Probabilities Of Moderate Deviations Research Articles

On the Asymptotic Behavior of Probabilities of Moderate Deviations for Combinatorial Sums

On the Asymptotic Behavior of Probabilities of Moderate Deviations for Combinatorial Sums

Moderate deviation probabilities for empirical distribution of the branching random walk

Given a super-critical branching random walk {Zn}n≥0 on R, let Zn(A) be the number of particles located in some Borel set A⊂R at generation n. Under some mild conditions, it's well-known (e.g., [4]) that Zn(nA)/Zn(R) converges almost surely to ν(A) as n→∞, where ν(⋅) is the standard Gaussian measure on R. Large deviation probabilities of Zn(nA)/Zn(R) have been well studied (see [10] and [27]). In this work, we investigate its moderate deviation probabilities, i.e. the convergence rate ofP(Zn(nA)/Zn(R)≥ν(A)+Δn) as n→∞, where {Δn}n≥0 is some positive sequence tending to zero.

On asymptotic behaviour for probabilities of moderate deviations of combinatorial sums

We investigate an asymptotic behaviour for probabilities of moderate deviations of combinatorial sums of independent random variables having moments of order p > 2. We find zones in which these probabilities are equivalent to the tail of the standard normal law. The width of the zone is a function from the logarithm of a combinatorial variant for Lyapunov’s ratio. The author earlier obtained similar results under Bernstein’s and Linnik’s conditions. The truncations method is used in proofs of the results.

Large and moderate deviations for record numbers in some non–nearest neighbor random walks

The deviation principles of record numbers in random walk models have not been completely investigated, especially for the non–nearest neighbor cases. In this paper, we derive the asymptotic probabilities of large and moderate deviations for the number of “weak records” (or “ladder points”) in two kinds of one–dimensional non–nearest neighbor random walks. The proofs depend only on the direct analysis of random walks. We illustrate that the traditional method of analyzing the local time of Brownian motions, which is often adopted for the simple random walks, would lead to wrong conjectures for our cases.

Deviations for random sums indexed by the generations of a branching process

Applying the results about harmonic moments of classical Galton-Watson process, we obtain the deviations for random sums indexed by the generations of a branching process. Our results show that the decay rates of large deviations and moderate deviations depend heavily on the degree of the heavy tail and the asymptotic distributions depend heavily on the normalizing constants. If the underlying Galton-Watson process belongs to the Schr?der case, both large deviation and moderate deviation probabilities show three decay rates, where the critical case depends heavily on the Schr?der index. Else if the Galton-Watson process belongs to the B?ttcher case, there are only two decay rate for both large deviation and moderate deviation probabilities. Simulations are also given to illustrate our results.

Lower deviation and moderate deviation probabilities for maximum of a branching random walk

Étant donnée une marche aléatoire branchante surcritique sur ${\mathbb{R}}$ issue de l’origine, on note $M_{n}$ la position maximale des individus à la $n$-ème génération. Sous des conditions raisonnables, il a été prouvé dans (Ann. Probab. 41 (2013) 1362–1426) que lorsque $n\rightarrow \infty $, $M_{n}-x^{*}n+\frac{3}{2\theta ^{*}}\log n$ converge en loi pour certaines constantes appropriées $x^{*}$ et $\theta ^{*}$. Dans cet article, nous envisageons la déviation modérée, autrement dit, les taux de convergence de \begin{equation*}\mathbb{P}\biggl(M_{n}\leq x^{*}n-\frac{3}{2\theta ^{*}}\log n-\ell _{n}\biggr),\end{equation*} pour toute positive suite $(\ell _{n})$ telle que $\ell _{n}=O(n)$ et $\ell _{n}\uparrow \infty $. En particulier, nous obtenons la déviation inférieure de $M_{n}$; c’est-à-dire, le taux de convergence de $\mathbb{P}(M_{n}\leq xn)$ avec $x<x^{*}$ dans le cas Böttcher où le nombre d’enfants est au moins deux. Nous appliquons également ces techniques à l’étude de la petite déviation de la limite de la martingale dérivée. Notre résultats complètent ceux dans (Ann. Inst. Henri Poincaré Probab. Stat. 52 (2016) 233–260) et (Electron. Commun. Probab. 23 (2018) 1–12).

