By Mu-Fa Chen
The 1st and purely e-book to make this examine to be had within the West Concise and available: proofs and different technical issues are saved to a minimal to aid the non-specialist each one bankruptcy is self-contained to make the ebook easy-to-use
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Extra resources for Eigenvalues, Inequalities, and Ergodic Theory (Probability and its Applications)
For birth–death processes, the coupling by reﬂection is ρ-optimal for any translation-invariant metric ρ on Z+ having the property that uk := ρ(0, k + 1) − ρ(0, k), k 0 is nonincreasing in k. 30 2 Optimal Markovian Couplings To see that the optimal coupling depends heavily on the metric ρ, note that the above metric ρ can be rewritten as ρ(i, j) = uk k<|i−j| for some positive nonincreasing sequence (uk ). In this way, for any positive sequence (uk ), we can introduce another metric as follows: uk − ρ˜(i, j) = k
Lindvall, 1999). Again, let ∆ denote the diagonals. (1) Let µ1 ≺ µ2 . Then inf c µ(F ˜ )=0 µ ˜ (∆c ) = 1 µ − µ2 Var . 2 1 (2) Let P1 and P2 be transition probabilities that satisfy P1 ≺ P2 . Then inf P (x1 ,x2 ; F c )=0 P (x1 , x2 ; ∆c ) = 1 P1 (x1 , ·) − P2 (x2 , ·) Var 2 for all x1 ≺ x2 . 35 (1). Y. P. Xu (1997). 38. Let ϕ ∈ C 2 (R2d \ ∆). Prove the existence of ϕ-optimal Markovian couplings for diﬀusions under some reasonable hypotheses. 39. Construct ϕ-optimal Markovian couplings. 4 Applications of coupling methods It should be helpful for readers, especially newcomers, to see some applications of couplings.
Intuitively, the construction of (Yt ) can be completed in two steps: Let y = x. 48 3 New Variational Formulas for the First Eigenvalue • Transport Xt from x to y in parallel along the line (x, y). • Make the mirror reﬂection of the transported image of Xt in the hyperplane that is perpendicular to the line (x, y) at y. Then, the mirror image gives us the process (Yt ). For the diﬀusion (Xt ) on manifold M with generator ∆, a process (Yt ) can be constructed in a similar way. S. Kendall (1986) and M.