In this research, we consider cyber risk in insurance using a quantum approach, with a focus on the differences between reported cyber claims and the number of cyber attacks that caused them. Unlike the traditional probabilistic approach, quantum modeling makes it possible to deal with non-commutative event paths. We investigate the classification of cyber claims according to different cyber risk behaviors to enable more precise analysis and management of cyber risks. Additionally, we examine how historical cyber claims can be utilized through the application of copula functions for dependent insurance claims. We also discuss classification, likelihood estimation, and risk-loss calculation within the context of dependent insurance claim data.

For the partial sums Sn = Y1 + . . . + Yn of a centered stationary strongly mixing sequence {Yi} with finite second moment, the well-known sufficient conditions for the central limit theorem are that Var(Sn)/n is slowly varying as n → ∞ and the sequence {S2 n/σn} is uniformly integrable (σn = Var(Sn)). The conditions are checkable under various mixing conditions and they lead to the central limit theorem under the normalization σn(see, Denker (1986) and Peligrad (1986) for a survey). Dehling, Denker and Philipp (1986) proved an interesting central limit theorem using the nontraditional normalization ρn = E|Sn|. One of their results, Theorem 3, roughly states that if both sequences σn/n and ρn/ √ n are slowly varying as n → ∞, then the central limit theorem holds. On the other hand, Braverman, Mallows and Shepp (1995) showed that if the absolute moments of partial sums of i.i.d. symmetric variables are equal to those of normal variables, then the marginals have normal distribution. This fact suggested the conjecture that probably the absolute moments alone characterize the homogeneous process with independent increments (see Bryc (2002) for a discussion on this topic and related conjectures). Our main interest in this topic is to prove some of these conjectures and to apply them to understand the nature of the intricate normalization in Dehling, Denker and Philipp (1986).

Motivated by modelling epidemics like COVID-19, this paper proposes a generalized chain binomial process which integrates two types of infectives, those with symptoms and those without. Testing of infectives and vaccination of susceptibles are then incorporated as preventive protective measures. Our interest relates to the distribution of the state of the population at the end of infection and to the reproduction number $$R_0$$ with the associated extinction condition. The method uses the construction of a family of martingales and a branching approximation for large populations, respectively. A more general branching process for epidemics is also constructed and studied. Finally, some results obtained are illustrated by numerical examples.

We apply the Lie bracket approach to characterize the semi-derivations on the positive integers. The approach is motivated by the Stroock-Lie bracket identity commonly used in Malliavin calculus.

We discuss a continuous-time Markov branching model in which each individual can trigger an alarm according to a Poisson process. The model is stopped when a given number of alarms is triggered or when there are no more individuals present. Our goal is to determine the distribution of the state of the population at this stopping time. In addition, the state distribution at any fixed time is also obtained. The model is then modified to take into account the possible influence of death cases. All distributions are derived using probability-generating functions, and the approach followed is based on the construction of families of martingales.

AbstractIn this chapter we further comment on the sharpness of several results presented in this monograph, by presenting examples and counterexamples. We study first the moment properties of the renewal Markov chain introduced in Chapter 11. This allows us to show that the Maxwell–Woodroofe projective condition introduced in Chapter 4 is essentially optimal for the partial sums of a stationary sequence in L2 to satisfy the central limit theorem under the standard normalization √n. Moreover, we also investigate the sharpness of the Burkholder-type inequality developed in Chapter 3 via Maxwell–Woodroofe-type characteristics. In the last part of this chapter, we analyze several telescopic-type examples allowing us to elucidate the fact that a CLT behavior does not imply its functional form under any normalization. Even in the case when the variance of the partial sums is linear in n, the CLT does not necessarily imply the invariance principle.

In this paper we develop non-stationary martingale techniques for dependent data. We shall stress the non-stationary version of the projective Maxwell-Woodroofe condition, which will be essential for obtaining maximal inequalities and functional central limit theorem for the following examples: nonstationary \r{ho}-mixing sequences, functions of linear processes with non-stationary innovations, quenched version of the functional central limit theorem for a stationary sequence, evolutions in random media such as a process sampled by a shifted Markov chain.

