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Basic Theory of Hilbert \(C^*\) -Modules
This chapter covers the basic concepts and methods of Hilbert \(C^*\)... -
Quantum Markov Semigroups Based on Hilbert \(C^*\) -Modules
In this chapter, we introduce the concept of module operator semigroup and give characterizations. Then the classical solution and mild solution of... -
Hilbert C*- Modules and Quantum Markov Semigroups
This book explains the basic theory of Hilbert C*-module in detail, covering a wide range of applications from generalized index to module framework.... -
Kasprove’s Stabilization and Fredholm Generalized Index Theory
In this chapter, we show that every countably generated Hilbert A-module can be regarded as a complemented submodule of... -
Zuverlässigkeitstheorie und technische Systeme
Unter Zuverlässigkeit (engl. Reliability) versteht man in der Technik die Eignung eines Systems, während einer gewissen Zeitspanne die an es... -
Anhang A
Die stochastische Matrix lautet -
Preliminaries
Herewe briefly reviewseveral previously obtained results – mostly when the variance is infinite – that are used or closely related to the topics... -
Introduction
At the time when Spitzer introduced 𝑎(𝑥), several works appeared which studied problems closely related to or involving the potential function,... -
Bounds of the Potential Function
In this chapter, we study asymptotic properties of the potential function 𝑎(𝑥) of a recurrent r.w. under our basic setting. We obtain a lower bound... -
Moments
We saw in Sect. 5.3 the claim that the asymptotic moments can be derived by differentiating the... -
Potential Functions of Random Walks in ℤ with Infinite Variance Estimates and Applications
This book studies the potential functions of one-dimensional recurrent random walks on the lattice of integers with step distribution of infinite...
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Pál–Bell Equation and Moment Growth
The previous chapter largely dealt with the relationship between the NTE and the NBP. The NTE is a linear equation and so there is limited... -
Some Background Markov Process Theory
Before we embark on our journey to explore the NTE in a stochastic context, we need to lay out some core theory of Markov processes that will appear... -
The Two-Sided Exit Problem for Relatively Stable Walks
This chapter is a continuation of Chapter 6. We use the same notation as therein. As in Chapter 6, we shall be primarily concerned with the event -
The Two-Sided Exit Problem – General Case
Remark 6.1.6 For general random walks (not restricted to arithmetic ones), Kesten and Maller [47] gave an analytic equivalent for the r.w. S to exit... -
Asymptotically Stable Random Walks Killed Upon Hitting a Finite Set
This section is concerned with the potential function for the r.w. S killed upon hitting a finite set. For its description, we do not need (AS). The... -
Martingale Convergence and Laws of Large Numbers
As usual, we will work in the setting that our branching Markov process,... -
Spines and Skeletons
We have seen in Chap. 6 that a natural way to study the long-term behaviour of the NBP is via spine... -
Generational Evolution
The eigenvalue problem in Theorem 4.1 is not the only one that offers insight into the evolution of... -
Survival at Criticality
We will remain in the setting of the Asmussen–Hering class of BMPs, i.e., assuming (G2), and insist throughout this chapter that we are in the...