):

Knight (Knight 1970) showed that for any orthogonal sequence of continuous local martingales, by rescaling time for each via its associated predictable process, we form a multivariate sequence of independent standard Brownian motions. Then Meyer (Meyer 1971) extended Knight’s theorem to the case of point processes, showing that given a simple multivariate point process \({N_i ; i = 1, 2, \dots, n}\), the multivariate point process obtained by rescaling each \(N_i\) according to its compensator is a sequence of independent Poisson processes, each having intensity 1. Since then, alternative proofs and variations of this result have been given by (Brown and Nair 1988; Kurtz 1980; Papangelou 1972; Aalen and Hoem 1978; Brémaud 1972) Papangelou (Papangelou 1972) gave the following interpretation in the univariate case:

Roughly, moving in \([0, \infty)\) so as to meet expected future points at a rate of one per time unit (given at each instant complete knowledge of the past), we meet them at the times of a Poisson process. […]

Generalizations of Meyer’s result to point processes on \(\mathbb{R}^d\) have been established by (Nair 1990; Merzbach and Nualart 1986; F. Schoenberg 1999). In each case, the method used has been to focus on one dimension of the point process, and rescale each point along that dimension according to the conditional intensity.

Going Multivariate

As seen in, e.g. (Ole E. Barndorff-Nielsen, Pedersen, and Sato 2001). How does multivariate time work then?


Aalen, Odd O., and Jan M. Hoem. 1978. Random Time Changes for Multivariate Counting Processes.” Scandinavian Actuarial Journal 1978 (2): 81–101.
Applebaum, David. 2009. Lévy Processes and Stochastic Calculus. 2nd ed. Cambridge Studies in Advanced Mathematics 116. Cambridge ; New York: Cambridge University Press.
Baddeley, A., R. Turner, J. Møller, and M. Hazelton. 2005. Residual Analysis for Spatial Point Processes (with Discussion).” Journal of the Royal Statistical Society: Series B (Statistical Methodology) 67 (5): 617–66.
Barndorff-Nielsen, Ole E., Jan Pedersen, and Ken-Iti Sato. 2001. Multivariate Subordination, Self-Decomposability and Stability.” Advances in Applied Probability 33 (1): 160–87.
Barndorff-Nielsen, Ole E, and Albert Shiryaev. 2010. Change of Time and Change of Measure. Vol. 13. Advanced Series on Statistical Science & Applied Probability. WORLD SCIENTIFIC.
Brémaud, Pierre. 1972. “A Martingale Approach to Point Processes.” University of California, Berkeley.
Brown, Timothy C., and M. Gopalan Nair. 1988. A Simple Proof of the Multivariate Random Time Change Theorem for Point Processes.” Journal of Applied Probability 25 (1): 210–14.
Caballero, M. E., and L. Chaumont. 2006. Conditioned Stable Lévy Processes and the Lamperti Representation.” Journal of Applied Probability 43 (4): 967–83.
Chaumont, Loïc, Henry Pantí, and Víctor Rivero. 2013. The Lamperti Representation of Real-Valued Self-Similar Markov Processes.” Bernoulli 19 (5B): 2494–2523.
Cheng, R. C. H., and M. A. Stephens. 1989. A Goodness-of-Fit Test Using Moran’s Statistic with Estimated Parameters.” Biometrika 76 (2): 385–92.
Çinlar, Erhan. 1980. On a Generalization of Gamma Processes.” Journal of Applied Probability 17 (2): 467–80.
Cox, D. R. 1955. Some Statistical Methods Connected with Series of Events.” Journal of the Royal Statistical Society: Series B (Methodological) 17 (2): 129–57.
Giesecke, K., H. Kakavand, and M. Mousavi. 2008. Simulating Point Processes by Intensity Projection.” In Simulation Conference, 2008. WSC 2008. Winter, 560–68.
Haslinger, Robert, Gordon Pipa, and Emery Brown. 2010. Discrete Time Rescaling Theorem: Determining Goodness of Fit for Discrete Time Statistical Models of Neural Spiking.” Neural Computation 22 (10): 2477–2506.
Knight, Frank B. 1970. An Infinitesimal Decomposition for a Class of Markov Processes.” The Annals of Mathematical Statistics 41 (5): 1510–29.
Kurtz, Thomas G. 1980. Representations of Markov Processes as Multiparameter Time Changes.” The Annals of Probability 8 (4): 682–715.
Lamperti, John. 1958. On the Isometries of Certain Function-Spaces.” Pacific J. Math 8 (3): 459–66.
———. 1967. Continuous-State Branching Processes.” Bull. Amer. Math. Soc 73 (3): 382–86.
———. 1972. Semi-Stable Markov Processes. I.” Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte Gebiete 22 (3): 205–25.
Leonenko, Nikolai N, Mark M Meerschaert, René L Schilling, and Alla Sikorskii. 2014. Correlation Structure of Time-Changed Lévy Processes.” Communications in Applied and Industrial Mathematics 6 (1).
Madan, Dilip B., and Eugene Seneta. 1990. The Variance Gamma (V.G.) Model for Share Market Returns.” The Journal of Business 63 (4): 511–24.
Merzbach, Ely, and David Nualart. 1986. A Characterization of the Spatial Poisson Process and Changing Time.” The Annals of Probability 14 (4): 1380–90.
Meyer, P. A. 1971. Demonstration simplifiee d’un theoreme de Knight.” In Séminaire de Probabilités V Université de Strasbourg, 191–95. Lecture Notes in Mathematics. Springer Berlin Heidelberg.
Nair, M. Gopalan. 1990. Random Space Change for Multiparameter Point Processes.” The Annals of Probability 18 (3): 1222–31.
Papangelou, F. 1972. Integrability of Expected Increments of Point Processes and a Related Random Change of Scale.” Transactions of the American Mathematical Society 165: 483–506.
Rao, C. R., and Y. Wu. 2001. On Model Selection.” In Institute of Mathematical Statistics Lecture Notes - Monograph Series, 38:1–57. Beachwood, OH: Institute of Mathematical Statistics.
Schoenberg, Frederic. 1999. Transforming Spatial Point Processes into Poisson Processes.” Stochastic Processes and Their Applications 81 (2): 155–64.
Schoenberg, Frederic Paik. 2002. On Rescaled Poisson Processes and the Brownian Bridge.” Annals of the Institute of Statistical Mathematics 54 (2): 445–57.
Seneta, Eugene. 2007. The Early Years of the Variance-Gamma Process.” In Advances in Mathematical Finance, edited by Michael C. Fu, Robert A. Jarrow, Ju-Yi J. Yen, and Robert J. Elliott, 3–19. Applied and Numerical Harmonic Analysis. Boston, MA: Birkhäuser.
Vere-Jones, David, and Frederic Paik Schoenberg. 2004. Rescaling Marked Point Processes.” Australian & New Zealand Journal of Statistics 46 (1): 133–43.

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