Let $(X _i)_{i\geq1}$ be a stationary sequence. Denote $m=\lfloor n^α\rfloor, 0< α< 1,$ and $ k=\lfloor n/m \rfloor,$ where $\lfloor a \rfloor$ stands for the integer part of $a.$ Set $S_{j}^\circ = \sum_{i=1}^m X_{m(j-1)+i}, 1\leq j \leq k,$ and $ (V_k^\circ)^2 = \sum_{j=1}^k (S_{j}^\circ)^2.$ We prove a Cramér type moderate deviation expansion for $\mathbb{P}( \sum_{j=1}^k S_{j}^\circ /V_k^\circ \geq x)$ as $n\to \infty.$ Applications to mixing type sequences, contracting Markov chains, expanding maps and confidence intervals are discussed.
In Stein's method, the exchangeable pair approach is commonly used to estimate the approximation errors in normal approximation. In this paper, we establish a Cramér-type moderate deviation theorem of normal approximation for unbounded exchangeable pairs. As applications, Cramér-type moderate deviation theorems for the sums of local statistics and general Curie--Weiss model are obtained.
In this paper, we study the self-normalized Cram\a'{e}r-type moderate deviations for centered independent random variables $X_1, X_2,...$ with $0<E |X_i|^3 <\infty$. The main results refine Theorems 1.1 and 1.2 of Wang (2011), the Berry-Esseen bound (2.11) and Corollaries 2.2 and 2.3 of Jing, Shao and Wang (2003) under stronger moment conditions.
A Cramér-type moderate deviation theorem quantifies the relative error of the tail probability approximation. It provides theoretical justification when the limiting tail probability can be used to estimate the tail probability under study. Chen Fang and Shao (2013) obtained a general Cramér-type moderate result using Stein's method when the limiting was a normal distribution. In this paper, Cramér-type moderate deviation theorems are established for nonnormal approximation under a general Stein identity, which is satisfied via the exchangeable pair approach and Stein's coupling. In particular…
We establish Cramér-type moderate deviation theorems for sums of locally dependent random variables and combinatorial central limit theorems. Under some mild exponential moment conditions, optimal error bounds and convergence ranges are obtained. Our main results are more general or shaper than the existing results in the literature. The main results follows from a more general Cramér-type moderate deviation theorem for dependent random variables without any boundedness assumptions, which is of independent interest. The proofs couple Stein's method with a recursive argument.
Let $(ξ_i,\mathcal{F}_i)_{i\geq1}$ be a sequence of martingale differences. Set $S_n=\sum_{i=1}^nξ_i $ and $[ S]_n=\sum_{i=1}^n ξ_i^2.$ We prove a Cramér type moderate deviation expansion for $\mathbf{P}(S_n/\sqrt{[ S]_n} \geq x)$ as $n\to+\infty.$ Our results partly extend the earlier work of [Jing, Shao and Wang, 2003] for independent random variables.
We establish Cramér type moderate deviation (MD}) results for heavy trimmed L-statistics; we obtain our results under a very mild smoothness condition on the inversion $F^{-1}$ ($F$ is the underlying distribution of i.i.d. observations) near two points, where trimming occurs, we assume also some smoothness of weights of the L-statistic. Our results complement previous work on Cramér type large deviations (LD) for trimmed L-statistics by Gribkova (2016) and Callaert et al. (1982).
Let $(η_i)_{i\geq1}$ be a sequence of $ψ$-mixing random variables. Let $m=\lfloor n^α\rfloor, 0< α< 1, k=\lfloor n/(2m) \rfloor,$ and $Y_j = \sum_{i=1}^m η_{m(j-1)+i}, 1\leq j \leq k.$ Set $ S_k^o=\sum_{j=1}^{k } Y_j $ and $[S^o]_k=\sum_{i=1}^{k } (Y_j )^2.$ We prove a Cramér type moderate deviation expansion for $\mathbb{P}(S_k^o/\sqrt{[ S^o]_k} \geq x)$ as $n\to \infty.$ Our result is similar to the recent work of Chen\textit{ et al.}\ [Self-normalized Cramér-type moderate deviations under dependence. Ann.\ Statist.\ 2016; \textbf{44}(4): 1593--1617] where the authors established Cramér type…
We give a Cramér moderate deviation expansion for martingales with differences having finite conditional moments of order $2+ρ, ρ\in (0,1],$ and finite one-sided conditional exponential moments. The upper bound of the range of validity and the remainder of our expansion are both optimal. Consequently, it leads to a "half-side" moderate deviation principle for martingales. It is worth mentioning that our result is new even for independent random variables. Moreover, applications to quantile coupling inequality, $β$-mixing and $ψ$-mixing sequences are discussed.
