By Ravi Montenegro, Prasad Tetali

Presents an creation to the analytical facets of the speculation of finite Markov chain blending occasions and explains its advancements. This ebook appears to be like at numerous theorems and derives them in easy methods, illustrated with examples. It comprises spectral, logarithmic Sobolev concepts, the evolving set method, and problems with nonreversibility.

**Read Online or Download Mathematical Aspects of Mixing Times in Markov Chains (Foundations and Trends in Theoretical Computer Science) PDF**

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**Extra info for Mathematical Aspects of Mixing Times in Markov Chains (Foundations and Trends in Theoretical Computer Science)**

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EZ g(EZ). 2 284 Evolving Set Methods It is fairly easy to translate these to mixing time bounds. 6 it is appropriate to let f (z) = 1−z z for L bounds. 5) τ2 ( ) ≤ 2x(1 − x)(1 − C√z(1−z) (x)) π∗ 1 1+ 2 /4 dx 4π∗ x(1 − x)(1 − C√ (x)) z(1−z) 1+3π∗ 1 with the ﬁrst integral requiring x 1 − C√z(1−z) 1+x to be convex. 2 r By making the change of variables x = 1+r and applying a few pessimistic approximations one obtains a result more strongly resembling spectral proﬁle bounds: 1 1 log √ √ 1 − C z(1−z) π∗ 1/ 2 dr τ2 ( ) ≤ √ 2r(1 − C z(1−z) (r)) π∗ 4/ 2 dr √ r(1 − C (r)) 4π∗ z(1−z) For total variation distance related results are in terms of Cz(1−z) (r), and Cz log(1/z) (r) for relative entropy.

PΛ(S, y) = ΛK(S, Proof. PΛ(S, y) = z∈S ˆ y) = ΛK(S, S y Q(S, y) π(z) P(z, y) = π(S) π(S) ˆ S ) π(y) = π(y) K(S, π(S ) π(S) The ﬁnal equality Q(S, y)/π(y). is because S K(S, S ) = S y y K(S, S Q(S, y) π(S) ) = P rob(y ∈ S ) = With duality it becomes easy to write the n step transitions in terms ˆ of the walk K. ˆ n . 4. Let E {x} then ˆ n πS (y) , Pn (x, y) = E n where πS (y) = set S by π. 1S (y)π(y) π(S) denotes the probability distribution induced on Proof. ˆ n )({x}, y) = E ˆ n πS (y) Pn (x, y) = (Pn Λ)({x}, y) = (ΛK n The ﬁnal equality is because Λ(S, y) = πS (y).

3. 15. λP ≥ 1 λˆ , MA P ρP ≥ 1 ρˆ , MA P ΛP (r) ≥ 1 Λ ˆ (r). MA P The log-Sobolev and spectral proﬁle mixing time bounds of P are ˆ thus at worst a factor M A times larger than those of P. ˆ along with the If the distribution π = π ˆ then a Nash inequality for P, 1 relation EP (f, f ) ≥ A EPˆ (f, f ), immediately yields a Nash inequality for P. It is not immediately clear how to compare Nash inequality bounds if π = π ˆ . 5) gives for P. EP (f, f ) to EPˆ (f, f ) and Varπ (f ) to Varπˆ (f ) in the original proofs of the mixing times.