P(Z k = 1) = 1/k = 1 − P(Z k = 0) (k > 1) and put T n := P 16k6n kZ k . It is then known that
T n /n converges weakly to a real random variable D with density proportional to the Dickman
function, defined by the delay-differential equation u% 0 (u) + %(u − 1) = 0 (u > 1) with initial condition %(u) = 1 (0 6 u 6 1). Improving on earlier work, we propose asymptotic formulae with remainders for the corresponding local and **almost** **sure** **limit** **theorems**, namely

1. **Almost** **sure** **limit** **theorems**
In this section, we discuss a version of Eagleson’s result that applies to **almost** **sure** **limit** **theorems**. Given two probability measures m 1 and m 2 , the goal will be to construct
a coupling between these two measures that respects the orbit structure of the space, as in [Kor17]. Then, it will readily follow that an **almost** **sure** **limit** theorem with respect to m 1 implies one with respect to m 2 . Our argument works for general maps but, contrary

[13] Lai T. L., Wei C. Z., 1982. Least squares estimates in stochastic regression models with applications to identification and control of dynamic systems. Ann. Statist. 10, 154-166.
[14] Lifshits M., 2002. **Almost** **sure** **limit** theorem for martingales. **Limit** **Theorems** in Probability and Statistics II, (I. Berkes, E. Cs´ aki, M. Cs¨ org˝ o, eds.) J. Bolyai Mathematical Society, Budapest, 367-390.

This paper is devoted to the proof of two new results concerning functions of Gaussian vectors. The first one (Lemma 1 of Section 2) is a moment bound for “off-diagonal” sums of products of functions of Gaussian vectors in a general frame. It is an extension of an important lemma by Taqqu (1977, Lemma 4.5). This result is useful for obtaining **almost** **sure** convergence and tightness of Gaussian subordinated functionals and statistics, see Remark 1 below. The proof of Lemma 1 uses the Hermitian decomposition of L 2 function

The results of this chapter are related to those in, e.g. [ 11 , 27 , 35 , 53 , 55 , 70 , 79 , 80 , 89 , 90 ] in which different scaling regimes occur for various classes of LRD models, in particular, heavy- tailed duration models. Isotropic scaling limits (case γ = 1) of random grain and random balls models in arbitrary dimension were discussed in Kaj et al. [ 53 ] and Biermé et al. [ 11 ]. The monograph [ 69 ] provides a nice discussion of **limit** behavior of heavy-tailed duration models whose spatial version is the random grain model in ( 7.2 ). From an application view- point, probably the most interesting is the study of different scaling regimes of superposed network traffic models [ 27 , 35 , 55 , 70 ]. In these studies, it is assumed that traffic is generated by independent sources and the problem concerns the **limit** distribution of the aggregated traffic as the time scale T and the number of sources M both tend to infinity, possibly at different rate. The present chapter extends the above-mentioned work, by considering the **limit** behavior of the aggregated workload process:

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As we saw, free products provide a great source of examples for local **limit** **theorems**. There are several equivalent ways of deﬁning relatively hyperbolic groups (see Section 2 for more details). If Ω is a collection of subgroups of Γ, we say that Γ is hyperbolic relative to Ω if it acts geometrically ﬁnitely on a proper geodesic hyperbolic space X such that the stabilizers of the parabolic **limit** points are exactly the subgroups in Ω. The elements of Ω are called peripheral subgroups or (maximal) parabolic subgroups. We ﬁx a collection Ω0 of representatives of conjugacy classes of Ω. According to [7, Proposition 6.15], such a collection is ﬁnite. If Γ is a free product of the form Γ = Γ1 ∗ ... ∗ Γn, then Γ is relatively hyperbolic with respect to conjugacy classes of the free factors Γi and one can choose Ω0 = {Γ1, ..., Γn}.

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k=0 r(k)f (X k )] is not always easy, we will
consider bounds for this quantity derived from a ”subgeometric” condition re- cently introduced in Douc et al. (2004), which might be seen, in the subgeomet- rical case, as an analog to the Foster-Lyapunov drift condition for geometrically ergodic Markov Chains. We obtain, using these drift conditions, explicit bounds for the (f, r)-modulated expectation of moments of the regeneration times in terms of the constants in A1, the sequence r and the constants appearing in the drift conditions. With these results, we obtain **limit** **theorems** for addi- tive functionals and deviations inequalities, under conditions which are easy to check.

