Cramer Condition Research Articles

Limit theorems for chains with unbounded variable length memory which satisfy Cramer condition

We consider a class of variable length Markov chains with a binary alphabet in which context tree is defined by adding finite trees with uniformly bounded height to the vertices of an infinite comb tree. Such type of Markov chain models the spike neuron patterns and also extends the class of persistent random walks. The main interest is the limiting properties of the empirical distribution of symbols from the alphabet. We obtain the strong law of large numbers, central limit theorem, and exact asymptotics for large and moderate deviations. The presence of an intrinsic renewal structure is the subject of discussion in the literature. Proofs are based on the construction of a renewals of the chain and the applying corresponding properties of the compound (or generalized) renewal processes.

Confidence intervals with higher accuracy for short and long-memory linear processes

In this paper an easy to implement method of stochastically weighing short and long-memory linear processes is introduced. The method renders asymptotically exact size confidence intervals for the population mean which are significantly more accurate than their classic counterparts for each fixed sample size n. It is illustrated both theoretically and numerically that the randomization framework of this paper produces randomized (asymptotic) pivotal quantities, for the mean, which admit central limit theorems with smaller magnitudes of error as compared to those of their leading classic counterparts. An Edgeworth expansion result for randomly weighted linear processes whose innovations do not necessarily satisfy the Cramer condition, is established. Numerical illustrations and applications to real world data are also included.

On Exact Large Deviation Principles for Compound Renewal Processes

We consider two large deviation principles (LDPs): the “ordinary” LDP (when the “strong” Cramer condition is met) and the “extended” LDP when only the standard Cramer condition is met and the devia...

Large deviations of generalized renewal process

AbstractLet (ξ(i), η(i)) ∈ ℝd+1, 1 ≤ i < ∞, be independent identically distributed random vectors, η(i) be nonnegative random variables, the vector (ξ(1), η(1)) satisfy the Cramer condition. On the base of renewal process, NT = max{k : η(1) + … + η(k) ≤ T} we define the generalized renewal process ZT = $\begin{array}{} \sum_{i=1}^{N_T} \end{array}$ξ(i). Put IΔT(x) = {y ∈ ℝd : xj ≤ yj < xj + ΔT, j = 1, …, d}. We find asymptotic formulas for the probabilities P(ZT ∈ IΔT(x)) as ΔT → 0 and P(ZT = x) in non-lattice and arithmetic cases, respectively, in a wide range of x values, including normal, moderate, and large deviations. The analogous results were obtained for a process with delay in which the distribution of (ξ(1), η(1)) differs from the distribution on the other steps. Using these results, we prove local limit theorems for processes with regeneration and for additive functionals of finite Markov chains, including normal, moderate, and large deviations.

The Asymptotic Behavior of the Mean Sojourn Time for a Random Walk Above a Receding Curvilinear Boundary

We study the asymptotic behavior of the mean sojourn time for a random walk above a receding curved boundary in the case where the jump distribution satisfies the Cramer condition.

Local theorems for arithmetic compound renewal processes when Cramer's condition holds

We continue the study of the compound renewal processes (c.r.p.), where the moment Cramer's condition holds (see [1]-[10], where the study of c.r.p. was started). In the paper arithmetic c.r.p. Z(n) are studied. In such processes random vector {\xi} = ({\tau},{\zeta}) has the arithmetic distribution, where {\tau} > 0 defines the distance between jumps, {\zeta} defines the values of jumps. For this processes the fine asymptotics in the local limit theorem for probabilities P(Z(n) = x) has been obtained in Cramer's deviation region of x $\in$ Z. In [6]-[10] the similar problem has been solved for non-lattice c.r.p., when the vector {\xi} = ({\tau},{\zeta}) has the non-lattice distribution.

The large deviation principle for a compound Poisson process

For a compound Poisson process, under the moment Cramer condition, the extended large deviation principle is established in the space of functions of bounded variation with the Borovkov metric.

The extended large deviation principle for a process with independent increments

Considering a process with independent increments under the moment Cramer condition, we establish the extended large deviation principle in the space of functions without discontinuities of the second kind which is endowed with the Borovkov metric.

A weak Cramér condition and application to Edgeworth expansions

We introduce a new, weak Cramér condition on the characteristic function of a random vector which does not only hold for all continuous distributions but also for discrete (non-lattice) ones in a generic sense. We then prove that the normalized sum of independent random vectors satisfying this new condition automatically verifies some small ball estimates and admits a valid Edgeworth expansion for the Kolmogorov metric. The latter results therefore extend the well known theory of Edgeworth expansion under the standard Cramér condition, to distributions that are purely discrete.

The Cramér condition for the Curie–Weiss model of SOC

We pursue the study of the Curie–Weiss model of self-organized criticality we designed in (Ann. Probab. 44 (2016) 444–478). We extend our results to more general interaction functions and we prove that, for a class of symmetric distributions satisfying a Cramer condition (C) and some integrability hypothesis, the sum $S_{n}$ of the random variables behaves as in the typical critical generalized Ising Curie–Weiss model. The fluctuations are of order $n^{3/4}$ and the limiting law is $k\exp(-\lambda x^{4})dx$ where $k$ and $\lambda$ are suitable positive constants. In (Ann. Probab. 44 (2016) 444–478), we obtained these results only for distributions having an even density.

