# Change of time

### stochastic processes derived by varying the rate of time's passage, which is more convenient than you'd think

TBD. Various notes on a.e. continuous monotonic changes of index in order to render a process “simple” in some sense.

In Warping and registration problems you try to align two or more processes. But here, the target is some “null”, basic process. This special case is often more computationally tractable or statistically well behaved.

## To explore

Lamperti representation for continuous state branching processes,

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

To understand:

## Subordinator

I’m going to follow Applebaum’s presentation (Appl09), which is brusque without being incomprehensible or unmotivated.

A subordinator is just a one-dimensional Lévy process which happens to be non-decreasing. i.e. A subordinator is an a.s. non-decreasing stochastic process $\Lambda(t), t \in \mathbb{R}$ with state space $\mathbb{R}$ such that

1. the increments are stationary

\begin{equation*} \Lambda(t)-\Lambda(s) \sim \Lambda(t-s), \,\forall t \geq s \end{equation*}
2. The increments are independent

\begin{equation*} \Lambda(t)-\Lambda(s)\perp \Lambda(t-s) \,\forall t \geq s \end{equation*}
1. The process is stochastically continuous
\begin{equation*} \lim_{t\searrow s} \mathbb{P} (|\Lambda(t)-\Lambda(s)|>\epsilon)=0, \,\forall \epsilon \gt 0 \end{equation*}
1. The increments are non-negative

\begin{equation*} \mathbb{P}(\Lambda(t)-\Lambda(s)\lt 0)=0\forall t \geq s \end{equation*}

The first three are standard Lévy process stuff. The last is only for subordinators.

Some definitions additionally require the increment distribution is a.s. positive, rather than non-negative, or that there are no atoms at zero in the increment distribution.

Curiously, upon giving that definition, many proceed to immediately assert that such a process is a model for a random change of time. This sounds not insane per se, but doesn’t have much in the way of narrative flow. TBD: explain why one would bother doing such an arbitrary thing as changing time in such a fashion.

Anyway I hope to use these to get a handle on time-changed residual tests and Lamperti representations. TBC.

The subordinator may be extended to multiple dimensions by requiring that each dimension is a.s. increasing. TBC.

## Point process transforms

As used in point process residual tests.

A summary in VeSc04 of the point process flavour:

Knight (Knig70) 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 (Meye71) extended Knight’s theorem to the case of point processes, showing that given a simple multivariate point process ${N_i ; i = 1, 2, . . . , n}$, 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 Brém72, Papa72, AaHo78, Kurt80 and BrNa88. Papangelou (Papa72) 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 MeNu86, Nair90 and Scho99. 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 BaPS01 and others. How does multivariate time work then?

## Refs

AaHo78
Aalen, O. O., & Hoem, J. M.(1978) Random time changes for multivariate counting processes. Scandinavian Actuarial Journal, 1978(2), 81–101. DOI.
Appl09
Applebaum, D. (2009) Lévy processes and stochastic calculus. (2nd ed.). Cambridge ; New York: Cambridge University Press
BTMH05
Baddeley, A., Turner, R., Møller, J., & Hazelton, M. (2005) Residual analysis for spatial point processes (with discussion). Journal of the Royal Statistical Society: Series B (Statistical Methodology), 67(5), 617–666. DOI.
BaPS01
Barndorff-Nielsen, O. E., Pedersen, J., & Sato, K.-I. (2001) Multivariate subordination, self-decomposability and stability. Advances in Applied Probability, 33(1), 160–187. DOI.
Brém72
Brémaud, P. (1972) A martingale approach to point processes. . University of California, Berkeley
BBVK02
Brown, E., Barbieri, R., Ventura, V., Kass, R., & Frank, L. (2002) The time-rescaling theorem and its application to neural spike train data analysis. Neural Computation, 14(2), 325–346. DOI.
BrNa88
Brown, T. C., & Nair, M. G.(1988) A Simple Proof of the Multivariate Random Time Change Theorem for Point Processes. Journal of Applied Probability, 25(1), 210–214. DOI.
CaCh06
Caballero, M. E., & Chaumont, L. (2006) Conditioned Stable Lévy Processes and the Lamperti Representation. Journal of Applied Probability, 43(4), 967–983.
ChPR13
Chaumont, L., Pantí, H., & Rivero, V. (2013) The Lamperti representation of real-valued self-similar Markov processes. Bernoulli, 19(5B), 2494–2523. DOI.
ChSt89
Cheng, R. C. H., & Stephens, M. A.(1989) A goodness-of-fit test using Moran’s statistic with estimated parameters. Biometrika, 76(2), 385–392. DOI.
Cox55
Cox, D. R.(1955) Some Statistical Methods Connected with Series of Events. Journal of the Royal Statistical Society. Series B (Methodological), 17(2), 129–164.
GiKM08
Giesecke, K., Kakavand, H., & Mousavi, M. (2008) Simulating point processes by intensity projection. In Simulation Conference, 2008. WSC 2008. Winter (pp. 560–568). DOI.
HaPB10
Haslinger, R., Pipa, G., & Brown, E. (2010) Discrete Time Rescaling Theorem: Determining Goodness of Fit for Discrete Time Statistical Models of Neural Spiking. Neural Computation, 22(10), 2477–2506. DOI.
Knig70
Knight, F. B.(1970) An Infinitesimal Decomposition for a Class of Markov Processes. The Annals of Mathematical Statistics, 41(5), 1510–1529. DOI.
Kurt80
Kurtz, T. G.(1980) Representations of Markov Processes as Multiparameter Time Changes. The Annals of Probability, 8(4), 682–715. DOI.
Lamp67
Lamperti, J. (1967) Continuous-state branching processes. Bull. Amer. Math. Soc, 73(3), 382–386.
Merzbach, E., & Nualart, D. (1986) A Characterization of the Spatial Poisson Process and Changing Time. The Annals of Probability, 14(4), 1380–1390. DOI.
Meye71
Meyer, P. A.(1971) Demonstration simplifiee d’un theoreme de Knight. In Séminaire de Probabilités V Université de Strasbourg (pp. 191–195). Springer Berlin Heidelberg DOI.
Nair90
Nair, M. G.(1990) Random Space Change for Multiparameter Point Processes. The Annals of Probability, 18(3), 1222–1231. DOI.
Papa72
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. DOI.
RaWu01
Rao, C. R., & Wu, Y. (2001) On model selection. In Institute of Mathematical Statistics Lecture Notes - Monograph Series (Vol. 38, pp. 1–57). Beachwood, OH: Institute of Mathematical Statistics
Scho99
Schoenberg, F. (1999) Transforming spatial point processes into Poisson processes. Stochastic Processes and Their Applications, 81(2), 155–164. DOI.
Scho02
Schoenberg, F. P.(2002) On Rescaled Poisson Processes and the Brownian Bridge. Annals of the Institute of Statistical Mathematics, 54(2), 445–457. DOI.
VeSc04
Vere-Jones, D., & Schoenberg, F. P.(2004) Rescaling Marked Point Processes. Australian & New Zealand Journal of Statistics, 46(1), 133–143. DOI.