Alternative forms of fractional Brownian motion

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Abstract

It is pointed out that two contradictory definitions of fractional Brownian motion are well-established, one prevailing in the probabilistic literature, the other in the econometric literature. Each is associated with a different definition of nonstationary fractional time series, arising in functional limit theorems based on such series. These various definitions have occasionally led to some confusion. The paper discusses the definitions and attempts a clarification.

Introduction

We define (standard) Brownian motion B(r),r∈R, to be a real-valued Gaussian process withEB(r)=0,r∈R,EB(r1)B(r2)=min(r1,r2),r1,r2⩾0.B(r) has independent increments, and indeed for integers j=0,±1,…, the sequence of incrementsb(j)=B(j+1)−B(j)are independent and identically distributed (i.i.d.) standard Gaussian variates. The sample paths of B(r) are almost all continuous, B(r) is 12−ss (self-similar with index 12); a process, X(r),r∈R, is said to be Hss if the finite-dimensional joint distributions of X(ar) are identical to those of aHX(r), for all a>0.

Functional central limit theorems (weak invariance principles) entail weak convergence of random variables to Brownian motion in a suitable metric space. A good introduction to this topic is given by Heyde (1981) with a particularly detailed treatment by Billingsley (1968). Denote by (X,δ) a complete separable metric space with metric δ and probability measures μi,i⩾0, on the Borel sets of X. We say that μn converges weakly to μ0 in (X,δ) if for every bounded continuous function f in X,limn→∞∫fdμn=∫fdμ0. We can construct a probability space {Ω,I,P} with random elements ξn,n⩾0, of X having distributions μn respectively. If μn converges weakly to μ0 then we write ξnξ0. Two metric spaces which feature considerably in the theory of weak convergence are C, the space of continuous functions on [0,1] endowed with the uniform topology, and D, the space of functions on [0,1] that are continuous on the right with finite left limits, endowed with the Skorohod (1956) J1 topology.

Consider partial sums of a sequence ut,t=1,2,… of random variables. LetS0=0,Sn=u1+⋯+un,n⩾1,and define the polygonal line functionSn(r)=S[nr]+(nr−[nr])u[nr]+1,0⩽r⩽1.Notice that Sn(r)∈C and S[nr]D. In case the ut are independent and identically distributed (i.i.d.) with zero mean and unit variance then Donsker's theorem (see Donsker (1951), Prohorov (1956), Skorohod (1956)) asserts thatn12xn(r)⇒B(r),asn→∞,0⩽r⩽1,for xn(r)=S[nr] or Sn(r). Clearly xn(r) can be centered and scaled to cope with alternative values of Eut and Var(ut). The proof of (1.6) on both C and D entails establishing convergence of finite-dimensional distributions and tightness, for which a sufficient condition (see Billingsley (1968), p. 128) is that for some γ>0,α>1E{|xn(t)−xn(t1)|γ|xn(t2)−xn(t)|γ}⩽K|t2−t1|α,for some K<∞ and all t,t1,t2 such that 0⩽t1tt2⩽1.

The convergence (1.6) has been extended to many classes of dependent random variables. Brown (1971) and subsequent authors considered martingale difference ut. Much literature has allowed for autocorrelation in ut. Suppose now that ut is covariance stationary and has (without loss of generality) mean zero, and lag-j autocovariance γ(j)=Eutut+j. Definef(0)=1j=−∞γ(j),assuming that0<f(0)<∞.Under a variety of conditions such that Eq. (1.7) holds, for example with ut a linear process (e.g. Hannan (1979), Phillips and Solo (1992)), various mixing or functions-of-mixing processes (e.g. McLeish (1977), Herrndorf (1984), Wooldridge and White (1988)), or with vector valued ut (e.g. Phillips and Durlauf (1986)), we have{2πf(0)n}−1/2xn(r)⇒B(r),asn→∞,0⩽r⩽1.A leading motivation for much of this work has been its application to limit distribution theory for statistics that arise when investigating the possibility that an observed time series has a unit root, against the alternative that it has autoregressive stationarity or explosivity; often application of functional limit theorems of the form (1.6) or (1.8), and the continuous mapping theorem (see Billingsley (1968)), leads to limit distributions that are nonstandard functionals of Brownian motion.

In case ut has absolutely continuous spectral distribution, f(0) is the ordinate of the spectral density function, f(λ), at λ=0. The property (1.7) can be viewed as a mild form of short-range dependence condition (while it is also possible to focus on behaviour at alternative frequencies λ). Some of the work establishing Eq. (1.8) has allowed for forms of nonstationarity requiring f(0) to have a broader interpretation, but nevertheless (1.7) still conveys a sense of weak dependence. While many standard time series models for ut, including stationary and invertible mixed autoregressive moving averages, satisfy Eq. (1.7), there has been considerable interest in ones which do not, and these lead to an interest in forms of fractional Brownian motion.

Section snippets

“Type I” fractional Brownian motion

Mandelbrot and Van Ness (1968) introduced fractional Brownian motion BH(r) which we present in the slightly modified form of Samorodnitsky and Taqqu (1994) (see also Taqqu (1979)), for 0<H<1:BH(r)=1A(H)R[{(r−s)+}H−1/2−{(−s)+}H−1/2]dB(s),r∈R,where (t)+=max(t,0) andA(H)=12H+D(H)12,D(H)=0{(1+s)H−1/2−sH−1/2}2ds.We term BH(r) “Type I” fractional Brownian motion.

For H=12 Eq. (2.1) is interpreted asBH(r)=0rdB(s),r⩾0,BH(r)=−r0dB(s),r<0,so that Eq. (1.1) is satisfied. For H≠12BH(r) can be formally

“Type II” fractional Brownian motion

Levy (1953), Mandelbrot and Van Ness (1968) mention an alternative definition of fractional Brownian motion, as a Holmgren–Riemann–Liouville fractional integral, which we write asWH(r)=(2H)120r(r−s)H−1/2dB(s),r⩾0,WH(r)=−(2H)12r0(s−r)H−1/2dB(s),r<0.We call WH(r) “Type II fractional Brownian motion”. Clearly WH(r) is again Gaussian with almost surely continuous sample paths, and for H=12, , reduce to , , thus nesting B(r) to the same extent as does BH(r). Also we haveEWH(r)=0,EWH2(r)=|r|2H,r∈R,

Conclusions

This paper has discussed two alternative definitions of nonstationary fractional processes which have arisen in the literature, the first prompted by Eq. (3.11), the other given in Eq. (3.8). It was pointed out that associated functional central limit theorems lead to different types of fractional integrals, which we labelled “Type I” and “Type II” fractional Brownian motion. This distinction has sometimes been overlooked in the econometric literature, leading to the definition (3.11) being

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    Research supported by ESRC Grant R000235892. The second author's research was also supported by a Leverhulme Trust Personal Research Professorship. We are grateful to two referees for comments that improved the presentation; the first author has also benefited from a useul discussion with P. Baldi.

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