Here we introduce the notion of asymptotic weakly associated dependence conditions, the practical applications of which will be discussed in the next chapter. The theoretical importance of this class of random variables is that it leads to the functional CLT without the need to estimate rates of convergence of mixing coefficients. More precisely, because of the maximal moment inequalities established in the previous chapter, we are able to prove tightness for a stochastic process constructed from a negatively dependent sequence. Furthermore, we establish the convergence of the partial sums process, either to a Gaussian process with independent increments or to a diffusion process with deterministic time-varying volatility. We also provide the multivariate form of these functional limit theorems. The results are presented in the non-stationary setting, by imposing Lindeberg’s condition. Finally, we give the stationary form of our results for both asymptotic positively and negatively associated sequences of random variables.

The aim of this chapter is to prove Khintchine–Marcinkiewicz–Zygmund or Rosenthal-type moment inequalities for the partial sums or for the maximum of partial sums, in terms of weak dependence coefficients. We start with general probability and moment inequalities for Lipschitz functions of weakly negatively dependent variables. Besides being of interest in themselves, they will be key tools for proving moment inequalities for the partial sums associated with weakly negatively dependent variables. Some parts of this chapter are devoted to the weak law of large numbers, useful to get the convergence of quadratic characteristics such as the quadratic variation and also to obtain an alternative characterization of the notion of weakly negatively dependent vectors.

AbstractIn this chapter, we treat several examples of stationary processes which are asymptotically negatively dependent and for which the results of Chapter 9 apply. Many systems in nature are complex, consisting of the contributions of several independent components. Our first examples are functions of two independent sequences, one negatively dependent and one interlaced mixing. For instance, the class of asymptotic negatively dependent random variables is used to treat functions of a determinantal point process and a Gaussian process with a positive continuous spectral density. Another example is point processes based on asymptotically negatively or positively associated sequences and displaced according to a Gaussian sequence with a positive continuous spectral density. Other examples include exchangeable processes, the weighted empirical process, and the exchangeable determinantal point process.

The purposes of this chapter are to introduce the notion of weakly associated (negatively or positively) random variables and to develop tools that will allow us, in the chapters to follow, to give estimations of moments of partial sums, maximal inequalities, and asymptotic results with both Gaussian and non-gaussian limits. As we shall see, these results shed light on the asymptotic behavior of numerous examples such as exchangeable variables, certain Gaussian processes, empirical processes, various classes of Markov chains, and determinantal processes. They are also useful to study stochastic processes that are functionals of the two independent processes mentioned above.

Gordin’s seminal paper (1969) initiated a line of research in which limit theorems for stationary sequences are proved via appropriate approximations by stationary martingale difference sequences followed by an application of the corresponding limit theorem for such sequences. In this chapter, we first review different ways to get suitable martingale approximations and then derive the central limit theorem and its functional form for strictly stationary sequences under various types of projective criteria. More general normalizations than the traditional ones will be also investigated, as well as the functional moderate deviation principle. We shall also address the question of the functional form of the central limit theorem for not necessarily stationary sequences. The last part of this chapter is dedicated to the moderate deviations principle and its functional form for stationary sequences of bounded random variables satisfying projective-type conditions.

This book has its origin in the need for developing and analyzing mathematical models for phenomena that evolve in time and influence each another, and aims at a better understanding of the structure and asymptotic behavior of stochastic processes. This monograph has double scope. First, to present tools for dealing with dependent structures directed toward obtaining normal approximations. Second, to apply the normal approximations presented in the book to various examples. The main tools consist of inequalities for dependent sequences of random variables, leading to limit theorems, including the functional central limit theorem (CLT) and functional moderate deviation principle (MDP). The results will point out large classes of dependent random variables which satisfy invariance principles, making possible the statistical study of data coming from stochastic processes both with short and long memory. Over the course of the book different types of dependence structures are considered, ranging from the traditional mixing structures to martingale-like structures and to weakly negatively dependent structures, which link the notion of mixing to the notions of association and negative dependence. Several applications have been carefully selected to exhibit the importance of the theoretical results. They include random walks in random scenery and determinantal processes. In addition, due to their importance in analyzing new data in economics, linear processes with dependent innovations will also be considered and analyzed.

In this chapter, we analyze the asymptotic behavior of the partial sums process associated with examples of stationary sequences in a random time scenery. The examples considered are stationary sequences sampled by shifted renewal Markov chains and random walks in a strictly stationary scenery. The asymptotic behavior of the partial sums process is essentially investigated with the help of the weak invariance principles stated in Chapter 4. More precisely, for the partial sums process associated with a stationary process sampled by a renewal Markov chain stated at zero, due to the non-stationarity of the underlying sequence, the functional CLT is obtained as an application of the functional CLT for non-stationary sequences developed in Section 4.4. In the case where we are sampling a strictly stationary random scenery by a random walk, stationarity is preserved, and the invariance principle is then derived by using the functional CLT under Maxwell–Woodroofe condition.

