We prove a moderate deviation principle for Minkowski sums of i.i.d. random compact sets in a Banach space.

In the present paper we prove moderate deviations for a Curie–Weiss model with external magnetic field generated by a dynamical system, as introduced by Dombry and Guillotin-Plantard in [C. Dombry and N. Guillotin-Plantard, Markov Process. Related Fields 15 (2009) 1–30]. The results extend those already obtained for the Curie–Weiss model without external field by Eichelsbacher and Löwe in [P. Eichelsbacher and M. Löwe, Markov Process. Related Fields 10 (2004) 345–366]. The Curie–Weiss model with...

We derive necessary and sufficient conditions for a sum of i.i.d. random variables ${\sum }_{i=1}^n{X}_i/b_n$ – where $\frac{b_n}{n}\downarrow 0$, but $\frac{b_n}{\sqrt{n}}\uparrow \infty$ – to satisfy a moderate deviations principle. Moreover we show that this equivalence is a typical moderate deviations phenomenon. It is not true in a large deviations regime.

We derive necessary and sufficient conditions for a sum of i.i.d. random variables ${\sum }_{i=1}^n{X}_i/b_n$ – where $\frac{b_n}{n}\downarrow 0$, but $\frac{b_n}{\sqrt{n}}\uparrow \infty$ – to satisfy a moderate deviations principle. Moreover we show that this equivalence is a typical moderate deviations phenomenon. It is not true in a large deviations regime.

In this paper we derive the moderate deviation principle for stationary sequences of bounded random variables under martingale-type conditions. Applications to functions of ϕ-mixing sequences, contracting Markov chains, expanding maps of the interval, and symmetric random walks on the circle are given.

The purpose of this paper is to investigate moderate deviations for the Durbin–Watson statistic associated with the stable first-order autoregressive process where the driven noise is also given by a first-order autoregressive process. We first establish a moderate deviation principle for both the least squares estimator of the unknown parameter of the autoregressive process as well as for the serial correlation estimator associated with the driven noise. It enables us to provide a moderate deviation...

Let X1,...,Xn1 be a random sample from a population with mean µ1 and variance ${\sigma }_1^2$, and X1,...,Xn1 be a random sample from another population with mean µ2 and variance ${\sigma }_2^2$ independent of {Xi,1 ≤ i ≤ n1}. Consider the two sample t-statistic $T=\frac{\overline{X}-\overline{Y}-({\mu }_1-{\mu }_2)}{\sqrt{s_1^2/n_1+s_2^2/n_2}}$. This paper shows that ln P(T ≥ x) ~ -x²/2 for any x := x(n1,n2) satisfying x → ∞, x = o(n1 + n2)1/2 as n1,n2 → ∞ provided 0 < c1 ≤ n1/n2 ≤ c2 < ∞. If, in addition, E|X1|3 < ∞, E|Y1|3 < ∞, then $\frac{P(T\ge x)}{1-\Phi \left(x\right)}\to 1$ holds uniformly in x ∈ (O,o((n1 + n2)1/6))

In this paper, we give sufficient conditions to establish central limit theorems and moderate deviation principle for a class of support estimates of empirical and Poisson point processes. The considered estimates are obtained by smoothing some bias corrected extreme values of the point process. We show how the smoothing permits to obtain Gaussian asymptotic limits and therefore pointwise confidence intervals. Some unidimensional and multidimensional examples are provided.

Dans cet article, nous étudions les résultats de grandes déviations associés au couple $(X^{\epsilon },{\nu }^{\epsilon })$, solution de l’E.D.S. interprétée au sens d’Itô :$$d{X}_t^{\epsilon }=\sqrt{\epsilon }{\sigma }_{{\nu }^{\epsilon }\left(t\right)}\left(X_t^{\epsilon }\right)d{W}_t+b_{{\nu }^{\epsilon }\left(t\right)}\left(X_t^{\epsilon }\right)dt;X_0^{\epsilon }=x\in {ℝ}^d$$avec des conditions assez générales sur les coefficients et dans les deux cas suivants :Premier cas : ${\nu }^{\epsilon }$ est indépendant du mouvement brownien $W$ et satisfait à un principe de grandes déviations ;Deuxième cas : ${\nu }^{\epsilon }$ est un processus markovien avec un nombre fini d’états $\{1,...,n\}$ vérifiant$$\mathbb{P}\{{\nu }^{\epsilon }(t+\Delta )=j/{\nu }^{\epsilon }\left(t\right)=i,X^{\epsilon }\left(t\right)=x\}=d_{ij}\left(x\right)\Delta +o\left(\Delta \right)$$uniformément dans ${ℝ}^d$ pourvu que $\Delta \downarrow 0,1\le i,j\le n,i\ne j$.Ces résultats sont des extensions de ceux de Bezuidenhout...