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Construction of stationary self-similar generalized fields by random wavelet expansion. (English) Zbl 1008.60055
The wavelet expansion operator \(\Psi_{g,H}\) with mother wavelet \( g\) of index \(H\) is defined by \[ (\Psi_{g,H}\varphi)(u,v)= \int_{\mathbb R^n} e^{Hu}g(e^ux+v)\varphi(x) dx,\quad \varphi\in{\mathcal S}_k(\mathbb R^n). \] Let \(W=\sum_{i}\delta(u-u_i)\delta(v-v_i)\) and define the random image \(I = {\Psi_{g,H}}^* W\), where \((u_i, v_i)\) means the location and \(H\) the distance of \(i\)th figure. For the scaling operator \(S_t\varphi(x)=t^{-n}\varphi(t^{-1}x)\) and the translation operator \(T_v\varphi(x)=\varphi(x-v)\), it holds \[ (\Psi_{g,H}T_{v_0}S_{e^{u_0}}\varphi)(u,v)= e^{-Hu_0}(U_{u_0,v_0}\Psi_{g,H}\varphi)(u,v), \] where \((U_{u_0,v_0}f)(u,v)=f(u+u_0, v+e^u v_0).\) Three types of stationary self-similar random fields are constructed as the above “random images” with randomly located figures: (1) a multiple-stochastic integral over the products domain of scale and translate, (2) an iterated one over the scale domain, (3) an iterated one over the translate domain.

MSC:
60G18 Self-similar stochastic processes
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