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Applied Probability

From Random Sequences to Stochastic Processes

Authors: Prof. Valérie Girardin, Prof. Nikolaos Limnios

Publisher: Springer International Publishing

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This textbook addresses postgraduate students in applied mathematics, probability, and statistics, as well as computer scientists, biologists, physicists and economists, who are seeking a rigorous introduction to applied stochastic processes. Pursuing a pedagogic approach, the content follows a path of increasing complexity, from the simplest random sequences to the advanced stochastic processes. Illustrations are provided from many applied fields, together with connections to ergodic theory, information theory, reliability and insurance. The main content is also complemented by a wealth of examples and exercises with solutions.

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Table of Contents

Frontmatter

1. Independent Random Sequences

Abstract

Random sequences are understood here as real random processes indexed by discrete time. Advanced notions on sequences of real random variables are presented. Apart for their intrinsic value, they are thought as a tool box for the other chapters. They will be completed in Chap. 4 by definitions and properties common to all real random processes, either indexed by continuous or by discrete time.

Valérie Girardin, Nikolaos Limnios

2. Conditioning and Martingales

Abstract

In this chapter, conditional distributions and expectations are presented, following steps of increasing difficulty. Conditional distributions are first supposed to exist, which is true for all random variables taking values in \(\mathbb {R}^d\). Still, conditional expectation is defined in the most general case. A section is dedicated to determining practically conditional distributions and expectations, and another to the linear model in which a random phenomenon is assumed to be linearly related to other simultaneously observed phenomena.

Valérie Girardin, Nikolaos Limnios

3. Markov Chains

Abstract

This chapter investigates the homogeneous discrete-time Markov chains with countable—finite or denumerable—state spaces, also called discrete. Markov chains generalize sequences of independent random variables to variables linked by a simple dependence relation. They model for example, phase transitions of substances between solid, liquid and gaseous states, passages of systems between up and down states, etc. A random sequence \((X_n)_{n\in \scriptstyle \mathbb {N}}\) is a Markov chain if its future values depend on its previous values only through its present value, the so-called Markov property. The index n is interpreted as a time, even when it is the n-th trial or step in a process.

Valérie Girardin, Nikolaos Limnios

4. Continuous Time Stochastic Processes

Abstract

A stochastic process represents a system, usually evolving along time, which incorporates an element of randomness, as opposed to a deterministic process.

Valérie Girardin, Nikolaos Limnios

5. Markov and Semi-Markov Processes

Abstract

This chapter is devoted to jump Markov processes and finite semi-Markov processes. In both cases, the index is considered as the calender time, continuously counted over the positive real line. Markov processes are continuous-time processes that share the Markov property with the discrete-time Markov chains. Their future evolution conditional to the past depends only on the last occupied state. Their extension to the so-called semi-Markov processes naturally arises in many types of applications. The future evolution of a semi-Markov process given the past depends on the occupied state too, but also on the time elapsed since the last transition.

Valérie Girardin, Nikolaos Limnios

Backmatter

Title: Applied Probability
Authors: Prof. Valérie Girardin
Prof. Nikolaos Limnios
Copyright Year: 2018
Publisher: Springer International Publishing
Electronic ISBN: 978-3-319-97412-5
Print ISBN: 978-3-319-97411-8
DOI: https://doi.org/10.1007/978-3-319-97412-5