arXiv subjects

Ion Grama

Publications and source records attributed to Ion Grama.

At least 55 records · Page 3Linked to original sources

arXiv2016

Asymptotic of the distribution and harmonic moments for a supercritical branching process in a random environment

Let $(Z_n)$ be a supercritical branching process in an independent and identically distributed random environment $ξ$. We show the exact decay rate of the probability $\mathbb{P}(Z_n=j | Z_0 = k)$ as $n \to \infty$, for each $j \geq k,$ assuming that $\mathbb{P} (Z_1 = 0) =0$. We also determine the critical value for the existence of harmonic moments of the random variable $W=\lim_{n\to\infty}\frac{Z_n}{\mathbb E (Z_n|ξ)}$ under a simple moment condition.

math.PR↗

arXiv2016

Conditioned limit theorems for products of random matrices

Consider the product $G_{n}=g_{n} ... g_{1}$ of the random matrices $g_{1},...,g_{n}$ in $GL(d,\mathbb{R}) $ and the random process $ G_{n}v=g_{n}... g_{1}v$ in $\mathbb{R}^{d}$ starting at point $v\in \mathbb{R}^{d}\smallsetminus \{0\} .$ It is well known that under appropriate assumptions, the sequence $(\log \Vert G_{n}v\Vert)_{n\geq 1}$ behaves like a sum of i.i.d.\ r.v.'s and satisfies standard classical properties such as the law of large numbers, law of iterated logarithm and the central limit theorem. Denote by $\mathbb{B}$ the closed unit ball in $\mathbb{R}^{d}$ and by $\mathbb{B}^{c}$ its complement. For any $v\in \mathbb{B}^{c}$ define the exit time of the random process $G_{n}v$ from $\mathbb{B}^{c}$ by $τ_{v}=\min \{n\geq 1:G_{n}v\in \mathbb{B}\} .$ We establish the asymptotic as $n \to \infty $ of the probability of the event $\{τ_{v}>n\} $ and find the limit law for the quantity $\frac{1}{\sqrt{n}} \log \Vert G_{n}v\Vert $ conditioned that $τ_{v}>n.$

math.PR↗

arXiv2016

Berry-Esseen's bound and Cramér's large deviation expansion for a supercritical branching process in a random environment

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.

math.PR↗

arXiv2016

Limit theorems for affine Markov walks conditioned to stay positive

Consider the real Markov walk $S_n = X_1+ \dots+ X_n$ with increments $\left(X_n\right)_{n\geq 1}$ defined by a stochastic recursion starting at $X_0=x$. For a starting point $y>0$ denote by $τ_y$ the exit time of the process $\left( y+S_n \right)_{n\geq 1}$ from the positive part of the real line. We investigate the asymptotic behaviour of the probability of the event $τ_y \geq n$ and of the conditional law of $y+S_n$ given $τ_y \geq n$ as $n \to +\infty$.

math.PR↗

arXiv2015

Martingale inequalities of type Dzhaparidze and van Zanten

Freedman's inequality is a supermartingale counterpart to Bennett's inequality. This result shows that the tail probabilities of a supermartingale is controlled by the quadratic characteristic and a uniform upper bound for the supermartingale difference sequence. Replacing the quadratic characteristic by $\textrm{H}_k^y:= \sum_{i=1}^k\left(\mathbf{E}(ξ_i^2 |\mathcal{F}_{i-1}) +ξ_i^2\textbf{1}_{\{|ξ_i|> y\}}\right),$ Dzhaparidze and van Zanten (\emph{Stochastic Process. Appl.}, 2001) have extended Freedman's inequality to martingales with unbounded differences. In this paper, we prove that $\textrm{H}_k^y$ can be refined to $\textrm{G}_k^{y} :=\sum_{i=1}^k \left( \mathbf{E}(ξ_i^2\textbf{1}_{\{ξ_i \leq y\}} |\mathcal{F}_{i-1}) + ξ_i^2\textbf{1}_{\{ξ_i> y\}}\right).$ Moreover, we also establish two inequalities of type Dzhaparidze and van Zanten. These results extend Sason's inequality (\emph{Statist. Probab. Lett.}, 2012) to the martingales with possibly unbounded differences and establish the connection between Sason's inequality and De la Peña's inequality (\emph{Ann.\ Probab.,} 1999). An application to self-normalized deviations is given.

math.PR↗

arXiv2015

Sharp large deviation results for sums of independent random variables

We show sharp bounds for probabilities of large deviations for sums of independent random variables satisfying Bernstein's condition. One such bound is very close to the tail of the standard Gaussian law in certain case; other bounds improve the inequalities of Bennett and Hoeffding by adding missing factors in the spirit of Talagrand (1995). We also complete Talagrand's inequality by giving a lower bound of the same form, leading to an equality. As a consequence, we obtain large deviation expansions similar to those of Cramér (1938), Bahadur-Rao (1960) and Sakhanenko (1991). We also show that our bound can be used to improve a recent inequality of Pinelis (2014).

math.PR↗

arXiv2015

A generalization of Cramér large deviations for martingales

In this note, we give a generalization of Cramér's large deviations for martingales, which can be regarded as a supplement of Fan, Grama and Liu (Stochastic Process. Appl., 2013). Our method is based on the change of probability measure developed by Grama and Haeusler (Stochastic Process. Appl., 2000).

