单粒子跟踪实验中路径均方位移的渐近分布
The asymptotic distribution of the pathwise mean squared displacement in single particle tracking experiments
摘要：
光学显微镜的最新进展通过围绕衍射屏障工作并允许观察纳米生物结构，在微生物学中产生了新的研究前沿。微观流变学是研究复杂流体的性质，例如生物学中发现的那些，通过小嵌入颗粒的动力学，通常是乳胶珠。然后，生物药理学家使用基于记录的样品路径的统计数据来推断流体的流变特性。在生物物理学文献中，表征扩散率的主要统计量是示踪粒子的所谓均方位移（MSD）。尽管MSD发挥了核心作用，但其在不同情况下的渐近分布尚未确定。在本文中，我们解决这个问题。我们采取路径的方法，并假设粒子运动经历高斯，平稳增量随机过程。我们发现，随着样本和增量滞后大小趋于无穷大，MSD显示高斯或非高斯极限分布，以及不同的收敛速度，取决于扩散指数参数。基于分数布朗运动和（积分）分数Ornstein-Uhlenbeck过程，我们用解析和计算的方法说明了我们的结果.
英文摘要：
Recent advances in light microscopy have spawned new research frontiers in microbiology by working around the diffraction barrier and allowing for the observation of nanometric biological structures. Microrheology is the study of the properties of complex fluids, such as those found in biology, through the dynamics of small embedded particles, typically latex beads. Statistics based on the recorded sample paths are then used by biophysicists to infer rheological properties of the fluid. In the biophysical literature, the main statistic for characterizing diffusivity is the so-named mean squared displacement (MSD) of the tracer particles. Notwithstanding the central role played by the MSD, its asymptotic distribution in different cases has not yet been established. In this paper, we tackle this problem. We take a pathwise approach and assume that the particle movement undergoes a Gaussian, stationary-increment stochastic process. We show that as the sample and the increment lag sizes go to infinity, the MSD displays Gaussian or non-Gaussian limiting distributions, as well as distinct convergence rates, depending on the diffusion exponent parameter. We illustrate our results analytically and computationally based on fractional Brownian motion and the (integrated) fractional Ornstein-Uhlenbeck process.
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