Syllabus for Advanced Probability II,
This course is an advanced treatment of interdependent random variables
and random functions, with twin emphases on extending the limit theorems
of probability fro
Solution to Homework #3, 36-754
25 February 2006
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 (
Solution to Homework #2, 36-754
7 February 2006
Exercise 5.3 (The Logistic Map as a MeasurePreserving Transformation)
The logistic map with a = 4 is a measure-preserving transformation, and the measure it preserves has the density 1/ x (1 x)
(on the unit
Solution to Homework #1, 36-754
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
Abramowitz, Milton and Irene A. Stegun (eds.) (1964). Handbook of Mathematical Functions . Washington, D.C.: National Bureau of Standards. URL
Algoet, Paul (1992). Universal Schemes for Prediction, Gambling a
Large Deviations for
This last chapter revisits large deviations for stochastic dierential equations in the small-noise limit, rst raised in Chapter 22.
Section 35.1 establishes the LDP for the Wiener process (Sc
Large Deviations for
Weakly Dep endent
This chapter proves the Grtner-Ellis theorem, establishing an
LDP for not-too-dependent processes taking values in topological
vector spaces. Most of our earlier LDP
Large Deviations for
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 continuous-time Markov
Large Deviations for I ID
Sequences: The Return of
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
General Theory of Large
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
Entropy Rates and
Section 29.1 introduces the entropy rate the asymptotic entropy per time-step of a stochastic process and shows that it is
well-dened; and similarly for information, divergence, etc. rates.
Shannon Entropy and
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
A stochastic process is mixing if its values at widely-separated
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
Decomp osition of
Stationary Pro cesses into
Ergo dic Comp onents
This chapter is concerned with the decomposition of asymptoticallymean-stationary processes into ergodic components.
Section 26.1 shows how to write the stationary distribution a
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
The Almost-Sure Ergo dic
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
Large Deviations for
Small-Noise Sto chastic
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
More on Sto chastic
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
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,
Diusions and the Wiener
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
Convergence of Random
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
Convergence of Feller
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
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
The Strong Markov
Prop erty and Martingale
Section 13.1 introduces the strong Markov property independence of the past and future conditional on the state at random
Section 13.2 describes the martingale problem for Ma
Generators of Markov
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
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
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
Markov Pro cesses
This lecture begins our study of Markov processes.
Section 9.1 is mainly ideological: it formally denes the Markov
property for one-parameter processes, and explains why it is a natural generalization of both complete determini
More on Continuity
Section 8.1 constructs separable modications of reasonable but
non-separable random functions, and explains how separability relates to non-denumerable properties like continuity.
Section 8.2 constructs versions of our favorit
Continuity of Sto chastic
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.