Conditional Limit Theorem Research Articles

A conditional limit theorem for a bivariate representation of a univariate random variable and conditional extreme values

We first consider a real random variable X represented through a random pair (R,T) and a deterministic function u as X = R⋅u(T). Under quite weak assumptions we prove a limit theorem for (R,T) given X>x, as x tends to infinity. The novelty of our paper is to show that this theorem for the representation of the univariate random variable X permits us to obtain in an elegant manner conditional limit theorems for random pairs (X,Y) = R⋅(u(T),v(T)) given that X is large. Our approach allows to deduce new results as well as to recover under considerably weaker assumptions results obtained previously in the literature. Consequently, it provides a better understanding and systematization of limit statements for the conditional extreme value models.

A Conditional Approach to Panel Data Models with Common Shocks

This paper studies the effects of common shocks on the OLS estimators of the slopes’ parameters in linear panel data models. The shocks are assumed to affect both the errors and some of the explanatory variables. In contrast to existing approaches, which rely on using results on martingale difference sequences, our method relies on conditional strong laws of large numbers and conditional central limit theorems for conditionally-heterogeneous random variables.

Decomposable branching processes with a fixed extinction moment

The asymptotic behavior as n→∞ of the probability of the event that a decomposable critical branching process Z(m) = (Z 1(m),..., Z N(m)), m = 0, 1, 2,..., with N types of particles dies at moment n is investigated, and conditional limit theorems are proved that describe the distribution of the number of particles in the process Z(·) at moment m < n given that the extinction moment of the process is n. These limit theorems can be considered as statements describing the distribution of the number of vertices in the layers of certain classes of simply generated random trees of fixed height.

Limit properties of subcritical CMJ processes about the coming generation

In this paper, we consider subcritical branching processes with overlapping generations in continuous time, which is sometimes referred to as subcritical CMJ processes in continuous time. CMJ processes depict populations of individuals who can produce offsprings at different ages, and time interval of next two reproductions is no longer of exponential type. We mainly study count of the coming at time t , which is called { H t }. The coming generation is potential population that are sure to be born at sometime in future. With help of renewal theorem, we obtain some conditional limit theorems for { H t }.

Some preliminary results on conditionally ψ-mixing sequences of random variables

From the ordinary notion of -mixing for a sequence of random variables, a new concept called conditionally -mixing is proposed. That conditionally -mixing neither implies nor is implied by -mixing is illustrated by examples. Certain criteria for checking conditionally -mixing property as well as basic properties are derived, and several conditional covariance inequalities are obtained. By means of these properties and inequalities, a conditional central limit theorem stated in terms of conditional characteristic functions is established.

Nonequilibrium Markov Processes Conditioned on Large Deviations

We consider the problem of conditioning a Markov process on a rare event and of representing this conditioned process by a conditioning-free process, called the effective or driven process. The basic assumption is that the rare event used in the conditioning is a large deviation-type event, characterized by a convex rate function. Under this assumption, we construct the driven process via a generalization of Doob's $h$-transform, used in the context of bridge processes, and show that this process is equivalent to the conditioned process in the long-time limit. The notion of equivalence that we consider is based on the logarithmic equivalence of path measures and implies that the two processes have the same typical states. In constructing the driven process, we also prove equivalence with the so-called exponential tilting of the Markov process, which is used with importance sampling to simulate rare events, and which gives rise, from the point of view of statistical mechanics, to a nonequilibrium version of the canonical ensemble. Other links between our results and the topics of bridge processes, quasi-stationary distributions, stochastic control, and conditional limit theorems are mentioned.

A quenched weak invariance principle

Dans cet article, nous étudions le théorème central limite conditionnel presque sûr, ainsi que sa forme fonctionnelle, pour des suites stationnaires de variables aléatoires réelles satisfaisant une condition de type projectif. Nous donnons des applications de ces résultats aux processus fortement mélangeants ainsi qu’à des chaînes de Markov nonirréductibles. Les preuves sont essentiellement basées sur une approximation normale de suites doublement indexées de variables aléatoires de type martingale.

Conditional limit theorems for critical continuous-state branching processes

We study the conditional limit theorems for critical continuous-state branching processes with branching mechanism ψ(λ) = λ1+α L(1/λ), where α ∈ [0, 1] and L is slowly varying at ∞. We prove that if α ∈ (0, 1], there are norming constants Q t → 0 (as t ↑ +∞) such that for every x > 0, P x (Q t X t ∈ · |X t > 0) converges weakly to a non-degenerate limit. The converse assertion is also true provided the regularity of ψ at 0. We give a conditional limit theorem for the case α = 0. The limit theorems we obtain in this paper allow infinite variance of the branching process.

