Limit Theorems For Distributions Research Articles

Limit theorems for power-series distributions with finite radius of convergence

Получены достаточные условия слабой сходимости распределений случайных величин $(1-x)\xi_x$ при $x\to1-$ к предельному гамма-распределению. Случайная величина $\xi_x$ имеет распределение типа степенного ряда с радиусом сходимости $1$ и параметром $x\in(0,1)$. Доказаны предельные теоремы для вероятностей $\mathbf P\{\xi_x=k\}$. Выведены асимптотические разложения локальных вероятностей для сумм независимых одинаково распределенных случайных величин, имеющих распределение случайной величины $\xi_x$, в схеме серий, когда $x\to1-$. Для соответствующей обобщенной схемы размещения получены локальные предельные теоремы, оценивающие совместные распределения заполнений ячеек.

Limit Theorems for Power-Series Distributions with Finite Radius of Convergence

Sufficient conditions for the weak convergence of the distributions of the random variables $(1-x)\xi_x$ as $x\to1-$ to the limiting gamma-distribution are put forward. The random variable $\xi_x$ ...

Mean Width of Regular Polytopes and Expected Maxima of Correlated Gaussian Variables

An old conjecture states that among all simplices inscribed in the unit sphere, the regular one has the maximal mean width. We restate this conjecture probabilistically and prove its asymptotic version. We also show that the mean width of the regular simplex with 2n vertices is remarkably close to the mean width of the regular crosspolytope with the same number of vertices. We establish several formulas conjectured by S. Finch on the projection length W of the regular cube, simplex, and crosspolytope onto a line with random direction. Finally, we prove distributional limit theorems for W as the dimension of the regular polytope goes to ∞. Bibliography: 25 titles.

Limit theory for the Gilbert graph

For a given homogeneous Poisson point process in Rd two points are connected by an edge if their distance is bounded by a prescribed distance parameter. The behavior of the resulting random graph, the Gilbert graph or random geometric graph, is investigated as the intensity of the Poisson point process is increased and the distance parameter goes to zero. The asymptotic expectation and covariance structure of a class of length-power functionals are computed. Distributional limit theorems are derived that have a Gaussian, a stable or a compound Poisson limiting distribution. Finally, concentration inequalities are provided using the convex distance.

Temporal Distributional Limit Theorems for Dynamical Systems

Suppose \(\{T^t\}\) is a Borel flow on a complete separable metric space X, \(f:X\rightarrow \mathbb R\) is Borel, and \(x\in X\). A temporal distributional limit theorem is a scaling limit for the distributions of the random variables \(X_T:=\int _0^t f(T^s x)ds\), where t is chosen randomly uniformly from [0, T], x is fixed, and \(T\rightarrow \infty \). We discuss such laws for irrational rotations, Anosov flows, and horocycle flows.

Exact convergence rate of the local limit theorem for branching random walks on the integer lattice

Consider branching random walks on the integer lattice Zd, where the branching mechanism is governed by a supercritical Galton–Watson process and the particles perform a symmetric nearest-neighbor random walk whose increments equal to zero with probability r∈[0,1). We derive exact convergence rate in the local limit theorem for distributions of particles. When r=0, our results correct and improve the existing results on the convergence speed conjectured by Révész (1994) and proved by Chen (2001). As a byproduct, we obtain exact convergence rate in the local limit theorem for some symmetric nearest-neighbor random walks, which is of independent interest.

On repetitions of long tuples in a Markov chain

AbstractLet X{Xwhen s, n → ∞. Limit theorems for distributions of some random variables connected with ξ̄(s, n) are derived as corollaries.

Distributional Behavior of Time Averages of Non- $$L^1$$ L 1 Observables in One-dimensional Intermittent Maps with Infinite Invariant Measures

In infinite ergodic theory, two distributional limit theorems are well-known. One is characterized by the Mittag-Leffler distribution for time averages of \(L^1(m)\) functions, i.e., integrable functions with respect to an infinite invariant measure. The other is characterized by the generalized arc-sine distribution for time averages of non-\(L^1(m)\) functions. Here, we provide another distributional behavior of time averages of non-\(L^1(m)\) functions in one-dimensional intermittent maps where each has an indifferent fixed point and an infinite invariant measure. Observation functions considered here are non-\(L^1(m)\) functions which vanish at the indifferent fixed point. We call this class of observation functions weak non-\(L^1(m)\) function. Our main result represents a first step toward a third distributional limit theorem, i.e., a distributional limit theorem for this class of observables, in infinite ergodic theory. To prove our proposition, we propose a stochastic process induced by a renewal process to mimic a Birkoff sum of a weak non-\(L^1(m)\) function in the one-dimensional intermittent maps.

Richter’s local limit theorem and Black–Scholes type formulas

We prove a Black–Scholes type formula when the geometric Brownian motion originates from approximations by multinomial distributions. It is shown that the variance appearing in the Black–Scholes formula for option pricing can be structured according to occurrences of different types of events at each time instance using a local limit theorem for multinomial distributions in Richter (1956). The general approach has first been developed in Kan (2005).

