Stopped random walks. Limit theorems and applications.

*(English)*Zbl 0634.60061
Applied Probability, Vol. 5. New York etc.: Springer-Verlag. IX, 199 p.; DM 88.00 (1988).

This clearly written book, useful for researcher and student and containing the author’s research results of recent years, deals with the following part of random walk theory.

Let \(X_ 1,X_ 2,..\). be i.i.d. random variables, \(S_ 0=0\), \(S_ n=S(n)=X_ 1+...+X_ n\), \(n\geq 1\), and let \(N_ t\), \(t\geq 0\), be positive integer random variables with \(N_ t\to \infty\) as \(t\to \infty\). Mostly, but not always, the \(N_ t\) are stopping times for the \(S_ n\)-process. The central part of the book is the study of limit theorems for \(S(N_ t)\) as \(t\to \infty:\) laws of large numbers and central limit theorems with convergence of moments, laws of iterated logarithm and uniform integrability. These theorems are applied in proving other limit theorems, e.g. in renewal theory on \({\mathbb{R}}\) and for \(M_ n=\max (S_ 0,...,S_ n).\)

Chapter I gives general theorems on convergence of random variables \(Y(N_ t)\) with random index \(N_ t\), the Anscombe-Rényi theorem, convergence of moments in strong law and central limit theorem with deterministic and random index and existence of moments and uniform integrability of \(S(N_ t)\). Chapter II gives the fundamentals of renewal theory with \(X_ i\geq 0\) and of random walk theory.

Chapter III is concerned with renewal theory on \({\mathbb{R}}\) for random walks with positive drift. Limit theorems of the types mentioned above are proved for the stopping time \(\nu_ t=\min \{n:\) \(S_ n>t\}\) and, where they make sense, for S(\(\nu\) (t)). The asymptotics of the renewal function and of \(S(\nu_ t)-t\), and their refinements, are dealt with. An important part here is played by ladder variable techniques. Some subjects of Chapter IV are: limit theorems for \(V(\tau_ t)\) where (U(n),V(n)) is a random walk in \({\mathbb{R}}^ 2 \)with E U(1)\(>0\) and \(\tau_ t=\min \{n:\) \(U(n)>t\}\), applications, limit theorems for \(M_ n\) by representing \(M_ n\) as a random sum of ladder steps, and first passage across general boundaries. Chapter V extends part of the previous results to functional limit theorems.

Let \(X_ 1,X_ 2,..\). be i.i.d. random variables, \(S_ 0=0\), \(S_ n=S(n)=X_ 1+...+X_ n\), \(n\geq 1\), and let \(N_ t\), \(t\geq 0\), be positive integer random variables with \(N_ t\to \infty\) as \(t\to \infty\). Mostly, but not always, the \(N_ t\) are stopping times for the \(S_ n\)-process. The central part of the book is the study of limit theorems for \(S(N_ t)\) as \(t\to \infty:\) laws of large numbers and central limit theorems with convergence of moments, laws of iterated logarithm and uniform integrability. These theorems are applied in proving other limit theorems, e.g. in renewal theory on \({\mathbb{R}}\) and for \(M_ n=\max (S_ 0,...,S_ n).\)

Chapter I gives general theorems on convergence of random variables \(Y(N_ t)\) with random index \(N_ t\), the Anscombe-Rényi theorem, convergence of moments in strong law and central limit theorem with deterministic and random index and existence of moments and uniform integrability of \(S(N_ t)\). Chapter II gives the fundamentals of renewal theory with \(X_ i\geq 0\) and of random walk theory.

Chapter III is concerned with renewal theory on \({\mathbb{R}}\) for random walks with positive drift. Limit theorems of the types mentioned above are proved for the stopping time \(\nu_ t=\min \{n:\) \(S_ n>t\}\) and, where they make sense, for S(\(\nu\) (t)). The asymptotics of the renewal function and of \(S(\nu_ t)-t\), and their refinements, are dealt with. An important part here is played by ladder variable techniques. Some subjects of Chapter IV are: limit theorems for \(V(\tau_ t)\) where (U(n),V(n)) is a random walk in \({\mathbb{R}}^ 2 \)with E U(1)\(>0\) and \(\tau_ t=\min \{n:\) \(U(n)>t\}\), applications, limit theorems for \(M_ n\) by representing \(M_ n\) as a random sum of ladder steps, and first passage across general boundaries. Chapter V extends part of the previous results to functional limit theorems.

Reviewer: A.J.Stam

##### MSC:

60G50 | Sums of independent random variables; random walks |

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

60Fxx | Limit theorems in probability theory |

60K05 | Renewal theory |