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One of the most beautiful theorems of probability theory is that simple random walk is recurrent in one and two dimensions, and transient in three and higher dimensions. This theorem was first proved by Polya at the
beginning of the 20th century. It is
conjectured, but not yet proven, that similar results are valid for reinforced random walks. In
this course, my goal is to present a clear and simple picture with understandable proofs of known results in this area,
in order to arrive at the level of
current research, and to explain ideas which could possibly lead to a solution of these conjectures. The course begins with a presentation of the basic theory, including a review of the necessary elementary notions from
probability theory. The second part
of the course deals with the two main reinforcement schemes, together with the corresponding results and conjectures for these reinforced random walks. No formal prerequisites are assumed; however, I expect from the
listeners a mathematical maturity at t
he undergraduate
major level, together with an intuitive understanding of discrete probability concepts and limiting procedures in analysis. In order to get an idea of whether you might enjoy this course, you could
try to
solve the following
problem:
Consider a game with two players, which proceeds as follows. The first player chooses one of the eight sequences
000, 001, 010, 011, 100, 101, 110, 111
and reveals her choice to the
second player. The second player th
en chooses one of the seven remaining sequences and reveals her choice to the first player. Then, a coin with 0 on one side and 1 on the other
side is flipped again and again, producing a sequence of 0's and 1's; we
assume that the coin is fair. The fl
ipping stops as soon as one of the two sequences chosen by the players appears in succession in the sequence of flips, and the player whose sequence appears (first) is declared the winner.
Question 1. Would you
rather be the first player, or the se
cond one?
Question 2. How many times do you expect to win (say, in 100 games), using the best possible strategy?
Question 3. How much would you be willing to pay per game, if you are allowed the choice in
question 1 and receive $1 each time
you win (and nothing if you lose)?
COURSE FORMAT: Lecture
Level: GRAD Credit: 1 Gen Ed Area Dept: NONE Grading Mode: Graded
Prerequisites: MATH516 AND MATH514
Last Updated on MAR-19-2002
Copyright Wesleyan University, Middletown, Connecticut, 06459