To Örebro University

Fractional Brownian motion (fBM) belongs to the class of long-range dependent systems with self-similarity property and has been widely used in different applications. Mathematically speaking, the scaled fBm is fully characterised by its scaling parameter σ > 0 and Hurst parameter H ∈ (0, 1). But in spite of many obvious advantages, for many real-life data with long-range dependence, the classical fBM with Gaussian property cannot be considered an appropriate model. For example the classical fBM cannot model the real time series with apparent constant time periods (called also trapping events), which are often observed in data sets recorded within various fields. One of the possible solutions is the time-changed fBM BHLt with α-stable L ´evy subordinator (Lt)t≥0.

We construct consistent estimators of the parameters (σ, α, H) for the time-changed fBM Xt =BHLt . Our approach is based on the limit theory for stationary increments of a linear fractional stable motion [1]. We use these techniques, combine negative power variation statistics and their empirical expectations and covariances to obtain consistent estimates of (σ, α, H). We show that Xt is a symmetric H/α-stable L ´evy process and for p < α we deduce the law of large numbers. The above law of large numbers immediately gives a consistent estimator of the self-similarity parameter H/α of Xt. In order to estimate the other parameters of the model we use the some identities, which has been shown in [2]. Finally we present the statistical inference and prove some weak limit theorems for the all parameters (σ, α, H) using classical delta-method.

References:

[1] Mazur, S., Otryakhin D. and Podolskij M., Estimation of the linear fractional stable motion, Bernoulli, 26(1), (2020) 226-252.

[2] Dang, T.T.N., Istas, J., Estimation of the Hurst and the stability indices of a H-self-similar stable process, Electronic Journal of Statistics, 11(2), (2018) 4103-4150.