Limit theorems for strongly and intermediately supercritical branching processes in random environment with linear fractional offspring distributions

In the present paper, we characterize the behavior of supercritical branching processes in random environment with linear fractional offspring distributions, conditioned on having small, but positive values at some large generation. As it has been noticed in previous works, there is a phase transition in the behavior of the process. Here, we examine the strongly and intermediately supercritical regimes The main result is a conditional limit theorem for the rescaled associated random walk in the intermediately case.

On the maximal displacement of critical branching random walk

We consider a branching random walk initiated by a single particle at location 0 in which particles alternately reproduce according to the law of a Galton-Watson process and disperse according to the law of a driftless random walk on the integers. When the offspring distribution has mean 1 the branching process is critical, and therefore dies out with probability 1. We prove that if the particle jump distribution has mean zero, positive finite variance $\eta^{2}$, and finite $4+\varepsilon$ moment, and if the offspring distribution has positive variance $\sigma^{2}$ and finite third moment then the distribution of the rightmost position $M$ reached by a particle of the branching random walk satisfies $P\{M \geq x\}\sim 6\eta^{2}/ (\sigma^{2}x^{2})$ as $x \rightarrow \infty$. We also prove a conditional limit theorem for the distribution of the rightmost particle location at time $n$ given that the process survives for $n$ generations.

Comparison and anti-concentration bounds for maxima of Gaussian random vectors

Slepian and Sudakov–Fernique type inequalities, which compare expectations of maxima of Gaussian random vectors under certain restrictions on the covariance matrices, play an important role in probability theory, especially in empirical process and extreme value theories. Here we give explicit comparisons of expectations of smooth functions and distribution functions of maxima of Gaussian random vectors without any restriction on the covariance matrices. We also establish an anti-concentration inequality for the maximum of a Gaussian random vector, which derives a useful upper bound on the Lévy concentration function for the Gaussian maximum. The bound is dimension-free and applies to vectors with arbitrary covariance matrices. This anti-concentration inequality plays a crucial role in establishing bounds on the Kolmogorov distance between maxima of Gaussian random vectors. These results have immediate applications in mathematical statistics. As an example of application, we establish a conditional multiplier central limit theorem for maxima of sums of independent random vectors where the dimension of the vectors is possibly much larger than the sample size.

Total variation approximations and conditional limit theorems for multivariate regularly varying random walks conditioned on ruin

We study a new technique for the asymptotic analysis of heavy-tailed systems conditioned on large deviations events. We illustrate our approach in the context of ruin events of multidimensional regularly varying random walks. Our approach is to study the Markov process described by the random walk conditioned on hitting a rare target set. We construct a Markov chain whose transition kernel can be evaluated directly from the increment distribution of the associated random walk. This process is shown to approximate the conditional process of interest in total variation. Then, by analyzing the approximating process, we are able to obtain asymptotic conditional joint distributions and a conditional functional central limit theorem of several objects such as the time until ruin, the whole random walk prior to ruin, and the overshoot on the target set. These types of joint conditional limit theorems have been obtained previously in the literature only in the one dimensional case. In addition to using different techniques, our results include features that are qualitatively different from the one dimensional case. For instance, the asymptotic conditional law of the time to ruin is no longer purely Pareto as in the multidimensional case.

SOME RESULTS ON CONDITIONALLY UNIFORMLY STRONG MIXING SEQUENCES OF RANDOM VARIABLES

From the ordinary notion of uniformly strong mixing for a se- quence of random variables, a new concept called conditionally uniformly strong mixing is proposed and the relation between uniformly strong mix- ing and conditionally uniformly strong mixing is answered by examples, that is, uniformly strong mixing neither implies nor is implied by condi- tionally uniformly strong mixing. A couple of equivalent definitions and some of basic properties of conditionally uniformly strong mixing ran- dom variables are derived, and several conditional covariance inequalities are obtained. By means of these properties and conditional covariance inequalities, a conditional central limit theorem stated in terms of condi- tional characteristic functions is established, which is a conditional version of the earlier result under the non-conditional case.

