By Huber, Mark Lawrence

ISBN-10: 1482232456

ISBN-13: 9781482232455

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**Example text**

If x ∈ A and C ⊆ B, then P(X1 ∈ C|X0 = x) ≥ ερ (C). MARKOV CHAINS AND APPROXIMATE SIMULATION 15 In other words, a chain is a Harris chain if there is a subset of states A that the chain returns to with positive probability after a finite number of steps from any starting state. Moreover, if the state is somewhere in A, then with probability ε the next state does not depend on exactly which element of A the state is at. The notions of recurrence and aperiodicity are needed to make the stationary distribution limiting as well.

Since pseudorandom sequences are iid uniform, this is an essential tool in building Monte Carlo methods. Moreover, the density of X can be unnormalized, which is essential to the applications of Monte Carlo. A simple example of this theorem is in one dimension, where ν is Lebesgue measure. This theorem says that to draw a continuous random variable X with density f , choose a point (X,Y ) uniformly from the area between the x-axis and the density, and then throw away the Y value. Here it is easy to see that for any a ∈ R, P(X ≤ a a) = −∞ f (x) dx as desired.

Continuing in this fashion, the {Bi } sequence of random bits allows construction of U1 ,U2 ,U3 , . , each of which is independent and has Unif([0, 1]) distribution. Therefore a single U ∼ Unif([0, 1]) can be viewed as an infinite sequence of iid Unif([0, 1]) random variables! The point is not to do this in practice, but by realizing this equivalence, all the randomness used by any randomized algorithm can be summarized by a single uniform, which simplifies the notation considerably. In other words, any randomized algorithm can be written as a deterministic function φ (x,U), where x is the deterministic input to the problem, and U ∼ Unif([0, 1]) represents all the randomness used by the algorithm.

### Perfect simulation by Huber, Mark Lawrence

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