(a) A symmetric one-dimensional random walk takes steps with equal probability each,…

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Fourier analysis and integral transforms · Probability, statistics and stochastic methods

Error, Chance and the Machine, Ch Random walks and Markov chains

Problem

(a) A symmetric one-dimensional random walk takes steps ±1\pm 1 with equal probability 12\tfrac12 each, independently at every step. Its single-step characteristic function is λ(θ)=⟨eiθX⟩=cos⁡θ\lambda(\theta) = \langle e^{i\theta X}\rangle = \cos\theta, and the probability of being at xx after nn steps is Pn(x)=12π∫−ππe−ixθλ(θ)n dθP_n(x) = \frac{1}{2\pi}\int_{-\pi}^{\pi} e^{-ix\theta}\lambda(\theta)^n\,\mathrm{d}\theta Find P8(0)P_8(0), the probability of being back at the origin after 88 steps. (b) A three-state chain has transition matrix P=(0.50.30.20.20.60.20.30.30.4)\mathsf{P} = \begin{pmatrix} 0.5 & 0.3 & 0.2 \\ 0.2 & 0.6 & 0.2 \\ 0.3 & 0.3 & 0.4\end{pmatrix} Starting from p0=(1,0,0)\mathbf{p}_0 = (1,0,0), repeated left-multiplication by P\mathsf{P} (the power iteration used for eigenvectors) converges to the stationary distribution π\boldsymbol\pi, the left eigenvector of eigenvalue 11. Find π\boldsymbol\pi exactly by solving πP=π\boldsymbol\pi\mathsf{P} = \boldsymbol\pi together with ∑iπi=1\sum_i \pi_i = 1.

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