Alternative forms of fractional Brownian motion☆
Introduction
We define (standard) Brownian motion , to be a real-valued Gaussian process withB(r) has independent increments, and indeed for integers j=0,±1,…, the sequence of incrementsare independent and identically distributed (i.i.d.) standard Gaussian variates. The sample paths of B(r) are almost all continuous, B(r) is (self-similar with index ); a process, , is said to be H−ss 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 , 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 . We can construct a probability space with random elements , of X having distributions μn respectively. If μn converges weakly to μ0 then we write ξn⇒ξ0. 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 of random variables. Letand define the polygonal line functionNotice 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 thatfor 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 for some K<∞ and all t,t1,t2 such that 0⩽t1⩽t⩽t2⩽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. Defineassuming thatUnder 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 haveA 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:where (t)+=max(t,0) andWe term BH(r) “Type I” fractional Brownian motion.
For Eq. (2.1) is interpreted asso that Eq. (1.1) is satisfied. For 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 asWe call WH(r) “Type II fractional Brownian motion”. Clearly WH(r) is again Gaussian with almost surely continuous sample paths, and for , , reduce to , , thus nesting B(r) to the same extent as does BH(r). Also we have
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
References (35)
- et al.
Distant long-range dependent sums and regression estimation
Stochastic Process. Appl.
(1995) The central limit theorem in time series regression
Stochastic Process. Appl.
(1979)- Abramowitz, M., Stegun, I., 1970. Handbook of Mathematical Functions. Dover, New...
- Akonom, J., Gourieroux, C., 1987. A functional central limit theorem for fractional processes. Technical Report #8801,...
- Billingsley, P., 1968. Convergence of Probability Measures. Wiley, New...
Martingale central limit theorems
Ann. Math. Statist.
(1971)- et al.
Spurious regression between I(1) processes with long memory errors
J. Time Ser. Anal.
(1997) - et al.
Inference for unstable long-memory processes with applications to fractional unit autoregressions
Ann. Statist.
(1995) The invariance principle for stationary processes
Theory Probab. Appl.
(1970)Gaussian and their subordinated self-similar random generalized fields
Ann. Probab.
(1979)
Non-central limit theorems for non-linear functions of Gaussian fields
Z. Wahrscheinlichkeitstheorie und verwandte Gebiete
On convergence to semi-stable Gaussian processes
Theory Probab. Appl.
A functional central limit theorem for weakly dependent sequences of random variables
Ann. Probab.
Invariance principles in statistics
Rev. Internat. Statist. Institute
Cited by (166)
Robust testing for explosive behavior with strongly dependent errors
2024, Journal of EconometricsEfficient tapered local Whittle estimation of multivariate fractional processes
2021, Journal of Statistical Planning and InferenceFixed-bandwidth CUSUM tests under long memory
2021, Econometrics and StatisticsCitation Excerpt :Section 4 concludes. Assumption 1 requires that the partial sum of the process converges to a fractional Brownian motion of type I. For a detailed discussion of fractional Brownian motions of type I and type II cf. (Marinucci and Robinson, 1999). The numerator and denominator of the limiting distribution are not independent random variables for d > 0 (see McElroy and Politis, 2012).
Aggregation of Seasonal Long-Memory Processes
2021, Econometrics and StatisticsWEAK CONVERGENCE to DERIVATIVES of FRACTIONAL BROWNIAN MOTION
2022, Econometric TheoryModeling trends and periodic components in geodetic time series: a unified approach
2024, Journal of Geodesy
- ☆
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.