Download Harmonic Analysis on Finite Groups: Representation Theory, by Tullio Ceccherini-Silberstein PDF

By Tullio Ceccherini-Silberstein

Ranging from a number of concrete difficulties equivalent to random walks at the discrete circle and the finite ultrametric house, this ebook develops the mandatory instruments for the asymptotic research of those tactics. Its subject matters diversity from the elemental concept wanted for college kids new to this zone, to complicated issues equivalent to the speculation of Green's algebras, the whole research of the random matchings, and a presentation of the presentation thought of the symmetric crew. This self-contained, particular research culminates with case-by-case analyses of the cut-off phenomenon chanced on via Persi Diaconis.

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Extra resources for Harmonic Analysis on Finite Groups: Representation Theory, Gelfand Pairs and Markov Chains

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Xk−1 ∈X ν(x0 )p(x0 , x1 ) · · · p(xk−1 , x) = p(x, x ). 3 Markov chains 17 ξ0 , ξ1 , . . , ξn , . . and (x0 , x1 , . . , xn , . ); see, for a detailed account on measure theory, Shiryaev’s book [199] and, for more on Markov chains, the books by Billingsley [27] and Stroock [215]. For the point of view of ergodic theory and dynamical systems we refer to Petersen’s monograph [177] In this book we are mainly interested in the asymptotic behavior of the distributions ν (k) ’s as k → ∞. The simplified approach will largely suffice in this setting.

10 Lumpable Markov chains 45 Let P be the transition matrix of the simple random walk on X. Set • W0 = {f : X → C : constant} • W1 = {f : X → C : f (x) = −f (x0 ), ∀x ∈ X1 } • W2 = {f : X → C : f (x0 ) = 0 and x∈X1 f (x) = 0}. (1) Show that W0 , W1 and W2 are eigenspaces of P and that the corresponding eigenvalues are λ0 = 1, λ1 = −1 and λ2 = 0. (2) Show that P is {{x0 }, X1 }-lumpable and that the lumped transition matrix has only the eigenvalues 1 and −1. 7 Let P be the transition matrix for the simple random walk on the discrete circle C4 .

2 Let P be an ergodic stochastic matrix and π its stationary distribution. Then there exist A > 0, 0 < c < 1 and n1 ∈ N such that, for all n ≥ n1 , max |p(n) (x, y) − π(y)| ≤ A(1 − c)n . 9). Indeed, if n = n0 k + r with 0 ≤ r < n0 , then nn0 = k + nr0 and (1 − ε)[n/n0 ]−1 = (1 − ε)k−1 = (1 − ε) n0 − n0 −1 n r = (1 − ε)− n0 −1 [(1 − ε) n0 ]n r 1 1 so that we can take A = (1 − ε)−2 , 1 − c = (1 − ε) n0 and n1 = n0 . 3 Let p be the transition probability kernel for the simple random walk on the discrete circle C2m+1 .

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