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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 the 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 then 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 flipping 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
second
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: NONE
Last Updated on MAR-18-2003
Copyright Wesleyan University, Middletown, Connecticut, 06459