Random walk in random and non-random environments [antikvár]

IntroductionThe first examinee is saying: Sir, I did not have time enough to study everything but I learned very carefully the first chapter of your handout.Very good - says the professor - you will be a great specialist. You know what a specialist is. A specialist knows more and more about less and less. Finally he knows everything about nothing.The second examinee is saying: Sir, I did not have enough time but I read your handout without taking care of the details. Very good - answers the professor -you will be a great polymath. You know what a polymath is. A polymath knows less and less about more and more. Finally he knows nothing about everything.Recalling this old joke and realizing that the biggest part of this book is devoted to the study of the properties of the simple symmetric random walk (or equivalently, coin tossing) the reader might say that this is a book for specialists written by a specialist. The most trivial plea of the author is to say that this book does not tell everything about coin tossing and even the author does not know everything about it. Seriously speaking I wish to explain my reasons for writing such a book.You know that the first probabilists (Bernoulli, Pascal, etc.) investigated the properties of coin tossing sequences and other simple games only. Later on the progress of the probability theory went into two different directions:(i)to find newer and deeper properties of the coin tossing sequence,(ii)to generalize the results known for a coin tossing sequence to more complicated sequences or processes.Nowadays the second direction is much more popular than the first one. In spite of this fact this book mostly follows direction (i). I hope that(a)using the advantage of the simple situation coming from concentrating on coin tossing sequences, the reader becomes familiar with the problems, results and partly the methods of proof of probability theory, especially those of the limit theorems, without suffering too much from technical tools and difficulties,(b)since the random walk (especially in Z'') is the simplest mathematical model of the Brownian motion, the reader can find a simple way to the problems