In two weeks I will be attending the STM2016 workshop in Tokyo on spatial-temporal modeling. During the workshop I will be presenting some work with Nourddine Azzaoui and Gareth Peters on the simulation of a general class of non-stationary -stable processes. In this post, I want to provide some background to this work.
Processes with Independent Increments
To understand the class of processes we have been studying, it is helpful to begin with a review of stochastic processes with independent increments. Recall that a continuous parameter stochastic process is the process where is an interval of . A process with independent increments is then a process with random variables that for , the differences
are mutually independent and . Moreover, if the distribution of depends only on , a process with independent increments is said to have strict sense stationary increments.
To ensure that a given process with independent increments exists, we can apply the Kolmogorov existence theorem. The key thing we need to check is consistency. To see that we run into no problems, suppose that and set , and . If the process has independent increments then and are mutually independent. Moreover, since it also follows that is the sum of two independent random variables, as required.
Note that it is not necessary to assign a distribution to the ‘s and consider instead the process , where . A key example of a process with independent increments is Brownian motion.
Example: (Brownian Motion) Suppose that is real and normally distributed, with
with . The interval is set to and .
Symmetric -Stable Processes with Independent Increments
Consider the process , with finite dimensional distributions corresponding to the distributions of the random vectors , , . is said to be an -stable process if for any there is a number such that the random vectors satisfy the stability condition equality (in distribution)
where are independent copies of . Gaussian processes correspond to the case . For further details see Stable Non-Gaussian Random Processes.
In general, symmetric -stable random vectors () do not have closed-form joint probability density functions. However, they do admit a closed-form characteristic function, which is given by
The measure is finite, symmetric and uniquely determined on the Borel subsets of the unit sphere .
For , symmetric -stable are qualitatively different to Gaussian processes. In particular, they are not characterized by their mean and covariance functions. In fact, the covariance is infinite, which is due to the heavy tail property of -stable random variables.
In the case of symmetric -stable processes (), a natural notion of dependence is the covariation. Let and , denote . For a pair of symmetric -stable random variables, denoted by , the covariation is defined as
A general method to represent stochastic processes is via a stochastic integral, which is defined by the Riemann-Stieltjes integral
where is a step function, is an appropriate stochastic process, and is a random measure (i.e., a measure-valued random element). In some cases, the class of functions can be extended using completion arguments. However, for our purposes it is sufficient to stick to step functions.
Stochastic integral representations play a key role in the theory of symmetric -stable processes. In this case, the process is a symmetric -stable process. Sufficient conditions for the stochastic integral representation to hold are that and the following regularity conditions hold:
- The process possesses weak right limits.
- For any linear combination of elements of , the map is of bounded variation on .
Aside from being a key notion of dependence, the covariation plays an important role in defining a norm for symmetric -stable stochastic integrals. Denote as the linear space of step functions . Then, the map is a norm.
Symmetric -stable processes with independent increments play a particularly important role. Here, the random measure is defined as
Denote the weak limit of as . Cambanis and Miller have shown that , where corresponds to the scale parameter of the random variable . This result provides a basis for extending the space of functions to a wider class of functions (denoted by in Cambanis and Miller).
It is also possible to represent the covariation in terms of the function . Consider the two processes and , where and are step functions. Cambanis and Miller have shown Why is this important? In general, consider an -dimensional skeleton of an -stable process with . In this case, the characteristic function of can be written as
I’ll leave it there for now. In our current work, we are considering a class of symmetric -stable processes with non-independent increments.We are using a generalization of these results for processes with independent increments to provide an explicit representation of the process which can be used to develop simulation and estimation techniques.