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University Of Michigan  STAT 36754
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Syllabus
School: University Of Michigan
Course: Stochastic Processes
Syllabus for Advanced Probability II, Stochastic Processes 36754 Cosma Shalizi Spring 2006 This course is an advanced treatment of interdependent random variables and random functions, with twin emphases on extending the limit theorems of probability fro

Solutions3
School: University Of Michigan
Course: Stochastic Processes
Solution to Homework #3, 36754 25 February 2006 Exercise 10.1 I need one last revision of the denition of a Markov operator: a linear operator on L1 satisfying the following conditions. 1. If f 0 (a.e.), then Kf 0 (a.e.). 2. If f M (a.e.), then Kf M (

Solutions2
School: University Of Michigan
Course: Stochastic Processes
Solution to Homework #2, 36754 7 February 2006 Exercise 5.3 (The Logistic Map as a MeasurePreserving Transformation) The logistic map with a = 4 is a measurepreserving transformation, and the measure it preserves has the density 1/ x (1 x) (on the unit

Solutions1
School: University Of Michigan
Course: Stochastic Processes
Solution to Homework #1, 36754 27 January 2006 Exercise 1.1 (The product eld answers countable questions) Let D = S X S , where the union ranges over all countable subsets S of the index set T . For any event D D, whether or not a sample path x D depend

References
School: University Of Michigan
Course: Stochastic Processes
Bibliography Abramowitz, Milton and Irene A. Stegun (eds.) (1964). Handbook of Mathematical Functions . Washington, D.C.: National Bureau of Standards. URL http:/www.math.sfu.ca/cbm/aands/. Algoet, Paul (1992). Universal Schemes for Prediction, Gambling a

Lecture35
School: University Of Michigan
Course: Stochastic Processes
Chapter 35 Large Deviations for Stochastic Dierential Equations This last chapter revisits large deviations for stochastic dierential equations in the smallnoise limit, rst raised in Chapter 22. Section 35.1 establishes the LDP for the Wiener process (Sc

Lecture34
School: University Of Michigan
Course: Stochastic Processes
Chapter 34 Large Deviations for Weakly Dep endent Sequences: The GrtnerEllis Theorem a This chapter proves the GrtnerEllis theorem, establishing an a LDP for nottoodependent processes taking values in topological vector spaces. Most of our earlier LDP

Lecture32
School: University Of Michigan
Course: Stochastic Processes
Chapter 32 Large Deviations for Markov Sequences This chapter establishes large deviations principles for Markov sequences as natural consequences of the large deviations principles for IID sequences in Chapter 31. (LDPs for continuoustime Markov process

Lecture31
School: University Of Michigan
Course: Stochastic Processes
Chapter 31 Large Deviations for I ID Sequences: The Return of Relative Entropy Section 31.1 introduces the exponential version of the Markov inequality, which will be our ma jor calculating device, and shows how it naturally leads to both the cumulant gen

Lecture30
School: University Of Michigan
Course: Stochastic Processes
Chapter 30 General Theory of Large Deviations A family of random variables follows the large deviations principle if the probability of the variables falling into bad sets, representing large deviations from expectations, declines exponentially in some ap

Lecture29
School: University Of Michigan
Course: Stochastic Processes
Chapter 29 Entropy Rates and Asymptotic Equipartition Section 29.1 introduces the entropy rate the asymptotic entropy per timestep of a stochastic process and shows that it is welldened; and similarly for information, divergence, etc. rates. Section 29.

Lecture28
School: University Of Michigan
Course: Stochastic Processes
Chapter 28 Shannon Entropy and KullbackLeibler Divergence Section 28.1 introduces Shannon entropy and its most basic properties, including the way it measures how close a random variable is to being uniformly distributed. Section 28.2 describes relative

Lecture27
School: University Of Michigan
Course: Stochastic Processes
Chapter 27 Mixing A stochastic process is mixing if its values at widelyseparated times are asymptotically independent. Section 27.1 denes mixing, and shows that it implies ergodicity. Section 27.2 gives some examples of mixing processes, both determinis

Lecture26
School: University Of Michigan
Course: Stochastic Processes
Chapter 26 Decomp osition of Stationary Pro cesses into Ergo dic Comp onents This chapter is concerned with the decomposition of asymptoticallymeanstationary processes into ergodic components. Section 26.1 shows how to write the stationary distribution a

Lecture25
School: University Of Michigan
Course: Stochastic Processes
Chapter 25 Ergo dicity This lecture explains what it means for a process to be ergodic or metrically transitive, gives a few characterizes of these properties (especially for AMS processes), and deduces some consequences. The most important one is that sa

Lecture24
School: University Of Michigan
Course: Stochastic Processes
Chapter 24 The AlmostSure Ergo dic Theorem This chapter proves Birkho s ergodic theorem, on the almostsure convergence of time averages to expectations, under the assumption that the dynamics are asymptotically mean stationary. This is not the usual proo

Lecture22
School: University Of Michigan
Course: Stochastic Processes
Chapter 22 Large Deviations for SmallNoise Sto chastic Dierential Equations This lecture is at once the end of our main consideration of diffusions and stochastic calculus, and a rst taste of large deviations theory. Here we study the divergence between

