Weak stability and generalized weak convolution for random vectors and stochastic processes

1. The main notions.- 2. The main lemmas.- 2.1. General lemmas on the approximation of distribution of an arbitrary random variable by the normal distribution.- 2.2. Proof of lemmas 2.1-2.4.- 3. Theorems on large deviations for the distributions of sums of independent random variables.- 3.1. Theorems on large deviations under Bernstein's condition.- a) Sums of non-identically distributed random variables.- b) Sums of weighted random variables.- 3.2. A theorem of large deviations in terms of Lyapunov's fractions.- 4. Theorems of large deviations for sums of dependent random variables.- 4.1. Estimates of the kth order centered moments of random processes with mixing.- 4.2. Estimates of mixed cumulants of random processes with mixing.- 4.3. Estimates of cumulants of sums of dependent random variables.- 4.4. Theorems and inequalities of large deviations for sums of dependent random variables.- 5. Theorems of large deviations for polynomial forms, multiple stochastic integrals and statistical estimates.- 5.1. Estimates of cumulants and theorems of large deviations for polynomial forms, polynomial Pitman estimates and U-statistics.- 5.2. Cumulants of multiple stochastic integrals and theorems of large deviations.- 5.3. Large deviations for estimates of the spectrum of a stationary sequence.- 6. Asymptotic expansions in the zones of large deviations.- 6.1. Asymptotic expansion for distribution density of an arbitrary random variable.- 6.2. Estimates for characteristic functions.- 6.3. Asymptotic expansion in the Cramer zone for distribution density of sums of independent random variables.- 6.4. Asymptotic expansions in integral theorems with large deviations.- 7. Probabilities of large deviations for random vectors.- 7.1. General lemmas on large deviations for a random vector with regular behaviour of cumulants.- 7.2. Theorems on large deviations for sums of random vectors and quadratic forms.- a) Sums of non-identically distributed random vectors.- b) Sums of weighted random vectors.- c) Sums of random number of random vectors.- d) Quadratic forms.- Appendices.- Appendix 1. Proof of inequalities for moments and Lyapunov's fractions.- Appendix 2. Proof of the lemma on the representation of cumulants.- Appendix 3. Leonov - Shiryaev's formula.- References.

The stable distribution, in its many parametrizations, is central to many stochastic processes. Many random variables that occur in the study of Levy processes are related to it. Good progress has been made recently for simulating various quantities related to the stable law. In this note, we survey exact random variate generators for these distributions. Many distributional identities are also reviewed.

In this chapter we introduce some basic ideas of time series analysis and stochastic processes. Of particular importance are the concepts Of stationarity and the autocovariance and sample autocovariance functions. Some standard techniques are described for the estimation and removal of trend and seasonality (of known period) from an observed time series. These are illustrated with reference to the data sets in Section 1.1. The calculations in all the examples can be carried out using the programs supplied on the enclosed diskette. The data sets are contained in files with names ending in.DAT. For example, the Australian red wine sales are filed as WINE.DAT. Most of the topics covered in this chapter will be developed more fully in later sections of the book. The reader who is not already familiar with random variables and random vectors should first read Appendix A where a concise account of the required background is given.KeywordsAutocorrelation FunctionTime Series ModelQuadratic TrendStationary Time SeriesSeasonal ComponentThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Chapter 7 Implementing Representations of Uncertainty

In this chapter, we introduce some important inequalities about stochastic processes and Banach space valued random elements. Because the image space of the random elements is more complicated than those for random variables or vectors, the proofs are generally more involved or need specific approaches. Therefore, proofs of some inequalities will be omitted and some references are given.

Stable distributions are a class of distributions which allow skewness and heavy tail. Non‐Gaussian stable random variables play the role of normal distribution in the central limit theorem, for normalized sums of random variables with infinite variance. The lack of analytic formula for density and distribution functions of stable random variables has been a major drawback to the use of stable distributions, also in the case of inference in Bayesian framework. Buckle introduced priors for the parameters of stable random variables to obtain an analytic form of posterior distribution. However, many researchers tried to solve the problem, through the Markov chain Monte Carlo methods, e.g. [8] and their references. In this paper a new class of heavy‐tailed distribution is introduced, called skewed stable. This class has two main advantages: It has many inferential advantages, since it is a member of exponential family, so the Bayesian inference can be drawn similar to the exponential family of distributions and modelling skew data with stable distributions is dominated by this family. Finally, Bayesian inference for skewed stable arc compared to the stable distributions through a few simulations study.

A record among a sequence of iid random variables $$X_1,X_2,\dots $$ on the real line is defined as a member $$X_n$$ such that $$X_n>\max (X_1,\cdots ,X_{n-1})$$ . Trying to generalize this concept to random vectors, or even stochastic processes with continuous sample paths, we introduce two different concepts: A simple record is a stochastic process (or a random vector) $${\varvec{X}}_n$$ that is larger than $${\varvec{X}}_1,\cdots ,{\varvec{X}}_{n-1}$$ in at least one component, whereas a complete record has to be larger than its predecessors in all components. In particular, the probability that a stochastic process $${\varvec{X}}_n$$ is a record as n tends to infinity is studied, assuming that the processes are in the max-domain of attraction of a max-stable process. Furthermore, the conditional distribution of $${\varvec{X}}_n$$ given that $${\varvec{X}}_n$$ is a record is derived.

