“ La multil é ctica pretende cortar y modificar la
3.2. GRAFICACIÓN DE TRAYECTORIA DE EXCLUSIÓN ESPACIAL IDENTIFICANDO LA MULTILÉCTICA
3.2.2 Graficación de la trayectoria de exclusión espacial.
q≥ma2qq! < ∞, where N 0 ∼ N(0,1), we con- clude that lim k→∞supn≥1 ∞ X q=k+1 q!kft,n,qk2H⊗q →0.
4.3
Self-Similar Processes with Stationary Increments
In this section we introduce the concept of self-similar processes with stationary incre- ments. These will be important for understanding the noncentral limit theorem in follow- ing section because these are the only candidates for the limit of partial sum processes. This section is based off [GKS12].Definition 4.3.1. A stochastic process (Xt)t∈T is self-similar if there exists a H >0,
such that for all a >0, (Xat)t∈T d
= (aHX
t)t∈T. We call H the self-similarity parameter.
Definition 4.3.2. A stochastic process (Xt)t∈T has stationary increments if
(Xt+s−Xs)t∈T d
= (Xt−X0)t∈T
for all s∈T.
Consider the covariance function RH(t, s) := 1 2 |t| 2H +|s|2H − |t−s|2H . (4.3.1)
Proposition 4.3.3. For H ∈ (0,1), let X = (Xt)t∈T be a H-self-similar process with
stationary increments and E(X2
1) = 1. Then
(i) X0 = 0 almost surely,
(ii) E(Xt) = 0 for all t ∈T,
(iii) E(XsXt) =RH(s, t) for all s, t ∈T.
Proof. (i) If 0 ∈T, the self-similarity property implies that X0
d
=cHX
0 for all c >0, so
X0 = 0 almost surely.
(ii) Fix s ∈ T. Using self-similarity and the stationary increments property, for all nonzerot ∈T we have E(Xs) = E(Xt+s−Xt) = t+s t H −1 ! E(Xt).
For t = s, this implies that E(Xs) = 0 because H > 0. Therefore, E(Xt) = 0 for all
nonzerot ∈T. When t = 0, (i) implies E(X0) = 0.
(iii) The self-similarity property implies that Xt d
= tHX
1, so E(Xt2) = |t|2H. Using
the stationary increments property, we have E(XsXt) = 1 2 E(X 2 t) + E(X 2 s)−E (Xt−Xs)2
48 CHAPTER 4. CONVERGENCE OF PARTIAL SUM PROCESSES = 1 2 E(X 2 t) + E(X 2 s)−E (Xt−s) 2 = 1 2 |t| 2H +|s|2H − |t−s|2H , for all s, t ∈T.
If X is a H-self-similar process with stationary increments and finite variance, then using the triangle inequality we have that
2HE(Xt2)1/2 = E(X22t)1/2 ≤E((X2t−Xt)2)1/2+ E(Xt2)
1/2 = 2 E(X2
t)
1/2.
Thus, we have 2H ≤ 2, which implies that H ≤ 1. Note that when H = 1, using Proposition4.3.3 (iii), we have
E (Xt−tX1)2
= E(Xt2)−2tE(XtX1) +t2E(X12) = t 2−
t(t2+ 1−(1−t)2) +t2 = 0. Thus,Xt=tX1 almost surely. Since this is an uninteresting stochastic process, we have
will always assume that H ∈(0,1). Without loss of generality, we will also assume that E(X2
1) = 1.
Consider the increment process of X defined by Y = (Yn)n∈Z whereYn:=Xn+1−Xn.
We now introduce the notion of short range dependence and long range dependence. Definition 4.3.4. Let Y = (Yn)n∈Z be a covariance stationary stochastic process with
autocovariance function ρ(n) = E(Y0Yn).
(i) If 0<P
k∈Z|ρ(k)|<∞, then Y is said to exhibit short range dependence. (ii) IfP
k∈Z|ρ(k)|=∞, then Y is said to exhibit long range dependence.
Note that the case where P
k∈Z|ρ(k)|= 0 is excluded since in such a case, Yn and Ym are uncorrelated for n6=m. .
Proposition 4.3.5. Let Yn := Xn+1 −Xn for all n ∈ Z, where X is a H-self-similar
process with stationary increments. Then Y is weakly stationary with autocovariance function
ρ(n) = 1
2 |n+ 1|
2H −2|n|2H +|n−1|2H ∼H(2H−1)n2H−2.
