Properties of trajectories of a multifractional Rosenblatt process. (English. Russian original) Zbl 1248.60045

Shevchenko, Georgiĭ

Theory Probab. Math. Stat. 83, 163-173 (2011); translation from Teor. Jmovirn. Mat. Stat. 83, 138-147 (2010).

The multifractional Rosenblatt process with Hurst function \(H\) is defined by \[ X_t = \iint_{\mathbb{R}^2} \int_0^t (s-x)_+^{H(t)/2-1} (s-y)_+^{H(t)/2-1} ds \, W(dx) W(dy), \quad 0 \leq t \leq T, \] where \(\iint f(x,y) W(dx) W(dy)\) denotes double Wiener integral. It is assumed that \(H: [0, T] \to (\frac12, 1)\) is a Hölder continuous function with exponent \(\gamma > \max_{0 \leq t \leq T} H(t)\). This process \((X_t)\) has long range dependence, with non-Gaussian, “moderate” tail distribution. (A “moderate” tail distribution is heavier than normal, but lighter than other stable distributions.) The Hurst function is useful to model local properties that vary with time.
The first main result of the paper is that \((X_t)\) is a.s. continuous, in fact, Hölder continuous with exponent \(\gamma_X < \min_{a \leq t \leq b}H(t)\) on any interval \([a,b] \subset [0,T]\). The second main result is that \((X_t)\) is strongly localizable at every point \(t_0 \in [0,T]\), that is, the distribution of the process \(\left\{\delta^{-H(t_0)} \left(X_{t_0+\delta t} - X_{t_0}\right), t \geq 0 \right\}\) converges to the distribution of a Rosenblatt process with constant Hurst parameter \(H(t_0)\) in \(C[0,T]\) as \(\delta \to 0^+\). The third main result is that \((X_t)\) has a square integrable local time.

Reviewer: Tamas Szabados (Budapest)

MSC:

60G22	Fractional processes, including fractional Brownian motion
60J55	Local time and additive functionals
60B10	Convergence of probability measures
60G17	Sample path properties

Keywords:

Rosenblatt process; path properties; local time; localizability

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