Let $\{Z_n, n\geq 0\}$ be a supercritical branching process in an independent and identically distributed random environment. We prove Cramér moderate deviations and Berry-Esseen bounds for $\ln (Z_{n+n_0}/Z_{n_0})$ % under the annealed law, uniformly in $n_0 \in \mathbb{N}$, which extend the corresponding results by Grama et al. (Stochastic Process.\ Appl. 2017) established for $n_0=0$. The extension is interesting in theory, and is motivated by applications. A new method is developed for the proofs; some conditions of Grama et al. (2017) are relaxed in our present setting. An example of appl…
We derive Cramér type moderate deviations for stationary sequences of bounded random variables. Our results imply the moderate deviation principles and a Berry-Esseen bound. Applications to quantile coupling inequalities, functions of $ϕ$-mixing sequences, and contracting Markov chains are discussed.
Let {(X_i,Y_i)}_{i=1}^n be a sequence of independent bivariate random vectors. In this paper, we establish a refined Cramér type moderate deviation theorem for the general self-normalized sum \sum_{i=1}^n X_i/(\sum_{i=1}^n Y_i^2)^{1/2}, which unifies and extends the classical Cramér (1938) theorem and the self-normalized Cramér type moderate deviation theorems by Jing, Shao and Wang (2003) as well as the further refined version by Wang (2011). The advantage of our result is evidenced through successful applications to weakly dependent random variables and self-normalized winsorized mean. Speci…
We consider a stochastic differential equation and its Euler-Maruyama (EM) scheme, under some appropriate conditions, they both admit a unique invariant measure, denoted by $π$ and $π_η$ respectively ($η$ is the step size of the EM scheme). We construct an empirical measure $Π_η$ of the EM scheme as a statistic of $π_η$, and use Stein's method developed in \citet{FSX19} to prove a central limit theorem of $Π_η$. The proof of the self-normalized Cramér-type moderate deviation (SNCMD) is based on a standard decomposition on Markov chain, splitting $η^{-1/2}(Π_η(.)-π(.))$ into a martingale differ…
In this article we establish Cramér type moderate deviation results for (intermediate) trimmed means $T_n=n^{-1} \sum_{i=k_n+1}^{n-m_n}X_{i:n}$, where $X_{i:n}$ -- the order statistics corresponding to the first $n$ observations of a~sequence $X_1,X_2,\dots $ of i.i.d random variables with $df$ $F$. We consider two cases of intermediate and heavy trimming. In the former case, when $\max(α_n,β_n)\to 0$ ($α_n=k_n/n$, $β_n=m_n/n$) and $\min(k_n,m_n)\to\infty$ as $n\to\infty$, we obtain our results under a~natural moment condition and a~mild condition on the rate at which $α_n$ and $β_n$ tend to z…
Cramér type moderate deviation theorems quantify the accuracy of the relative error of the normal approximation and provide theoretical justifications for many commonly used methods in statistics. In this paper, we develop a new randomized concentration inequality and establish a Cramér type moderate deviation theorem for general self-normalized processes which include many well-known Studentized nonlinear statistics. In particular, a sharp moderate deviation theorem under optimal moment conditions is established for Studentized $U$-statistics.
Let $(Z_n)_{n\geq0}$ be a supercritical Galton-Watson process. The Lotka-Nagaev estimator $Z_{n+1}/Z_n$ is a common estimator for the offspring mean.In this paper, we establish some Cramér moderate deviation results for the Lotka-Nagaev estimator via a martingale method. Applications to construction of confidence intervals are also given.
Here we obtain the exact asymptotics for large and moderate deviations, strong law of large numbers and central limit theorem for chains with unbounded variable length memory.
In this paper we study the moderate deviations for the magnetization of critical Curie-Weiss model. Chen, Fang and Shao considered a similar problem for non-critical model by using Stein method. By direct and simple arguments based on Laplace method, we provide an explicit formula of the error and deduce a Cramér-type result.
In this paper, Cramér type moderate deviations for the maximum of the periodogram and its studentized version are derived. The results are then applied to a simultaneous testing problem in gene expression time series. It is shown that the level of the simultaneous tests is accurate provided that the number of genes $G$ and the sample size $n$ satisfy $G=\exp(o(n^{1/3}))$.