Conditional limit theorems for intermediately subcritical branching processes in random environment

Nous considérons un processus de branchement dans un environnement aléatoire dont la distribution des enfants des individus varie aléatoirement de façon indépendante d’une génération à l’autre. Dans le régime sous critique, une transition de phase apparaît. Cet article est consacré à l’étude de la région proche de la transition. Nous étudions le comportement asymptotique de la probabilité de survie ainsi que la taille de la population et la forme de l’environnement aléatoire sous la condition de non-extinction. Nous montrons finalement que conditionnée à la non-extinction, la population alterne des périodes de petite et de grande taille. Ce type de comportement apparaît sous la mesure moyennée uniquement dans ce régime sous critique proche de la transition.

Branching processes in a Markov random environment

AbstractThe paper is concerned with critical branching processes in a Markov random environment. A conditional functional limit theorem for the number of particles in a process and a conditional invariance principle are proved. The asymptotic tail behaviour for the distributions of the maximum and the total number of particles in a process is found.

CONDITIONAL CENTRAL LIMIT THEOREMS FOR A SEQUENCE OF CONDITIONAL INDEPENDENT RANDOM VARIABLES

A conditional version of the classical central limit theorem is derived rigorously by using conditional characteristic functions, and a more general version of conditional central limit theorem for the case of conditionally independent but not necessarily conditionally identically distributed random variables is established. These are done anticipating that the field of conditional limit theory will prove to be of significant applicability.

On Large Deviations of Maximum of a Cramér Random Walk and the Queueing Process

In this paper large deviations of maxima $M_n$ and ${\cal M}^{W}_n$ of a random walk and the queueing process segments $S_j = \sum_{i=1}^j X_i$, $W_j$ $j \le n$, $W_{j+1} := \max (0, W_j + X_{j+1})$ are considered in the case of random variables $X_i$ subjected to the Cramér right-hand condition. Some results of Borovkov and Korshunov on asymptotics of probabilities ${\bf P}(M_n > tn)$, ${\bf E}\,X_j < 0$, $n\rightarrow\infty$ are improved. Conditional functional limit theorems for segments of a random walk and the queueing process conditioned by $T(tn) = k$, $T(x) := \inf (k: S_k > x)$, $M_n > tn$, and by $\vrule width0pt height9.5pt depth3pt {\cal M}^{W}_n > tn$ are obtained.

Conditional Limit Theorem for the Maximum of a Random Walk in Random Environment

Let $(p_{i},q_{i}) $, $i\in {\bf Z}$, be a sequence of independent identically distributed pairs of random variables, where $p_{0}+q_{0}=1$ and, in addition, $p_{0}>0$ and $q_{0}>0 $ a.s. We consider a random walk in the random environment $(p_{i},q_{i}) $, $i\in {\bf Z}$. This means that, given the environment, a walking particle located at some moment $n$ at a state $i$ jumps at the moment $n+1$ either to the state $(i+1)$ with probability $p_{i}$ or to the state $(i-1)$ with probability $q_{i}$. It is assumed that the random variable $\log ( q_{0}/p_{0}) $ has zero mean and finite positive variance $\sigma^{2} $. Let $X_{n}$ be the state of working particle at the moment $n$. It is shown that given $X_{1}\ge 0,\ldots ,X_{n}\ge 0,$ the random variable $\sigma^{2}\max_{0\le i\le n}X_{i}/\log^{2}n$ converges in distribution, as $n\to \infty $ to a positive random variable whose distribution function is known.

On conditional extreme values of random vectors with polar representation

An effective approach for studying the asymptotics of bivariate random vectors is to search for the limits of conditional probabilities where the conditioning variable becomes large. In this context, elliptical and related distributions have been extensively investigated. A quite general model was presented by Fougeres and Soulier (Limit conditional distributions for bivariate vectors with polar representation in Stochastic Models, 2010), who derived a conditional limit theorem for random vectors (X, Y) with a polar representation R · (u(T), v(T)), where R, T are stochastically independent and R is in the Gumbel max-domain of attraction. We reformulate their assumptions, such that they have a simpler structure, display more clearly the geometry of the curves (u(t), v(t)) and allow us to deduce interesting generalizations into two directions: u has several global maxima instead of only one, the curve (u(t), v(t)) is no longer differentiable, but forms a “cusp”. The latter generalization yields results where only random norming leads to a non-degenerate limit statement. Ideas and results are elucidated by several figures.

Simulations and a conditional limit theorem for intermediately subcritical branching processes in random environment

Intermediately subcritical branching processes in random environment are at the borderline between two subcritical regimes and exhibit particularly rich behavior. In this paper, we prove a functional limit theorem for these processes. It is discussed together with two other recently proved limit theorems for the intermediately subcritical case and illustrated by several computer simulations.