Iterated random pulse processes and their spectral properties.

*(English)*Zbl 1080.60051The author introduces a new class of scaling pulse processes in \(\mathbb R^n\), called iterated random pulse (IRP) processes and studies their second-moment properties at finite and infinite resolutions. The models differ from fractal sum of pulse processes of B. B. Mandelbrot [in: Lévy flights and related topics in physics. Lect. Notes Phys. 450, 110–123 (1995; Zbl 0829.60032)]. Models with hierarchical clustering of the pulses arise naturally in various application fields [see D. Veneziano, G. E. Moglen, P. Furcolo and V. Iacobellis, Water Resour. Res. 36, 1143 (2000)].

The basic idea of IRP processes is as follows. Let \(Y(\Omega)\), \(\Omega\subset \mathbb R^n\), be the random measure of interest. When it exists, the measure density is denoted by \(X(t)\), \(t\in \mathbb R^n\). Like in other multi-scale models, \(Y(\Omega)\) is specified recursively as \(Y_j(\Omega)\), \(j=0,1,\dots\), where \(j\) denotes the resolution level. For any finite \(j\), the measure density \(X(t)\) exists and \(Y_j(\Omega) = \int_{\Omega} X_j(t)\,dt\). Hence, \(Y(\Omega)\) can be characterized through an initial random field \(X_0(t)\) and conditional random fields \(\{X_j(t)| X_k(t)| , k=0,1,\dots, j-1\}\) for \(j=1,2,\dots\). In general, the pulses that define \(X_j(t)\) may be random field and may be dependent among themselves, while the pulse locations may be clustered or regular in quite general ways. The paper focuses mainly on homogeneous IRP processes in which the pulses at any given level \(j\) are independent copies of a random function \(f_j(t)\) and their locations are described by a doubly stochastic Poisson process whose intensity \(\lambda_j(t)\) depends on the process at \(X_{j-1}(t)\). In the simplest case, the random functions \(f_j(t)\) are related as \(f_j(t) = r^{-\gamma} f_{j-1}(r t)\), where \(r>1\) is a support contraction factor, \(\gamma > -n\) is an affinity parameter, and \(n\) is the dimension of physical space.

The paper is organized as follows. Section 2 describes in greater detail two classes of IRP processes, which differ in the way the intensity \(\lambda_j(t)\) depends on \(X_{j-1}(t)\). In Section 3 recursive relations for the power spectral density \(S_j(\omega)\) of \(X_j(t)\) are obtained, whereas Section 4 analyzes the existence and behavior of the limit \(S(\omega) =\lim_{j\to \infty} S_j(\omega)\) for a class of IRP processes. Section 5 extends the models by allowing anisotropic affine relations among the pulses \(f_j(t)\) at different resolutions levels and non-homogeneous initial processes \(X_0(t)\). Other extensions are mentioned in the concluding section.

The basic idea of IRP processes is as follows. Let \(Y(\Omega)\), \(\Omega\subset \mathbb R^n\), be the random measure of interest. When it exists, the measure density is denoted by \(X(t)\), \(t\in \mathbb R^n\). Like in other multi-scale models, \(Y(\Omega)\) is specified recursively as \(Y_j(\Omega)\), \(j=0,1,\dots\), where \(j\) denotes the resolution level. For any finite \(j\), the measure density \(X(t)\) exists and \(Y_j(\Omega) = \int_{\Omega} X_j(t)\,dt\). Hence, \(Y(\Omega)\) can be characterized through an initial random field \(X_0(t)\) and conditional random fields \(\{X_j(t)| X_k(t)| , k=0,1,\dots, j-1\}\) for \(j=1,2,\dots\). In general, the pulses that define \(X_j(t)\) may be random field and may be dependent among themselves, while the pulse locations may be clustered or regular in quite general ways. The paper focuses mainly on homogeneous IRP processes in which the pulses at any given level \(j\) are independent copies of a random function \(f_j(t)\) and their locations are described by a doubly stochastic Poisson process whose intensity \(\lambda_j(t)\) depends on the process at \(X_{j-1}(t)\). In the simplest case, the random functions \(f_j(t)\) are related as \(f_j(t) = r^{-\gamma} f_{j-1}(r t)\), where \(r>1\) is a support contraction factor, \(\gamma > -n\) is an affinity parameter, and \(n\) is the dimension of physical space.

The paper is organized as follows. Section 2 describes in greater detail two classes of IRP processes, which differ in the way the intensity \(\lambda_j(t)\) depends on \(X_{j-1}(t)\). In Section 3 recursive relations for the power spectral density \(S_j(\omega)\) of \(X_j(t)\) are obtained, whereas Section 4 analyzes the existence and behavior of the limit \(S(\omega) =\lim_{j\to \infty} S_j(\omega)\) for a class of IRP processes. Section 5 extends the models by allowing anisotropic affine relations among the pulses \(f_j(t)\) at different resolutions levels and non-homogeneous initial processes \(X_0(t)\). Other extensions are mentioned in the concluding section.

Reviewer: Viktor Oganyan (Erevan)

##### MSC:

60G55 | Point processes (e.g., Poisson, Cox, Hawkes processes) |

62M15 | Inference from stochastic processes and spectral analysis |

86A60 | Geological problems |

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