Let $(Z_n)$ be a supercritical branching process in a random environment $ξ= (ξ_n)$. We establish a Berry-Esseen bound and a Cramér's type large deviation expansion for $\log Z_n$ under the annealed law $\mathbb P$. We also improve some earlier results about the harmonic moments of the limit variable $W=lim_{n\to \infty} W_n$, where $W_n =Z_n/ \mathbb{E}_ξ Z_n$ is the normalized population size.
The empirical mean of $n$ independent and identically distributed (i.i.d.) random variables $(X_1,\dots,X_n)$ can be viewed as a suitably normalized scalar projection of the $n$-dimensional random vector $X^{(n)}\doteq(X_1,\dots,X_n)$ in the direction of the unit vector $n^{-1/2}(1,1,\dots,1) \in \mathbb{S}^{n-1}$. The large deviation principle (LDP) for such projections as $n\rightarrow\infty$ is given by the classical Cramér's theorem. We prove an LDP for the sequence of normalized scalar projections of $X^{(n)}$ in the direction of a generic unit vector $θ^{(n)} \in \mathbb{S}^{n-1}$, as $n…
Two-sample $U$-statistics are widely used in a broad range of applications, including those in the fields of biostatistics and econometrics. In this paper, we establish sharp Cramér-type moderate deviation theorems for Studentized two-sample $U$-statistics in a general framework, including the two-sample $t$-statistic and Studentized Mann-Whitney test statistic as prototypical examples. In particular, a refined moderate deviation theorem with second-order accuracy is established for the two-sample $t$-statistic. These results extend the applicability of the existing statistical methodologies f…
Let $X_1,\dots, X_n$ be independent and identically distributed random vectors in $\mathbb{R}^d$. Suppose $\mathbb{E} X_1=0$, $\mathrm{Cov}(X_1)=I_d$, where $I_d$ is the $d\times d$ identity matrix. Suppose further that there exist positive constants $t_0$ and $c_0$ such that $\mathbb{E} e^{t_0|X_1|}\leq c_0<\infty$, where $|\cdot|$ denotes the Euclidean norm. Let $W=\frac{1}{\sqrt{n}}\sum_{i=1}^n X_i$ and let $Z$ be a $d$-dimensional standard normal random vector. Let $Q$ be a $d\times d$ symmetric positive definite matrix whose largest eigenvalue is 1. We prove that for $0\leq x\leq \varepsi…
The sub-linear expectation or called G-expectation is a nonlinear expectation having advantage of modeling non-additive probability problems and the volatility uncertainty in finance. Let $\{X_n;n\ge 1\}$ be a sequence of independent random variables in a sub-linear expectation space $(Ω, \mathscr{H}, \widehat{\mathbb E})$. Denote $S_n=\sum_{k=1}^n X_k$ and $V_n^2=\sum_{k=1}^n X_k^2$. In this paper, a moderate deviation for self-normalized sums, that is, the asymptotic capacity of the event $\{S_n/V_n \ge x_n \}$ for $x_n=o(\sqrt{n})$, is found both for identically distributed random variables…
We establish a moderate deviations principle (MDP) for the log-determinant $\log | \det (M_n) |$ of a Wigner matrix $M_n$ matching four moments with either the GUE or GOE ensemble. Further we establish Cramér--type moderate deviations and Berry-Esseen bounds for the log-determinant for the GUE and GOE ensembles as well as for non-symmetric and non-Hermitian Gaussian random matrices (Ginibre ensembles), respectively.
In this paper, we derive the moderate deviation principle for stationary sequences of bounded random variables with values in a Hilbert space. The conditions obtained are expressed in terms of martingale-type conditions. The main tools are martingale approximations and a new Hoeffding inequality for non adpated sequences of Hilbert-valued random variables. Applications to Cramer-Von Mises statistics, functions of linear processes and stable Markov chains are given.
Let $G$ be an infinite connected graph with vertex set $V$. Let $\{S_n: n \in \mathbb N_0 \}$ be the simple random walk on $G$ and let $\{ ξ(v) : v \in V \}$ be a collection of i.i.d. random variables which are independent of the random walk. Define the random walk in random scenery as $T_n = \sum_{k=0}^n ξ(S_k)$, and the normalization variables $V_n = (\sum_{k=0}^n ξ^2(S_k))^{1/2}$ and $L_{n,2} = (\sum_{v \in V} \ell^2_n(v))^{1/2}$. For $G= \mathbb Z^d$ and $G = \mathbb T_d$, the $d$-ary tree, we provide large deviations results for the self-normalized process $T_n \sqrt{n}/(L_{n,2}V_n)$ unde…
Let $(g_{n})_{n\geq 1}$ be a sequence of independent and identically distributed positive random $d\times d$ matrices and consider the matrix product $G_n: = g_n \ldots g_1$. Under suitable conditions, we establish the Berry-Esseen bounds on the rate of convergence in the central limit theorem and moderate deviation expansions of Cramér type, for the matrix norm $\| G_n \|$ of $G_n$, for its $(i,j)$-th entry $G_n^{i,j}$, and the and for its spectral radius $ρ(G_n)$.