Approximation of the expectation of the first exit time from an interval for a random walk

We obtain asymptotic expansions for the expectation of the first exit time from an expanding strip for a random walk trajectory. We suppose that the distribution of random walk jumps satisfies the Cramer condition on the existence of an exponential moment.

Probabilistic Inequalities for the Galton--Watson Processes

Probabilistic inequalities are deduced for all values of a breeding coefficient $A$ of a branching process. Previously, only the case A=1 was considered. Separately the cases are studied such that the Cramer condition is fulfilled and there exists only the finite number of moments. Thereafter one of the obtained inequalities is applied for deducing the Rosenthal-type moment inequality for $A\ge 1$.

Blackwell-type theorems for weighted renewal functions

We establish Blackwell-type theorems for weighted renewal functions under much weaker conditions, as compared to the available, on the weight sequence and the distributions of the jumps in the renewal process. The proofs are based on using integro-local limit theorems and large deviation bounds. For the jump distribution, we consider conditions of the four types: (a) it has finite second moment, (b) it belongs to the domain of attraction of a stable law, (c) its tails are locally regularly varying, and (d) it satisfies the moment Cramer condition. In cases (a)–(c), the weights are assumed to satisfy a broad regularity condition on their moving averages, whereas in case (d) the weights can change exponentially fast.

Lyapunov Exponents and Large Deviations Analysis of Eigenfunctions in Anderson Models on Graphs

We propose a new probabilistic approach to the analysis of decay of the Green's functions and the eigenfunctions of the Anderson Hamiltonians on count- able graphs. Our method is close in spirit to the Frac- tional Moment Method, but we show how the use of the fractional moments can be avoided, so that exponen- tial decay of the Green's functions can be established in some models where the fractional moments diverge, due to low regularity of the random potential. We elucidate the exceptional role of the Holder continuity condition, usual in the FMM, in terms of Cramer's condition in the large deviations problem for a suitably constructed rigorous path expansion.

New uniqueness conditions for the classical moment problem

New conditions for the well-posedness of the Hamburger and Stieltjes power moment problems are proposed. The proposed conditions are closely related in form and by the simplicity of their application to the Cramer condition, and by their accuracy and field of application to the Carleman conditions. The results are obtained by using ideas from the theory of extrapolation of spaces and operators that was constructed by Jawerth and Milman in the 1980s.

Subcritical branching processes in a random environment without the Cramer condition

A subcritical branching process in random environment (BPRE) is considered whose associated random walk does not satisfy the Cramer condition. The asymptotics for the survival probability of the process is investigated, and a Yaglom type conditional limit theorem is proved for the number of particles up to moment n given survival to this moment. Contrary to other types of subcritical BPRE, the limiting distribution is not discrete. We also show that the process survives for a long time owing to a single big jump of the associate random walk accompanied by a population explosion at the beginning of the process.

Functional Limit Theorems for Lévy Processes Satisfying Cramér's Condition

We consider a Levy process that starts from $x<0$ and conditioned on having a positive maximum. When Cramer's condition holds, we provide two weak limit theorems as $x$ goes to $-\infty$ for the law of the (two-sided) path shifted at the first instant when it enters $(0,\infty)$, respectively shifted at the instant when its overall maximum is reached. The comparison of these two asymptotic results yields some interesting identities related to time-reversal, insurance risk, and self-similar Markov processes.

On asymptotic behavior of probabilities of large and moderate deviations for some iterated stochastic processes

We derive logarithmic asymptotics for probabilities of large deviations for some iterated processes. We show that under appropriate conditions, these asymptotics are the same as those for sums of independent random variables. When these conditions do not hold, the asymptotics of large deviations for iterated processes are quite different. When the iterated process is a homogeneous process with independent increments in which time is replaced by random one, the behavior of large and moderate deviations is studied in the case of finite variance. For this case, the following one-sided moment restriction are considered: the Cramer condition, the Linnik condition, and the existence of moment of order p > 2 for a positive part. Bibliography: 6 titles.

On Greenwood goodness-of-fit test

Abstracts The Greenwood test is based on the sum of squares of disjoint spacings. For the goodness-of-fit problem within the class of symmetric tests, the Greenwood test is known to be optimal in terms of the Pitman asymptotic efficiency (AE), whereas it is much inferior to the log-spacings test in terms of the Bahadur AE. In this paper we extend these properties of Greenwood test showing that it remains optimal in terms of the weak intermediate and intermediate AE, but it is much inferior to those tests satisfying the Cramer condition in terms of the strong intermediate AE.

Integro-Local and Local Theorems on Normal and Large Deviations of the Sums of Nonidentically Distributed Random Variables in the Triangular Array Scheme

Gnedenko's local theorem and Stone–Shepp's integro-local theorem (see [L. A. Shepp, Ann. Math. Statist., 35 (1964), pp. 419–423], [C. Stone, Ann. Math. Statist., 36 (1965), pp. 546–551], [B. V. Gnedenko and A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, Addison-Wesley, Cambridge, MA, 1954]) for sums on independent identically distributed random variables are extended to the case when the summands are nonidentically distributed in a triangular array scheme. Under Cramer's condition on the random variables, we also obtain integro-local and local theorems that are valid in both normal and large deviations zones.