A variable has the Laplace density

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

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Problem

A variable XX has the Laplace density f(x)=12e−∣x∣f(x) = \tfrac12e^{-|x|}, whose characteristic function is ϕX(t)=1/(1+t2)\phi_X(t) = 1/(1+t^2). Let X1,…,XNX_1, \dots, X_N be independent copies and let σ2\sigma^2 be the variance of XX. ((a)) Use the small-tt expansion of ϕX\phi_X to find ⟨X⟩\langle X\rangle and σ2\sigma^2. Then carry the proof of the central limit theorem through for this density: show that ZN=∑iXi/(σN)Z_N = \sum_i X_i/(\sigma\sqrt N) has ϕZN(t)=(1+t2/2N)−N\phi_{Z_N}(t) = (1 + t^2/2N)^{-N}. ((b)) Show that ln⁡ϕZN(t)=−t2/2+t4/(8N)+O(N−2)\ln\phi_{Z_N}(t) = -t^2/2 + t^4/(8N) + O(N^{-2}), and so find the limit of ϕZN(t)\phi_{Z_N}(t). For N=50N = 50 and t=2t = 2, evaluate ϕZN(t)\phi_{Z_N}(t) exactly, the Gaussian value e−t2/2e^{-t^2/2}, and the two-term estimate exp⁡[−t2/2+t4/(8N)]\exp[-t^2/2 + t^4/(8N)]. Why is the leading correction of order 1/N1/N and not 1/N1/\sqrt N? ((c)) Now let XX have the density g(x)=∣x∣−3g(x) = |x|^{-3} for ∣x∣≥1|x|\ge1 and g(x)=0g(x) = 0 otherwise. Show that gg is normalised, that XX has a finite mean, and that ⟨X2⟩\langle X^2\rangle diverges. Say which step of the proof fails. It can be shown that near t=0t=0 the characteristic function of gg is ϕ(t)=1+t2ln⁡∣t∣+O(t2)\phi(t) = 1 + t^2\ln|t| + O(t^2). Use this to find the limit as N→∞N\to\infty of the characteristic function of ∑iXi/N\sum_i X_i/\sqrt N at a fixed t≠0t \ne 0, and say what the result means for the distribution of ∑iXi/N\sum_i X_i/\sqrt N.

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