Stein's method is applied to obtain a general Cramer-type moderate deviation result for dependent random variables whose dependence is defined in terms of a Stein identity. A corollary for zero-bias coupling is deduced. The result is also applied to a combinatorial central limit theorem, a general system of binary codes, the anti-voter model on a complete graph, and the Curie-Weiss model. A general moderate deviation result for independent random variables is also proved.
We investigate random walks in independent, identically distributed random sceneries under the assumption that the scenery variables satisfy Cramer's condition. We prove moderate deviation principles in dimensions two and larger, covering all those regimes where rate and speed do not depend on the actual distribution of the scenery. In the case of dimension four and larger we even obtain precise asymptotics for the annealed probability of a moderate deviation, extending a classical central limit theorem of Kesten and Spitzer. In dimension three and larger, an important ingredient in the proofs…
Let $(Z_{n})$ be a supercritical branching process in a random environment $ξ$, and $W$ be the limit of the normalized population size $Z_{n}/\mathbb{E}[Z_{n}|ξ]$. We show large and moderate deviation principles for the sequence $\log Z_{n}$ (with appropriate normalization). For the proof, we calculate the critical value for the existence of harmonic moments of $W$, and show an equivalence for all the moments of $Z_{n}$. Central limit theorems on $W-W_n$ and $\log Z_n$ are also established.
We use a new method via $p$-Wasserstein bounds to prove Cramér-type moderate deviations in (multivariate) normal approximations. In the classical setting that $W$ is a standardized sum of $n$ independent and identically distributed (i.i.d.) random variables with sub-exponential tails, our method recovers the optimal range of $0\leq x=o(n^{1/6})$ and the near optimal error rate $O(1)(1+x)(\log n+x^2)/\sqrt{n}$ for $P(W>x)/(1-Φ(x))\to 1$, where $Φ$ is the standard normal distribution function. Our method also works for dependent random variables (vectors) and we give applications to the combinat…
Moderate deviation principles for empirical measure processes associated with weakly interacting Markov processes are established. Two families of models are considered: the first corresponds to a system of interacting diffusions whereas the second describes a collection of pure jump Markov processes with a countable state space. For both cases the moderate deviation principle is formulated in terms of a large deviation principle (LDP), with an appropriate speed function, for suitably centered and normalized empirical measure processes. For the first family of models the LDP is established in …
Consider the random walk $G_n : = g_n \ldots g_1$, $n \geq 1$, where $(g_n)_{n\geq 1}$ is a sequence of independent and identically distributed random elements with law $μ$ on the general linear group ${\rm GL}(V)$ with $V=\mathbb R^d$. Under suitable conditions on $μ$, we establish Cramér type moderate deviation expansions and local limit theorems with moderate deviations for the coefficients $\langle f, G_n v \rangle$, where $v \in V$ and $f \in V^*$. Our approach is based on the Hölder regularity of the invariant measure of the Markov chain $G_n \!\cdot \! x = \mathbb R G_n v$ on the projec…
Let $(g_n)_{n\geq 1}$ be a sequence of independent and identically distributed random elements with law $μ$ on the general linear group $\textrm{GL}(V)$, where $V=\mathbb R^d$. Consider the random walk $G_n : = g_n \ldots g_1$, $n \geq 1$, and the coefficients $\langle f, G_n v \rangle$, where $v \in V$ and $f \in V^*$. Under suitable moment assumptions on $μ$, we prove the strong and weak laws of large numbers and the central limit theorem for $\langle f, G_n v \rangle$, which improve the previous results established under the exponential moment condition on $μ$. We further demonstrate the Be…
Let $p\in[1,\infty]$. Consider the projection of a uniform random vector from a suitably normalized $\ell^p$ ball in $\mathbb{R}^n$ onto an independent random vector from the unit sphere. We show that sequences of such random projections, when suitably normalized, satisfy a large deviation principle (LDP) as the dimension $n$ goes to $\infty$, which can be viewed as an annealed LDP. We also establish a quenched LDP (conditioned on a fixed sequence of projection directions) and show that for $p\in(1,\infty]$ (but not for $p=1$), the corresponding rate function is "universal", in the sense that …
