Stopped Random Walks

1. Front Matter

   Pages i-ix

   PDF

2. Introduction

   • Allan Gut

   Pages 1-7

3. Limit Theorems for Stopped Random Walks

   • Allan Gut

   Pages 8-45

4. Renewal Processes and Random Walks

   • Allan Gut

   Pages 46-73

5. Renewal Theory for Random Walks with Positive Drift

   • Allan Gut

   Pages 74-107

6. Generalizations and Extensions

   • Allan Gut

   Pages 108-146

7. Functional Limit Theorems

   • Allan Gut

   Pages 147-164

8. Back Matter

   Pages 165-199

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My first encounter with renewal theory and its extensions was in 1967/68 when I took a course in probability theory and stochastic processes, where the then recent book Stochastic Processes by Professor N.D. Prabhu was one of the requirements. Later, my teacher, Professor Carl-Gustav Esseen, gave me some problems in this area for a possible thesis, the result of which was Gut (1974a). Over the years I have, on and off, continued research in this field. During this time it has become clear that many limit theorems can be obtained with the aid of limit theorems for random walks indexed by families of positive, integer valued random variables, typically by families of stopping times. During the spring semester of 1984 Professor Prabhu visited Uppsala and very soon got me started on a book focusing on this aspect. I wish to thank him for getting me into this project, for his advice and suggestions, as well as his kindness and hospitality during my stay at Cornell in the spring of 1985. Throughout the writing of this book I have had immense help and support from Svante Janson. He has not only read, but scrutinized, every word and every formula of this and earlier versions of the manuscript. My gratitude to him for all the errors he found, for his perspicacious suggestions and remarks and, above all, for what his unusual personal as well as scientific generosity has meant to me cannot be expressed in words.

• Probability theory
• Random variable
• random walk
• renewal theory
• stochastic processes

• Department of Mathematics, Uppsala University, Uppsala, Sweden

  Allan Gut

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