Limit theory for some positive stationary processes with infinite mean

Nous prouvons des théorèmes limites et des lois du logarithme itéré unilatérales pour une classe de processus stochastiques positifs, mélangeants et stationnaires. Cette classe contient en particulier les processus obtenus par des observables nonintégrables de certaines applications dilatantes. Ceci est obtenu en généralisant la théorie de Darling–Kac à une famille appropriée de transformations préservant la mesure.

Distributions of time averages for weakly chaotic systems: The role of infinite invariant density

Distributions of time averaged observables are investigated using deterministic maps with N indifferent fixed points and N-state continuous time random walk processes associated with them. In a weakly chaotic phase, namely when separation of trajectories is subexponential, maps are characterized by an infinite invariant density. We find that the infinite density can be used to calculate the distribution of time averages of integrable observables with a formula recently obtained by Rebenshtok and Barkai. As an example we calculate distributions of the average position of the particle and average occupation fractions. Our work provides the distributional limit theorem for time averages for a wide class of nonintegrable observables with respect to the infinite invariant density, in other words it deals with the situation where the Darling-Kac-Aaronson theorem does not hold.

Aging generates regular motions in weakly chaotic systems

Using intermittent maps with infinite invariant measures, we investigate the universality of time-averaged observables under aging conditions. According to Aaronson's Darling-Kac theorem, in non-aged dynamical systems with infinite invariant measures, the distribution of the normalized time averages of integrable functions converges to the Mittag-Leffler distribution. This well known theorem holds when the start of observations coincides with the start of the dynamical processes. Introducing a concept of the aging limit where the aging time ${t}_{a}$ and the total measurement time $t$ goes to infinity while the aging ratio ${t}_{a}/t$ is kept constant, we obtain a novel distributional limit theorem of time-averaged observables integrable with respect to the infinite invariant density. Applying the theorem to the Lyapunov exponent in intermittent maps, we find that regular motions and a weakly chaotic behavior coexist in the aging limit. This mixed type of dynamics is controlled by the aging ratio and hence is very different from the usual scenario of regular and chaotic motions in Hamiltonian systems. The probability of finding regular motions in non-aged processes is zero, while in the aging regime it is finite and it increases when system ages.

Recurrence rates and hitting-time distributions for random walks on the line

We consider random walks on the line given by a sequence of independent identically distributed jumps belonging to the strict domain of attraction of a stable distribution, and first determine the almost sure exponential divergence rate, as r goes to zero, of the return time to (-r,r). We then refine this result by establishing a limit theorem for the hitting-time distributions of (x-r,x+r) with arbitrary real x.

Exponential chi-squared distributions in infinite ergodic theory

AbstractWe prove distributional limit theorems for random walk adic transformations obtaining ergodic distributional limits of exponential$\chi ^2$form.

О вероятности протяжки однобитовой разности через сложение и вычитание по модулю

О вероятности протяжки однобитовой разности через сложение и вычитание по модулю

Оценки скорости сходимости в предельных теоремах для совместных распределений части характеристик случайных двоичных отображений

Оценки скорости сходимости в предельных теоремах для совместных распределений части характеристик случайных двоичных отображений

Relative complexity of random walks in random sceneries

Relative complexity measures the complexity of a probability preserving transformation relative to a factor being a sequence of random variables whose exponential growth rate is the relative entropy of the extension. We prove distributional limit theorems for the relative complexity of certain zero entropy extensions: RWRSs whose associated random walks satisfy the $\alpha$-stable CLT ($1<\alpha\le2$). The results give invariants for relative isomorphism of these.

Thermodynamic phase transitions for Pomeau-Manneville maps

We study phase transitions in the thermodynamic description of Pomeau-Manneville intermittent maps from the point of view of infinite ergodic theory, which deals with diverging measure dynamical systems. For such systems, we use a distributional limit theorem to provide both a powerful tool for calculating thermodynamic potentials as also an understanding of the dynamic characteristics at each instability phase. In particular, topological pressure and Rényi entropy are calculated exactly for such systems. Finally, we show the connection of the distributional limit theorem with non-Gaussian fluctuations of the algorithmic complexity proposed by Gaspard and Wang [Proc. Natl. Acad. Sci. USA 85, 4591 (1988)].

Some distributional limit theorems for the maxima of Gaussian vector sequences

In this paper, not only the weak convergence is considered, as in the ASCLT in Theorem 2.3 the a.s. convergence is at stake for extreme-order statistics from the sequences of normal random vectors, which may be stationary or not. In particular, we consider an almost sure limit theorem for the maxima.

Distributional Response to Biases in Deterministic Superdiffusion

We report on a novel response to biases in deterministic superdiffusion. For its reduced map, we show using infinite ergodic theory that the time-averaged velocity (TAV) is intrinsically random and its distribution obeys the generalized arcsine distribution. A distributional limit theorem indicates that the TAV response to a bias appears in the distribution, which is an example of what we term a distributional response induced by a bias. Although this response in single trajectories is intrinsically random, the ensemble-averaged TAV response is linear.