Nonstationary self‐similar Gaussian processes as scaling limits of power‐law shot noise processes and generalizations of fractional Brownian motion
Nonstationary self‐similar Gaussian processes as scaling limits of power‐law shot noise processes and generalizations of fractional Brownian motion
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DOI：
10.1002/hf2.10028
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发表时间：
2019-04
期刊：
影响因子：
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通讯作者：
G. Pang;M. Taqqu
中科院分区：
文献类型：
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作者：
G. Pang;M. Taqqu
我们研究具有泊松到达和非平稳噪声的散粒噪声过程。在给定到达时间的情况下，噪声是条件独立的，但每个噪声的分布确实取决于其到达时间。我们在两种情况下建立了这类散粒噪声过程的尺度极限：(A)噪声的条件方差函数具有幂定律；(B)条件噪声分布是分段的。在这两种情况下，极限过程都是具有非平稳增量的自相似高斯过程。受这些过程的启发，我们通过时间域积分表示引入了一类新的具有非平稳增量的自相似高斯过程，它是分数布朗运动的自然推广。
We study shot noise processes with Poisson arrivals and nonstationary noises. The noises are conditionally independent given the arrival times, but the distribution of each noise does depend on its arrival time. We establish scaling limits for such shot noise processes in two situations: (a) the conditional variance functions of the noises have a power law and (b) the conditional noise distributions are piecewise. In both cases, the limit processes are self‐similar Gaussian with nonstationary increments. Motivated by these processes, we introduce new classes of self‐similar Gaussian processes with nonstationary increments, via the time‐domain integral representation, which are natural generalizations of fractional Brownian motions.