Let Z:={Zt,t≥0} be a stationary Gaussian process. We study two estimators of E[Z02], namely fˆT(Z):=1 T ∫0TZt2dt, and f˜n(Z):=1 n ∑i=1nZti2, where ti=iΔn, i=0,1,…,n, Δn→0 Tn:=nΔn→∞. prove that the are strongly consistent establish Berry-Esseen bounds for central limit theorem involving fˆT(Z) f˜n(Z). apply these results to asymptotically processes estimate drift parameter Ornstein-Uhlenbeck pro...

For sum of independent and identically distributed (i.i.d.) random variables {Xi}i=1, the Berry-Esseen theorem states that sup y∈< ∣∣∣∣Pr { 1 sn (X1 + X2 + · · ·+ Xn) ≤ y } − Φ(y) ∣∣∣∣ ≤ C ρ σ3 √ n , where σ and ρ are respectively the variance and the absolute third moment of the parent distribution, Φ(·) is the unit normal cumulative distribution function, and C is an absolute constant. In thi...

We prove a Berry–Esseen theorem and Edgeworth expansions for partial sums of the form $$\displaystyle S_N=\sum \nolimits _{n=1}^{N}f_n(X_n,X_{n+1})$$ , where $$\{X_n\}$$ is uniformly elliptic inhomogeneous Markov chain $$\{f_n\}$$ sequence bounded functions. The holds without additional assumptions, while order 1 hold when irreducible, which an optimal condition. For higher expansions, we then ...

A central limit theorem and a moderate deviations principle for the ratio of geometric p-generalized arithmetic mean are shown. Also Berry–Esseen-type upper bound on rate convergence in is proved. This has implications probabilistic version question whether inequality between can be reversed or improved up to multiplicative constants. The involved random vectors we study belong class distributi...

Abstract Let $\mu $ be a probability measure on $\mathrm {GL}_d(\mathbb {R})$ , and denote by $S_n:= g_n \cdots g_1$ the associated random matrix product, where $g_j$ are i.i.d. with law . Under assumptions that has finite exponential moment generates proximal strongly irreducible semigroup, we prove Berry–Esseen bound optimal rate $O(1/\sqrt n)$ for coefficients of $S_n$ settling long-standing...

The asymptotic behaviour of a closed BCMP network, with n queues and m n clients, is analyzed when n and m n become simultaneously large. Our method relies on Berry-Esseen type approximations coming in the Central Limit Theorem. We construct critical sequences m 0 n , which are necessary and sufficient to distinguish between saturated and non-saturated regimes for the network. Several applicati...

This paper studies the asymptotic distribution of descents $\des(w)$ in a permutation $w$, and its inverse, distributed according to Mallows measure. The measure is non-uniform probability on permutations introduced study ranked data. Under this measure, are weighted number inversions they contain, with weighting controlled by parameter $q$. main results Berry-Esseen theorem for $\des(w)+\des(w...