Titlebook: Random Processes with Applications to Circuits and Communications; Bernard C. Levy Textbook 2020 Springer Nature Switzerland AG 2020 Rando

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Convergence and Limit Theoremsarguments. Such arguments require extending to random variables the familiar convergence analysis of real numbers. We recall that if .., . ≥ 1 is a sequence of real numbers, it converges to a real limit . if for each . > 0, there exists and integer .(.) such that |.. − .| < . for all . ≥ .(.). The m

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Discrete-Time Finite Markov Chainss as elementary as possible, we restrict our attention to homogeneous, i.e., time-invariant, and finite chains, i.e., chains where the process .(.) can take only a finite number of values. Under these simplifying assumptions, Markov chains can be analyzed by using tools of linear algebra. Yet, the c

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Wiener Process and White Gaussian Noisens by smaller atoms in a liquid. This phenomenon, first studied by botanist Robert Brown in 1827, was investigated by Einstein, Langevin and other physicists..The Wiener process was first defined mathematically by MIT mathematician Norbert Wiener and its properties were later studied in detail by Pa

Implicit

Poisson Process and Shot Noisenuous sample paths and changes infrequently by unit increments. So, it is well adapted to model discrete random phenomena, such as photon counts in optics, the number of tasks processed by a computer system, or random spikes in a neural pathway. Like the Wiener process, the Poisson process has indep

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Processing and Frequency Analysis of Random Signalst of the attention is focused on WSS random signals, since such signals can be analyzed in the Fourier domain by using their power spectral density (PSD), which as explained in Sect. 8.3 provides a picture of the frequency composition of the signal of interest. The effect of linear filtering on the

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Ergodicityes could be replaced by time averages. A further observation was provided by Gibbs who observed that for Hamiltonian systems, Liouville’s theorem ensures that for a system in equilibrium with a large number of particles, the probability density of particles in phase space is constant, thus ensuring

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Scalar Markov Diffusions and Ito Calculuslmost surely, but their dynamics are richer in the sense that unlike the Wiener process, which is homogeneous in both time and space, the process dynamics depend on both the current time . and position .(.) = . of the process. The diffusion processes can be used to describe a great variety of physic