The Law of the Iterated Logarithm and Probabilities of Moderate Deviations of Sums of Dependent Random Variables

Sufficient conditions are obtained for the applicability of one of the classical forms of the law of iterated logarithm to a sequence of random variables without conditions of independence and the existence of any moments. Bibliography: 5 titles.

On Probabilities of Moderate Deviations of Empirical Measures for Contiguous Distributions

On Probabilities of Moderate Deviations of Empirical Measures for Contiguous Distributions

Asymptotically Efficient Importance Sampling for Bootstrap

The Large Deviation Principle is proved for the conditional probabilities of moderate deviations of weighted empirical bootstrap measures with respect to a fixed empirical measure. Using this LDP for the problem of calculation of moderate deviation probabilities of differentiable statistical functionals, it is shown that the importance sampling based on influence function is asymptotically efficient.

A note on martingale deviation bounds

Let {Yi}i=1n be a martingale difference sequence and Sn=∑i=1nYi. Probability deviation bounds for martingale difference sequences generally focus on upper bounds for probabilities of large deviations P(Sn>λ), particularly of maxima of Sn. In this article bounds for probabilities of moderate deviations P(Sn<λ) are studied. The motivation is estimating the probability that the cumulative drift of a Markov chain is moderate, and thus estimates derived from sampling the chain are reliable.

On Asymptotically Efficient Statistical Inference on a Signal Parameter

We consider the problems of confidence estimation and hypotheses testing of the parameter of a signal observed in a Gaussian white noise. For these problems, we point out lower bounds of asymptotic efficiency in the zone of moderate deviation probabilities. These lower bounds are versions of the local asymptotic minimax Hajek–Le Cam lower bound in estimation and the lower bound for the Pitman efficiency in hypotheses testing. The lower bounds are obtained both for logarithmic and sharp asymptotics of moderate deviation probabilities.

On the probabilities of moderate deviations for combinatorial sums

We investigate the asymptotic behavior of the probabilities of moderate deviations for combinatorial sums. We find a zone in which tails of distributions of combinatorial sums have the same asymptotic behavior as that of the standard normal law. The combinatorial sums have dependent increments. It follows from this that the classical method of characteristic functions cannot be applied. We use Esseen-type bounds for combinatorial sums that were recently obtained in another paper by the author. We show that the zone of the normal convergence is close to the best one, which is for the case of centered independent random variables. We consider the case of finite variations of summands. The case of infinite variations is also discussed.

Large Deviation Principle for Moderate Deviation Probabilities of Bootstrap Empirical Measures

We prove two Large Deviation Principles (LDP) in the zone of moderate deviation probabilities. First we establish the LDP for conditional distributions of moderate deviations of empirical bootstrap measures given an empirical probability measure. Then we establish the LDP for the joint distributions of an empirical measure and a bootstrap empirical measure. Using these LDPs, similar LDPs for differentiable statistical functionals can be established. The LDPs for moderate deviations of an empirical quantile process and an empirical bootstrap copula function are provided as illustrations of these results. Bibliography: 28 titles.

On the rate of convergence in the strong law of large numbers for martingales

The aim of this note is to establish the Baum–Katz type rate of convergence in the Marcinkiewicz–Zygmund strong law of large numbers for martingales, which improves the recent works of Stoica [Series of moderate deviation probabilities for martingales, J. Math. Anal. Appl. 336 (2005), pp. 759–763; Baum–Katz–Nagaev type results for martingales, J. Math. Anal. Appl. 336 (2007), pp. 1489–1492; A note on the rate of convergence in the strong law of large numbers for martingales, J. Math. Anal. Appl. 381 (2011), pp. 910–913]. Furthermore, we also study some relevant limit behaviours for the uniform mixing process. Under some uniform mixing conditions, the sufficient and necessary condition of the convergence of the martingale series is established.