math.PR↗

arXiv2015

Exponential inequalities for martingales with applications

The paper is devoted to establishing some general exponential inequalities for supermartingales. The inequalities improve or generalize many exponential inequalities of Bennett, Freedman, de la Peña, Pinelis and van de Geer. Moreover, our concentration inequalities also improve some known inequalities for sums of independent random variables. Applications associated with linear regressions, autoregressive processes and branching processes are provided. In particular, an interesting application of {de la Peña's} inequality to self-normalized deviations is also provided.

math.PR↗

arXiv2014

Cramér large deviation expansions for martingales under Bernstein's condition

By using the conjugate distribution technique of Cramér, we obtain some expansions of large deviation probabilities for martingales with differences satisfying the conditional Bernstein's condition. The expansions are of the same order as in the classical Cramér's large deviation result and are therefore optimal.

math.PR↗

arXiv2013

A Non-Local Means Filter for Removing the Poisson Noise

A new image denoising algorithm to deal with the Poisson noise model is given, which is based on the idea of Non-Local Mean. By using the "Oracle" concept, we establish a theorem to show that the Non-Local Means Filter can effectively deal with Poisson noise with some modification. Under the theoretical result, we construct our new algorithm called Non-Local Means Poisson Filter and demonstrate in theory that the filter converges at the usual optimal rate. The filter is as simple as the classic Non-Local Means and the simulation results show that our filter is very competitive.

stat.AP↗

arXiv2013

Sharp large deviation probabilities for sums of independent bounded random variables

We obtain some optimal inequalities on tail probabilities for sums of independent bounded random variables. Our main result completes an upper bound on tail probabilities due to Talagrand by giving a one-term asymptotic expansion for large deviations. This result can also be regarded as sharp large deviations of types of Cramér and Bahadur-Ranga Rao.

math.PR↗

arXiv2012

Convergence Theorems for the Non-Local Means Filter

In this paper, we establish convergence theorems for the Non-Local Means Filter in removing the additive Gaussian noise. We employ the techniques of "Oracle" estimation to determine the order of the widths of the similarity patches and search windows in the aforementioned filter. We propose a practical choice of these parameters which improve the restoration quality of the filter compared with the usual choice of parameters.

stat.AP↗

arXiv2012

Large deviation exponential inequalities for supermartingales

Let $(X_{i}, \mathcal{F}_{i})_{i\geq 1}$ be a sequence of supermartingale differences and let $S_k=\sum_{i=1}^k X_i$. We give an exponential moment condition under which $P(\max_{1\leq k \leq n} S_k \geq n)=O(\exp\{-C_1 n^α\}),$ $n\rightarrow \infty,$ where $α\in (0, 1)$ is given and $C_{1}>0$ is a constant. We also show that the power $α$ is optimal under the given condition. In particular, when $α=1/3$, we recover an inequality of Lesigne and Volný.

math.PR↗

arXiv2012

Hoeffding's inequality for supermartingales

We give an extension of Hoeffding's inequality to the case of supermartingales with differences bounded from above. Our inequality strengthens or extends the inequalities of Freedman, Bernstein, Prohorov, Bennett and Nagaev.

math.PR↗

arXiv2012

Optimal Weights Mixed Filter for Removing Mixture of Gaussian and Impulse Noises

According to the character of Gaussian, we modify the Rank-Ordered Absolute Differences (ROAD) to Rank-Ordered Absolute Differences of mixture of Gaussian and impulse noises (ROADG). It will be more effective to detect impulse noise when the impulse is mixed with Gaussian noise. Combining rightly the ROADG with Optimal Weights Filter (OWF), we obtain a new method to deal with the mixed noise, called Optimal Weights Mixed Filter (OWMF). The simulation results show that the method is effective to remove the mixed noise.

cs.CV↗

arXiv2012

A New Poisson Noise Filter based on Weights Optimization

We propose a new image denoising algorithm when the data is contaminated by a Poisson noise. As in the Non-Local Means filter, the proposed algorithm is based on a weighted linear combination of the bserved image. But in contract to the latter where the weights are defined by a Gaussian kernel, we propose to choose them in an optimal way. First some "oracle" weights are defined by minimizing a very tight upper bound of the Mean Square Error. For a practical application the weights are estimated from the observed image. We prove that the proposed filter converges at the usual optimal rate to the true image. Simulation results are presented to compare the performance of the presented filter with conventional filtering methods.

stat.AP↗

arXiv2011

Controlled Total Variation regularization for inverse problems

This paper provides a new algorithm for solving inverse problems, based on the minimization of the $L^2$ norm and on the control of the Total Variation. It consists in relaxing the role of the Total Variation in the classical Total Variation minimization approach, which permits us to get better approximation to the inverse problems. The numerical results on the deconvolution problem show that our method outperforms some previous ones.

cs.CV↗

arXiv2011

Removing Gaussian Noise by Optimization of Weights in Non-Local Means

A new image denoising algorithm to deal with the additive Gaussian white noise model is given. Like the non-local means method, the filter is based on the weighted average of the observations in a neighborhood, with weights depending on the similarity of local patches. But in contrast to the non-local means filter, instead of using a fixed Gaussian kernel, we propose to choose the weights by minimizing a tight upper bound of mean square error. This approach makes it possible to define the weights adapted to the function at hand, mimicking the weights of the oracle filter. Under some regularity conditions on the target image, we show that the obtained estimator converges at the usual optimal rate. The proposed algorithm is parameter free in the sense that it automatically calculates the bandwidth of the smoothing kernel; it is fast and its implementation is straightforward. The performance of the new filter is illustrated by numerical simulations.

stat.OT↗