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Lecture 1: Preliminaries and Hydrodynamics of Independent Random Walks
| Content Provider | Semantic Scholar |
|---|---|
| Copyright Year | 2012 |
| Abstract | P (Xn = xn|X0 = x0, . . . , Xm = xm) = P (Xn = xn|Xm = xm) = P (Xn−m = xn|X0 = xm) for all x0, . . . , xm ∈ E and n > m ≥ 0. When n = 1 and m = 0, the last quantity on the right-side represents a ‘transition probability’ of the Markov chain, denoted p(x, y) = P (X1 = y|X0 = x). We now construct a ‘continuous time Markov chain’ on E with ‘skeleton’ {Xn : n ≥ 0} and transition probability vanishing on the diagonal, that is p(x, x) = 0 for all x ∈ E. Let {λx : x ∈ E} be a collection of positive numbers, and let {Wn : n ≥ 0} be a collection of independent identically distributed exponential random variables with rate 1, independent in particular the skeleton discrete time chain. Define now the process {Zt : t ≥ 0} as follows: Initially, Z0 = X0 ∈ E. After time λ−1 X0W0, the process jumps to value X1, and after a subsequent time λ −1 X1 W1, the process jumps to value X2, and so on. Let Tk = ∑k i=0 λ −1 Xi Wi for k ≥ 0. Then, |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://math.arizona.edu/~sethuram/588/lecture1.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
