Optimal Change-Point Detection with Training Sequences in the Large and Moderate Deviations Regimes

Cramér moderate deviations for a supercritical Galton-Watson process

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.

Moderate deviation principles for kernel estimator of invariant density in bifurcating Markov chains models

Bitseki and Delmas (2021) have studied recently the central limit theorem for kernel estimator of invariant density in bifurcating Markov chains models. We complete their work by proving a moderate deviation principle for this estimator. Unlike the work of Bitseki and Gorgui (2021), it is interesting to see that the distinction of the two regimes disappears and that we are able to get moderate deviation principle for large values of the ergodic rate. It is also interesting and surprising to see that for moderate deviation principle, the ergodic rate begins to have an impact on the choice of th…

Central Limit Theorem and Moderate deviation for nonhomogenenous Markov chains

Our purpose is to prove central limit theorem for countable nonhomogeneous Markov chain under the condition of uniform convergence of transition probability matrices for countable nonhomogeneous Markov chain in Cesàro sense. Furthermore, we obtain a corresponding moderate deviation theorem for countable nonhomogeneous Markov chain by Gärtner-Ellis theorem and exponential equivalent method.

Self-normalized Cramér Type Moderate Deviations under Dependence

We establish a Cramér-type moderate deviation result for self-normalized sums of weakly dependent random variables, where the moment requirement is much weaker than the non-self-normalized counterpart. The range of the moderate deviation is shown to depend on the moment condition and the degree of dependence of the underlying processes. We consider two types of self-normalization: the big-block-small-block scheme and the interlacing or equal-block scheme. Simulation study shows that the latter can have a better finite-sample performance. Our result is applied to multiple testing and constructi…

Large and moderate deviation principles for recursive kernel density estimators defined by stochastic approximation method

In this paper we prove large and moderate deviations principles for the recursive kernel estimators of a probability density function defined by the stochastic approximation algorithm introduced by Mokkadem et al. [2009. The stochastic approximation method for the estimation of a probability density. J. Statist. Plann. Inference 139, 2459-2478]. We show that the estimator constructed using the stepsize which minimize the variance of the class of the recursive estimators defined in Mokkadem et al. (2009) gives the same pointwise LDP and MDP as the Rosenblatt kernel estimator. We provide results…

Large and moderate deviation principles for averaged stochastic approximation method for the estimation of a regression function

In this paper we prove large deviations principles for the averaged stochastic approximation method for the estimation of a regression function introduced by A. Mokkadem et al. [Revisiting Révész's stochastic approximation method for the estimation of a regression function, ALEA Lat. Amm. J. Probab. Math. Stat. 6 (2009), 63-114]. We show that the averaged stochastic approximation algorithm constructed using the weight sequence which minimize the asymptotic variance gives the same pointwise LDP as the Nadaraya-Watson kernel estimator. Moreover, we give a moderate deviations principle for these …

Moderate Deviations in Channel Coding

We consider block codes whose rate converges to the channel capacity with increasing block length at a certain speed and examine the best possible decay of the probability of error. We prove that a moderate deviation principle holds for all convergence rates between the large deviation and the central limit theorem regimes.

Moderate Deviations Analysis of Binary Hypothesis Testing

This paper is focused on the moderate-deviations analysis of binary hypothesis testing. The analysis relies on a concentration inequality for discrete-parameter martingales with bounded jumps, where this inequality forms a refinement to the Azuma-Hoeffding inequality. Relations of the analysis to the moderate deviations principle for i.i.d. random variables and to the relative entropy are considered.

Moderate Deviation Asymptotics for Variable-Length Codes with Feedback

We consider data transmission across discrete memoryless channels (DMCs) using variable-length codes with feedback. We consider the family of such codes whose rates are $ρ_N$ below the channel capacity $C$, where $ρ_N$ is a positive sequence that tends to zero slower than the reciprocal of the square root of the expectation of the (random) blocklength $N$. This is known as the moderate deviations regime and we establish the optimal moderate deviations constant. We show that in this scenario, the error probability decays sub-exponentially with speed $\exp(-(B/C)Nρ_N)$, where $B$ is the maximum …

Moderate Deviation Analysis for Classical-Quantum Channels and Quantum Hypothesis Testing

In this work, we study the tradeoffs between the error probabilities of classical-quantum channels and the blocklength $n$ when the transmission rates approach the channel capacity at a rate slower than $1/\sqrt{n}$, a research topic known as moderate deviation analysis. We show that the optimal error probability vanishes under this rate convergence. Our main technical contributions are a tight quantum sphere-packing bound, obtained via Chaganty and Sethuraman's concentration inequality in strong large deviation theory, and asymptotic expansions of error-exponent functions. Moderate deviation …

Moderate-Deviations of Lossy Source Coding for Discrete and Gaussian Sources

We study the moderate-deviations (MD) setting for lossy source coding of stationary memoryless sources. More specifically, we derive fundamental compression limits of source codes whose rates are $R(D) \pm ε_n$, where $R(D)$ is the rate-distortion function and $ε_n$ is a sequence that dominates $\sqrt{1/n}$. This MD setting is complementary to the large-deviations and central limit settings and was studied by Altug and Wagner for the channel coding setting. We show, for finite alphabet and Gaussian sources, that as in the central limit-type results, the so-called dispersion for lossy source co…

Finite-Length Bounds for Joint Source-Channel Coding with Markovian Source and Additive Channel Noise to Achieve Large and Moderate Deviation Bounds

We derive novel upper and lower finite-length bounds of the error probability in joint source-channel coding when the source obeys an ergodic Markov process and the channel is a Markovian additive channel or a Markovian conditional additive channel. These bounds are tight in the large and moderate deviation regimes.

全量子任务的中偏差展开

Moderate deviation expansion for fully quantum tasks

该研究推导部分平滑最大信息与假设检验相对熵的紧关系，刻画了量子信源编码等多种全量子通信任务在中偏差区域下通信代价与误差的有限块长权衡。