Limiting distribution for the maximal standardized increment of a random walk

Let X1,X2,… be independent identically distributed (i.i.d.) random variables with EXk=0, V arXk=1. Suppose that φ(t)≔logEetXk<∞ for all t>−σ0 and some σ0>0. Let Sk=X1+⋯+Xk and S0=0. We are interested in the limiting distribution of the multiscale scan statisticMn=max0≤i<j≤nSj−Sij−i. We prove that for an appropriate normalizing sequence an, the random variable Mn2−an converges to the Gumbel extreme-value law exp{−e−cx}. The behavior of Mn depends strongly on the distribution of the Xk’s. We distinguish between four cases. In the superlogarithmic case we assume that φ(t)<t2/2 for every t>0. In this case, we show that the main contribution to Mn comes from the intervals (i,j) having length l≔j−i of order a(logn)p, a>0, where p=q/(q−2) and q∈{3,4,…} is the order of the first non-vanishing cumulant of X1 (not counting the variance). In the logarithmic case we assume that the function ψ(t)≔2φ(t)/t2 attains its maximum m∗>1 at some unique point t=t∗∈(0,∞). In this case, we show that the main contribution to Mn comes from the intervals (i,j) of length d∗logn+alogn, a∈R, where d∗=1/φ(t∗)>0. In the sublogarithmic case we assume that the tail of Xk is heavier than exp{−x2−ε}, for some ε>0. In this case, the main contribution to Mn comes from the intervals of length o(logn) and in fact, under regularity assumptions, from the intervals of length 1. In the remaining, fourth case, the Xk’s are Gaussian. This case has been studied earlier in the literature. The main contribution comes from intervals of length alogn, a>0. We argue that our results cover most interesting distributions with light tails. The proofs are based on the precise asymptotic estimates for large and moderate deviation probabilities for sums of i.i.d. random variables due to Cramér, Bahadur, Ranga Rao, Petrov and others, and a careful extreme value analysis of the random field of standardized increments by the double sum method.

Exact moderate and large deviations for linear processes

Large and moderate deviation probabilities play an important role in many applied areas, such as insurance and risk analysis. This paper studies the exact moderate and large deviation asymptotics in non-logarithmic form for linear processes with independent innovations. The linear processes we analyze are general and therefore they include the long memory case. We give an asymptotic representation for probability of the tail of the normalized sums and specify the zones in which it can be approximated either by a standard normal distribution or by the marginal distribution of the innovation process. The results are then applied to regression estimates, moving averages, fractionally integrated processes, linear processes with regularly varying exponents and functions of linear processes. We also consider the computation of value at risk and expected shortfall, fundamental quantities in risk theory and nance.

The sharp lower bound of asymptotic efficiency of estimators in the zone of moderate deviation probabilities

For the zone of moderate deviation probabilities the local asymptotic minimax lower bound of asymptotic efficiency of estimators is established. The estimation parameter is multidimensional. The lower bound admits the interpretation as the lower bound of asymptotic efficiency in confidence estimation.

On the Hsu–Robbins–Erdős–Spitzer–Baum–Katz theorem for random fields

The by now classical results on convergence rates in the law of large numbers involving the sums ∑ n = 1 ∞ n α r − 2 P ( | S n | > n α ε ) , where r > 0 , α > 1 / 2 , such that α r ⩾ 1 has been extended to the case α = 1 / 2 by adding additional logarithms. All of this has been generalized to random fields by the first named author in [A. Gut, Marcinkiewicz laws and convergence rates in the law of large numbers for random variables with multidimensional indices, Ann. Probab. 6 (1978) 469–482; A. Gut, Convergence rates for probabilities of moderate deviations for sums of random variables with multidimensional indices, Ann. Probab. 8 (1980) 298–313]. The purpose of the present paper is to treat the case when the αʼs differ in the different directions of the field, as well as mixed cases with some αʼs equal to 1/2 with added logarithms and/or iterated ones.

Nonparametric signal detection with small type I and type II error probabilities

In the problem of signal detection in the heteroscedastic Gaussian white noise we show asymptotic minimaxity of kernel-based tests. The test statistics equal L 2- norms of kernel estimators. The sets of alternatives are defined by the sets of all signals such that L 2- norms of signals smoothed by the kernel exceed some constants $${\rho_\epsilon}$$ . The constants $${\rho_\epsilon}$$ depend on the power $${\epsilon}$$ of noise and $${\rho_\epsilon \to 0}$$ as $${\epsilon \to 0}$$ . The setup is considered in the zone of moderate deviation probabilities. We suppose that type I or type II error probabilities of tests tend to zero as $${\epsilon \to 0}$$ .

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.