Lecture20
School: University Of Michigan
Course: Stochastic Processes
Chapter 20 More on Sto chastic Dierential Equations Section 20.1 shows that the solutions of SDEs are diusions, and how to nd their generators. Our previous work on Feller processes and martingale problems pays o here. Some other basic properties of solut

Lecture18
School: University Of Michigan
Course: Stochastic Processes
Chapter 18 Stochastic Integrals with the Wiener Pro cess Section 18.1 addresses an issue which came up in the last lecture, namely the martingale characterization of the Wiener process. Section 18.2 gives a heuristic introduction to stochastic integrals,

Lecture17
School: University Of Michigan
Course: Stochastic Processes
Chapter 17 Diusions and the Wiener Pro cess Section 17.1 introduces the ideas which will occupy us for the next few lectures, the continuous Markov processes known as diusions, and their description in terms of stochastic calculus. Section 17.2 collects s

Lecture16
School: University Of Michigan
Course: Stochastic Processes
Chapter 16 Convergence of Random Walks This lecture examines the convergence of random walks to the Wiener process. This is very important both physically and statistically, and illustrates the utility of the theory of Feller processes. Section 16.1 nds t

Lecture15
School: University Of Michigan
Course: Stochastic Processes
Chapter 15 Convergence of Feller Pro cesses This chapter looks at the convergence of sequences of Feller processes to a limiting process. Section 15.1 lays some ground work concerning weak convergence of processes with cadlag sample paths. Section 15.2 st

Lecture14
School: University Of Michigan
Course: Stochastic Processes
Chapter 14 Feller Processes Section 14.1 fullls the demand, made last time, for an example of a Markov process which is not strongly Markovian. Section 14.2 makes explicit the idea that the transition kernels of a Markov process induce a kernel over sampl

Lecture13
School: University Of Michigan
Course: Stochastic Processes
Chapter 13 The Strong Markov Prop erty and Martingale Problems Section 13.1 introduces the strong Markov property independence of the past and future conditional on the state at random (optional) times. Section 13.2 describes the martingale problem for Ma

Lecture12
School: University Of Michigan
Course: Stochastic Processes
Chapter 12 Generators of Markov Pro cesses This lecture is concerned with the innitessimal generator of a Markov process, and the sense in which we are able to write the evolution operators of a homogeneous Markov process as exponentials of their generato

Lecture11
School: University Of Michigan
Course: Stochastic Processes
Chapter 11 Markov Examples Section 11.1 nds the transition kernels for the Wiener process, as an example of how to manipulate such things. Section 11.2 looks at the evolution of densities under the action of the logistic map; this shows how deterministic

Lecture10
School: University Of Michigan
Course: Stochastic Processes
Chapter 10 Alternate Characterizations of Markov Pro cesses This lecture introduces two ways of characterizing Markov processes other than through their transition probabilities. Section 10.1 addresses a question raised in the last class, about when being

Lecture09
School: University Of Michigan
Course: Stochastic Processes
Chapter 9 Markov Pro cesses This lecture begins our study of Markov processes. Section 9.1 is mainly ideological: it formally denes the Markov property for oneparameter processes, and explains why it is a natural generalization of both complete determini

Lecture08
School: University Of Michigan
Course: Stochastic Processes
Chapter 8 More on Continuity Section 8.1 constructs separable modications of reasonable but nonseparable random functions, and explains how separability relates to nondenumerable properties like continuity. Section 8.2 constructs versions of our favorit

Lecture07
School: University Of Michigan
Course: Stochastic Processes
Chapter 7 Continuity of Sto chastic Pro cesses Section 7.1 describes the leading kinds of continuity for stochastic processes, which derive from the modes of convergence of random variables. It also denes the idea of versions of a stochastic process. Sect

Lecture05
School: University Of Michigan
Course: Stochastic Processes
Chapter 5 Stationary OneParameter Pro cesses Section 5.1 describes the three main kinds of stationarity: strong, weak, and conditional. Section 5.2 relates stationary processes to the shift operators introduced in the last chapter, and to measurepreserv

Lecture04
School: University Of Michigan
Course: Stochastic Processes
Chapter 4 OneParameter Pro cesses, Usually Functions of Time Section 4.1 denes oneparameter processes, and their variations (discrete or continuous parameter, one or two sided parameter), including many examples. Section 4.2 shows how to represent one

Lecture03
School: University Of Michigan
Course: Stochastic Processes
Chapter 3 Building Innite Pro cesses from Regular Conditional Probability Distributions Section 3.1 introduces the notion of a probability kernel, which is a useful way of systematizing and extending the treatment of conditional probability distributions

Lecture02
School: University Of Michigan
Course: Stochastic Processes
Chapter 2 Building Innite Pro cesses from FiniteDimensional Distributions Section 2.1 introduces the nitedimensional distributions of a stochastic process, and shows how they determine its innitedimensional distribution. Section 2.2 considers the consi

Lecture01
School: University Of Michigan
Course: Stochastic Processes
Chapter 1 Basic Denitions: Indexed Collections and Random Functions Section 1.1 introduces stochastic processes as indexed collections of random variables. Section 1.2 builds the necessary machinery to consider random functions, especially the product el