Random function based spectral representation of stationary and non-stationary stochastic processes

Quantities governed by randomness correspond to functions on the probability space called random variables. The value taken by a random variable is subject to chance, and the associated likelihoods are described by a function called the distribution function. Two important classes of random variables are discussed, namely discrete variables and continuous variables. The law of averages, known also as the law of large numbers, states that the proportion of successes in a long run of independent trials converges to the probability of success in any one trial. This result provides a mathematical basis for a philosophical view of probability based on repeated experimentation. Worked examples involving random variables and their distributions are included, and the chapter terminates with sections on random vectors and on Monte Carlo simulation.

A random vector ${\bf X}$ is weakly stable if and only if for all $a,b \in {\mathbb R}$ there exists a random variable $\Theta$ such that $a{\bf X} + b {\bf X}' \stackrel{d}{=} {\bf X} \Theta$, where ${\bf X}'$ is an independent copy of ${\bf X}$ and $\Theta$ is independent of ${\bf X}$. This is equivalent (see [J. K. Misiewicz, K. Oleszkiewicz, and K. Urbanik, Studia Math., 167 (2005), pp. 195--213]) to the condition that for all random variables $Q_1, Q_2$ there exists a random variable $\Theta$ such that ${\bf X} Q_1 + {\bf X}' Q_2 \stackrel{d}{=} {\bf X} \Theta,$ where ${\bf X}, {\bf X}', Q_1, Q_2, \Theta$ are independent. In this paper we define a weak generalized convolution of measures defined by the formula ${\cal L}(Q_1) \otimes_{\mu} {\cal L}(Q_2) = {\cal L}(\Theta),$ if the former equation holds for ${\bf X}, Q_1, Q_2, \Theta$ and $\mu = {\cal L}(\bf X)$. We study here basic properties of this convolution and basic properties of distributions which are infinitely divisible in the sense of this ...

It was analyzed the current state of methodology for determining the estimated value of fire risk in buildings, structures and fire compartments of different classes of functional fire hazard (Annex to the Order of the Ministry of Emergency Situations of Russia on 30.06.2009 No. 382 as amended on 02.12.2015). It was founded that the main parameters used in the procedure are random variables and functions, describing stochastic processes in nature. In the same procedure there are used deterministic parameters of the calculated values, which can sometimes lead to the actual non-compliance with the normative values of individual fire risk. It is shown that the probability of evacuation, accepted under certain conditions equal to 0.999, shall be provided and calculated taking into account the random nature of the spread of fire hazards and processes evacuation during fires, fire safety characteristics of the system objects. It is proposed to improve the methodology for assessing individual fire risk for public facilities in the following areas: 1) improved methods of structure; 2) consideration of the stochastic nature of the input factors affecting the magnitude of the fire risk; 3) accounting for stochastic processes of evacuation, the emergence and spread of fire hazards in the way of evacuation; 4) extension of the list of input factors that affect the value of fire risk, based on their probability characteristics. In this case the object is given a list of fire scenarios, which is described as an event tree. To simplify accounting stochastic computational variables and processes dynamics hazards of fire and evacuation of magnitude Q v individual fire risk assessment proposed to be carried out in two stages. In the first stage, covering the calculated variants of fire scenarios, the probability of a successful evacuation Ver( t bl > t se + t e ), where t bl - while blocking dangerous factors of fire escape routes; t se - the start time of the evacuation; t e - time of the evacuation can be carried out Monte Carlo using simulation systems. In the second stage payment identified in the first phase of extreme situations is carried out using the specified models, fully taking into account the specifics of the individual movements of people that make up the flow of evacuees in this building, and possible dynamics of fire hazards taking into account peculiarities of functioning of its engineering systems. The proposals on improvement of individual risk methods allow to increase the reliability and accuracy of the estimates, although stipulate additional requirements in terms of knowledge of random variables of input factors distributions.

Numerous studies have attempted to determine whether stream fish communities are structured primarily by deterministic or stochastic processes. Previous work has assumed that stream fish communities will show either persistence about an equilibrium because of strong density-dependent processes or random variation in structure as a result of environmental stochasticity. In a 10-year study of a California stream, fish community structure changed under the influence of storm-induced high discharge events that impacted recruitment. Species' relative abundances were altered as pre-recruitment stream discharges differentially influenced year-class strength among species with contrasting life histories. Simulation of stream fish community assembly under flow-driven recruitment variation indicates that community structure will vary depending on how particular high-flow events affect species' relative abundances and ongoing density-dependent processes, including competition and predation. Results suggest that stream fish communities are likely to show alternate states rather than a single persistent equilibrium. However, community assembly will not be random but will depend on situation-specific interactions between density-independent and density-dependent processes.

For a random vector $X$ with a fixed distribution $\mu$ we construct a class of distributions ${\cal M}(\mu)= \{ \mu\circ\lambda: \lambda\in{\cal P}\}$, which is the class of all distributions of random vectors $X {\mit\Theta}$, where $ {\mit\Theta}$ is i

A time-variant reliability analysis method for non-linear limit-state functions with the mixture of random and interval variables

DISTANCE COVARIANCE FOR STOCHASTIC PROCESSESThe distance covariance of two random vectors is a measure of their dependence. The empirical distance covariance and correlation can be used as statistical tools for testing whether two random vectors are independent. We propose an analog of the distance covariance for two stochastic processes defined on some interval. Their empirical analogs can be used to test the independence of two processes.