In the first chapter we stated and proved various limit theorems for stopped random walks. These limit theorems shall, in subsequent chapters, be used in order to obtain results for random walks stopped according to specific stopping procedures as well as for the families of stopping times (random indices) themselves. However, before doing so we shall, in this chapter, survey some of the basic facts about random walks.

Renewal Processes and Random Walks

Branching Processes with Biological Applications

If dividing by $p$ is a mistake, multiply by $q$ and translate, and so you'll live to iterate. We show that if we define a Collatz-like map in this form then, under suitable conditions on $p$ and $q$, almost all orbits of this map attain almost bounded values. This generalizes a recent breakthrough result of Tao for the original Collatz map (i.e., $p=2$ and $q=3$). In other words, given an arbitrary growth function $N\mapsto f(N)$ we show that almost every orbit of such map with input $N$ eventually attains a value smaller than $f(N)$.

Generalized Collatz Maps with Almost Bounded Orbits.

An introduction to probability theory and its applications

An Introduction to Probability Theory and Its Applications

Stochastic processes

Introduction to Stochastic Processes

This book is devoted exclusively to a very special class of random processes, namely to random walk on the lattice points of ordinary Euclidean space. The author considered this high degree of specialization worth while, because of the theory of such random walks is far more complete than that of any larger class of Markov chains. The book will present no technical difficulties to the readers with some solid experience in analysis in two or three of the following areas: probability theory, real variables and measure, analytic functions, Fourier analysis, differential and integral operators. There are almost 100 pages of examples and problems.

Renewal Processes and Random Walks

Citations

Branching Processes with Biological Applications

Generalized Collatz Maps with Almost Bounded Orbits.

References

An introduction to probability theory and its applications

An Introduction to Probability Theory and Its Applications

Stochastic processes

Introduction to Stochastic Processes

Principles of random walk

Related Papers (5)

Stopped Random Walks: Limit Theorems and Applications

Limit Theorems for Stopped Random Walks

Renewal theorems for random walks in multidimensional time

Renewal theory for random walks on surface groups

Return statistics of simple random walks