In this chapter we investigate the question of central limit behavior and its functional form for the partial sums associated with a centered L2-stationary sequence of real-valued random variables (usually called the random scenery) sampled by a recurrent one-dimensional strongly aperiodic random walk. This question is handled under various conditions dependent on the random scenery. In particular, we assume that the random scenery either satisfies an asymptotic negative dependence condition, or is a function of a determinantal process and a Gaussian sequence, or satisfies a mild projective criterion. We first show that study of central limit behavior for such random walks in random scenery can be handled with results related to linear statistics developed in Chapter 12, provided the random walk has good properties. We then look extensively at the properties of a recurrent one-dimensional strongly aperiodic random walk. The functional form of the central limit theorem is also investigated.

The paper proposes to treat financial models using techniques of quantum mechanics. The methodology relies on the Dirac matrix formalism and the Feynman path integral approach. This leads us to reexamine in this framework the classical option pricing models of Cox-Ross-Rubinstein and Black-Scholes. Moreover, financial data are classified with respect to the spectrum of a certain observable and then analyzed to identify price jumps using supervised machine learning tools.

AbstractIn this chapter we establish different kinds of moment inequalities for partial sums and the maximum of partial sums of a large class of random variables, including martingale sequences, mixingales, and other dependent structures. All the bounds involve the moments of the conditional expectations of either the partial sums or the individual random variables. In most of the proofs martingale approximations are used. This method allows us to use the moment inequalities for the martingale part developed in Chapter 2. We start with a dyadic scheme useful for analysis of the variance of partial sums in the stationary setting. Then, we obtain Burkholder-type inequalities via Maxwell–Woodroofe-type characteristics and an extension of Doob’s maximal inequality for adapted sequences. A Rosenthal-type inequality for stationary sequences is also provided with bounds using conditional expectations of the partial sums. Maximal exponential inequalities are established involving either Maxwell–Woodroofe-type characteristics or the projective operators.

AbstractHere we discuss the Gaussian approximation for the empirical process under different kinds of dependence assumptions for the underlying stationary sequence. First, we state a general criterion to prove tightness of the empirical process associated with a stationary sequence of uniformly distributed random variables. This tightness criterion can be verified for many different dependence structures. For ρ-mixing sequences, by an application of a Rosenthal-type inequality, tightness is verified under the same condition leading to the usual CLT. For α-dependent sequences whose α-dependent coefficients decay polynomially to zero, it is shown to hold with the help of the Rosenthal inequality stated in Section 3.3. Since the asymptotic behavior of the finite-dimensional distributions of the empirical process is handled via the CLT developed in previous chapters, we then derive the functional CLT for the empirical process associated with the above-mentioned classes of stationary sequences. β-dependent sequences are also investigated by directly proving tightness of the empirical process.

AbstractThe aim of this chapter is to present useful tools for analyzing the asymptotic behavior of partial sums associated with dependent sequences, by approximating them with martingales. We start by collecting maximal and moment inequalities for martingales such as the Doob maximal inequality, the Burkholder inequality, and the Rosenthal inequality. Exponential inequalities for martingales are also provided. We then present several sufficient conditions for the central limit behavior and its functional form for triangular arrays of martingales. The last part of the chapter is devoted to the moderate deviations principle and its functional form for triangular arrays of martingale difference sequences.

AbstractWe start by stating the need for a Gaussian approximation for dependent structures in the form of the central limit theorem (CLT) or of the functional CLT. To justify the need to quantify the dependence, we introduce illustrative examples: linear processes, functions of stationary sequences, recursive sequences, dynamical systems, additive functionals of Markov chains, and self-interactions. The limiting behavior of the associated partial sums can be handled with tools developed throughout the book. We also present basic notions for stationary sequences of random variables: various definitions and constructions, and definitions of ergodicity, projective decomposition, and spectral density. Special attention is given to dynamical systems, as many of our results also apply in this context. The chapter also surveys the basic theory of the convergence of stochastic processes in distribution, and introduces the reader to tightness, finite-dimensional convergence, and the need for maximal inequalities. It ends with the concepts of the moderate deviations principle and its functional form.