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OF SPECTRAL DENSITY
HERMINE BIERM´ E, ALINE BONAMI, AND JOS´ E R. LE ´ ON
Abstract. We give a new proof and provide new bounds for the speed of convergence in the Central **Limit** **Theorems** of Breuer Major on stationary Gaussian time series, which generalizes to particular triangular arrays. Our assumptions are given in terms of the spectral density of the time series. We then consider generalized quadratic variations of Gaussian fields with stationary increments under the assumption that their spectral density is asymptotically self-similar and prove Central **Limit** **Theorems** in this context.

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ond moment, one expects that the central **limit** theorem holds for N −1/2 S N ,
as N → +∞. This, of course, requires some assumptions about the rate of the decay of correlations of the chain, as well as hypotheses about its dy- namics. If Ψ has an infinite second moment and its tails satisfy a power law, then one expects, again under some assumption on the transition prob- abilities, convergence of the laws of N −1/α S N , for an appropriate α to the

[2] R. J. Bhansali, L. Giraitis, and P. S. Kokoszka. Approximations and **limit** theory for quadratic forms of linear processes. Stochastic Process. Appl., 117(1):71–95, 2007. [3] P. Billingsley. Convergence of probability measures. Wiley Series in Probability and
Statistics: Probability and Statistics. John Wiley & Sons Inc., New York, second edition, 1999. A Wiley-Interscience Publication.

2E[(τ Hx + 1)f (Hx)]
τ x + 2 − f(x)
. 3.2. Many-to-one formulas. In order to compute our **limit** theorem, we need to control the second moment. As in [5], we begin by describing the population over the whole tree. Then we give a many-to-one formula for forks. Let T be the random set representing cells that have lived at a certain moment. It is defined by

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignemen[r]

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H l (θ, x) = θ + γ l H(θ, x). In this latter case, it is assumed that the best values for θ are
the solutions of the equation R H(θ, x)π(dx) = 0. Since the pioneer work of Gilks et al. (1998); Holden (1998); Haario et al. (2001); Andrieu and Robert (2001), the number of AMCMC algorithms in the literature has significantly increased in recent years. But de- spite many recent works on the topic, the asymptotic behavior of these algorithms is still not completely understood. **Almost** all previous works on the convergence of AMCMC are limited to the case when each kernel P θ is geometrically ergodic (see e.g.. Roberts and

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[MSS07] J.A. Mingo, P. ´ Sniady, R. Speicher Second order freeness and fluctuations of random matrices. II. Unitary random matrices. Adv. Math. 209 (2007), no. 1, 212–240.
[PSV77] G. C. Papanicolaou, D. Stroock, S. R. S. Varadhan Martingale approach to some **limit** **theorems**. Papers from the Duke Turbulence Conference (Duke Univ., Durham, N.C., 1976), Paper No. 6, Duke Univ. Math. Ser., Vol. III, Duke Univ., Durham, N.C., 1977.

Note that the speed of the central **limit** theorem is N −1/2 as for independent integrable
random variables, but differently from what happens for standard Wigner’s matrices. This phenomenon has also already been observed for adjacency matrix of random graphs [9, 24] and we will see below that it also holds for L´evy matrices. It suggests that the repulsive interactions exhibited by the eigenvalues of most models of random matrices with lighter tails than heavy tailed matrices no longer work here.

2 A Central **Limit** Theorem for Stochastic Approximation 8
θ ⋆ | ≤ δ}. In most contributions, rates of convergence are derived under the con-
dition (7) (see e.g. the recent works by Pelletier [23] and Lelong [21]). This framework is too restrictive to address the case of SA with controlled Markov chain dynamics when the ergodic properties of the transition kernels {Q θ , θ ∈ Θ}

given by ajQj, the compact group ZQ being endowed with its normalized Haar measure M, and clearly. In this article, we show roughly speaking that although the relatio[r]

Dans cette partie de notre travail, nous mettons en oeuvre les techniques du calcul de Malliavin combinées à la méthode de Stein aﬁn de determiner, dans un cadre gaussien, des bornes de [r]

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α = 2 and L(n) = 1, this is an instance of the classical central **limit** theorem. (2) When T preserves m and m has infinite mass, one obtains different limits. Let α ∈
[0, 1) (we exclude α = 1 to avoid degenerate **limit** laws). The Darling-Kac theorem gives examples of transformations T such that, for any function f which is integrable and has non-zero average for m, then S n f /(n α L(n)) converges in distribution with