Moreover, if H ∈ (0,1/2) then Y exhibits short range dependence, and if H ∈ (1/2,1)
then Y exhibits long range dependence. Proof. We have that
E(Yk+nYk) = E ((Xk+n+1−Xk+n) (Xk+1−Xk)) =RH(k+n+ 1, k+ 1)−RH(k+n+ 1, k)−RH(k+n, k+ 1) +RH(k+n, k) = 1 2 |n+ 1| 2H −2|n|2H +|n−1|2H ,
for all k, n ∈Z. Thereforeρ(n) = E(YnY0) = E(Yk+nYk), which implies that Y is weakly
4.3. SELF-SIMILAR PROCESSES WITH STATIONARY INCREMENTS 49 For n≥0, we can write ρ(n) =n2H−2L(n)/2, where
L(n) := n2 1 + 1 n 2H −2 + 1− 1 n 2H! .
Applying l’Hˆopital’s rule twice toL(n) shows thatL(n)→2H(2H−1) asn→ ∞. Thus, ρ(n)∼H(2H−1)n2H−2.
Thus, for some positive constant K, X n∈Z |ρ(n)|< K ∞ X n=0 n2H−2.
IfH ∈(0,1/2), this sum converges soY exhibits short range dependence. IfH∈(1/2,1), this sum diverges soY exhibits long range dependence.
The canonical example of a self-similar process with stationary increments is fractional Brownian motion. This class of processes also includes standard Brownian motion. Definition 4.3.6. Let H ∈ (0,1). A H-self-similar Gaussian process with stationary increments is known as fractional Brownian motion and H is called the Hurst pa- rameter.
From Proposition 4.3.3, it is clear that fractional Brownian motion is the only Gaus- sian process that is also a H-self-similar process with stationary increments, in the sense that every other such process is of the formσBH, for someσ >0. We will only work with the case whereσ= 1, which is also known as standard fractional Brownian motion. There are also non-Gaussian examples of self-similar processes with stationary increments, such as Hermite processes which will be introduced in later sections.
When H = 1/2, fractional Brownian motion reduces to standard Brownian motion on R. This immediately follows from the fact that centered Gaussian processes are determined by the covariance function which becomes E(Bt1/2Bs1/2) =t∧s, the covariance
function for standard Brownian motion.
The next result gives an alternative definition for fractional Brownian motion. Proposition 4.3.7. Let H ∈(0,1). A stochastic process is fractional Brownian motion if and only if it is a centered Gaussian process BH = (BtH)t∈R with covariance function
E(BtHBsH) = 1 2 |t|
2H +|s|2H − |t−s|2H
. (4.3.2)
Proof. One direction follows from Proposition4.3.3. For the other direction, suppose that BH is a centered Gaussian process with covariance functionR
H. Then sinceRH(at, as) =
a2HR
H(t, s) and centered Gaussian processes are determined by their covariance function,
it follows thatBH is similar. Similarly, it can be shown that E (BtH −BsH)2=|t−s|2H
so that it has stationary increments.
It is not immediately obvious that fractional Brownian motion exists. To establish existence, it suffices to show that RH is a valid covariance function, meaning that RH is
positive semi-definite. A proof of this fact can be found in [Nou12]. Next we show the continuity of the sample paths.
50 CHAPTER 4. CONVERGENCE OF PARTIAL SUM PROCESSES Proposition 4.3.8. Let BH be fractional Brownian motion with Hurst parameter H.
There exists a version of BH with locally H¨older continuous paths of order α < H.
Proof. Using the self-similarity and stationary increments property, we have E BtH −BsH q = E Bt−sH q = E B1H q |t−s|qH.
Then by the Kolmogorov continuity criteria, there exists a version of BH with H¨older
continuous paths of order α < (qH − 1)/q. Then letting q → ∞ gives the required result.
It is well-known that an appropriately normalized sum of independent and identi- cally distributed random variables can only converge in distribution to a stable random variable. In fact an analogous result holds in the case of convergence in finite dimension distribution for stochastic processes. Lamperti [Lam62] proved a theorem which says that the limit of any normalized partial sum must be a self-similarity. Moreover, it motivates the introduction of self-similar processes and their use in various applications.
Theorem 4.3.9. Suppose that (Xn)n∈Z is a stationary process and there exists a deter-
ministic sequence an → ∞ and a nonzero stochastic process (Zt)t≥0 such that
1 an bntc X k=1 Xk f f d → Zt.
Then (Zt)t≥0 is a continuous and H-self-similar process with stationary increments, for
some H > 0, and an = nHL(n), where L is a slowly varying function. Furthermore,
every self-similar process is the limit of such a partial sum process.
In the next section, we will see that the partial sum processes under the assumptions we have set out in Section 4.1 converges either to Brownian motion or the Hermite process based on the dependence structure of (Xn)n∈Z. This theorem explains why we
will assume that the autocovariance is in the form ρ(n)∼ nHL(n), where L is a slowly
varying function.