# Selfsimilar Processes

Paul Embrechts
Makoto Maejima
Pages: 152
https://www.jstor.org/stable/j.ctt7t1hk

1. Front Matter
(pp. i-vi)
(pp. vii-viii)
3. Preface
(pp. ix-xii)
Paul Embrechts and Makoto Maejima
4. Chapter One Introduction
(pp. 1-12)

Selfsimilar processes are stochastic processes that are invariant in distribution under suitable scaling of time and space (see Definition 1.1.1).

It is well known that Brownian motion is selfsimilar (see Theorem 1.2.1). Fractional Brownian motion (see Section 1.3 and Chapter 4), which is a Gaussian selfsimilar process with stationary increments, was first discussed by Kolmogorov [Ko140]. The first paper giving a rigorous treatment of general selfsimilar processes is due to Lamperti [Lam62], where a fundamental limit theorem was proved (see Section 2.1). Later, the study of non-Gaussian selfsimilar processes with stationary increments was initiated by Taqqu [Taq75], who extended a...

5. Chapter Two Some Historical Background
(pp. 13-18)

One reason for the fact that selfsimilar processes are important in probability theory is their connection to limit theorems. This was first observed by Lamperti [Lam62]. In the following, we say that a random variable isnondegenerateif it is not constant almost surely. The class of slowly varying functions, defined below, will be needed in the formulation of the next theorem.

Definition 2.1.1A positive, measurable function L is called slowly varying if for all x> 0,

${\lim }\limits_{t \to \infty } {{L\left({tx} \right)} \over {L\left(t \right)}} = 1.$

A positive, measurable function f is called regularly varying of index α∈ ℝ,if f(x) = xαL(x), where L is...

6. Chapter Three Selfsimilar Processes with Stationary Increments
(pp. 19-42)

Stable Lévy processes (including Brownian motion) are the only selfsimilar processes with independent and stationary increments. Consequently, one is interested in selfsimilar processes with just stationary increments or just independent increments. In this chapter, we discuss selfsimilar processes with stationary increments. Selfsimilar processes with independent increments are discussed in Chapter 5.

When {X(t),t≥ 0} isH-selfsimilar with stationary increments, we call itH-ss, si, for short.

The following results give some basic formulas and estimates on moments and the exponent of selfsimilarity.

Theorem 3.1.1Suppose that {X(t)} is H-ss, si, H >0and that X(t) is nondegenerate for each...

7. Chapter Four Fractional Brownian Motion
(pp. 43-56)

Although we have mentioned fractional Brownian motion in Section 1.3, we discuss this important process in more detail in this chapter.

When two stochastic processes {X(t)} and {Y(t)} satisfyP{X(t) =Y(t)} = 1 for allt≥ 0, we say that one is a modification of the other. It is well known that Brownian motion has a modification, the sample paths of which are continuous almost surely, but sample paths of any modification are nowhere differentiable. As it turns out, these facts remain true for fractional Brownian motion.

We say that a stochastic process {X(t), 0 ≤t≤...

8. Chapter Five Selfsimilar Processes with Independent Increments
(pp. 57-62)

In this chapter, we discuss selfsimilar processes with independent increments but not necessarily having stationary increments. We call {X(t),t≥ 0} which isH-selfsimilar with independent increments asH-ss, ii. Selfsimilar processes discussed in this chapter are ℝd-valued,d≥ 1.

As already seen in Theorem 1.4.2, if selfsimilar processes have independent and stationary increments, then their distributions are stable. Hence, the class of their marginal distributions is determined. However, this is no longer the case for selfsimilar processes without independent and stationary increments. Actually, as mentioned in [BarPer99], there is no simple characterization of the possible families of...

9. Chapter Six Sample Path Properties of Selfsimilar Stable Processes with Stationary Increments
(pp. 63-66)

When two stochastic processes {X(t)} and {Y(t)} satisfy$\{ X(t)\} \mathop = \limits^d \{ Y(t)\}$, we say that one is a version of the other. If they are modifications of each other, they are also versions of each other.

Typical sample path properties examined in the literature can be summarized as follows:

Property I: There exists a version with continuous sample paths.

Property II: Property I does not hold, but there is a version whose sample paths are right-continuous and have left limits (i.e. are so-called càdlàg).

Property III: Any version of the process is nowhere bounded, i.e. unbounded on every fine interval.

The processes...

10. Chapter Seven Simulation of Selfsimilar Processes
(pp. 67-80)

There are numerous textbooks on simulation. For our purposes, an excellent text is Ripley [Rip87], at the more introductory level Morgan [Mor84] and Ross [Ros91]. See also Rubinstein and Melamed [RubMe198]. For the simulation of α-stable processes, see Janicki and Weron [JanWer94]. The presentation of this chapter is mainly based on Asmussen [Asm99]. The latter also contains an excellent list of references related to the simulation of rare events. A short discussion on the simulation of long-memory processes is to be found in Beran [Ber94]. As so often in this field, Benoit Mandelbrot was involved very early on; see for...

11. Chapter Eight Statistical Estimation
(pp. 81-92)

In this chapter we briefly discuss some methods to detect the presence of a long-range dependence structure in a data set. In particular, we present ways to estimate the exponentH.More details and further estimation methods can be found in [Ber94], a text we follow closely. An excellent review of various existing methods, together with examples can be found on Murad Taqqu’s website: http://math.bu.edu/people/murad under the heading “Statistical methods for long-range dependence”.

The methods described are mainly useful as simple diagnostic tools. Since the results in this section are often difficult to interpret, for statistical inference, more efficient methods...

12. Chapter Nine Extensions
(pp. 93-100)

The definition of selfsimilarity in ℝdcan be extended to allow for scaling by linear operators on ℝdas follows [LahRoh82, HudMas82].

Definition 9.1.1And-valued stochastic process{X(t)}is called operator selfsimilar”if

(a) it is stochastically continuous at each t≥ 0,and

(b) for every a> 0there exists a linear operator B(a) on ℝdsuch that

$\matrix{ {\{ X(at)\} = \limits^d \{ B(a)X(t)\}.} \quad\quad\quad (9.1.1)$

WhenDis a linear operator on ℝd,aDwill denote the linear operator

$\exp \{ (\log a)D\} = \sum\limits_{j = 0}^\infty {{1 \over {j!}}} (\log a)^j D^j$

fora> 0. We say that an ℝd-valued stochastic process {X(t)} isproperif for allt> 0,L(X(t)) is full, namely it is not...

13. References
(pp. 101-108)
14. Index
(pp. 109-111)