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Sketch of the proof of Theorem 3. We only give the main steps to prove Inequality 3. Let us introduce the processes Unt and Hi f , defined by. Its predictable variation and its optional variation are respectively given by. It follows that. In this part we show how the Bernstein Inequality 3. We want to obtain a prognostic on the survival time adjusted on the covariates in a high-dimensional setting, and more precisely to estimate the unknown intensity Ao using a Lasso procedure.

The properties of the Lasso estimator are stated in terms of non-asymptotic oracle inequalities. Such an oracle inequality is a consequence of an appropriate Bernstein inequality. The Lasso procedure based on a direct application of the Bernstein inequality 3. This type of procedure has already been considered, especially by Hansen, Reynaud-Bouret and Rivoirard and Gai'ffas and Guilloux , but for different statistical models.

The baseline hazard function a0 is assumed to be known and the estimation of A0 reduces to the estimation of f0. For the sake of simplicity, we consider here that a0 is known, and we refer to Lemler for the general case. In this high-dimensional setting, the function fo is estimated using a weighted Lasso procedure. The weights u, are positive data-driven weights suitably chosen thanks to the empirical Bernstein's inequality 3. Main steps leading to the non-asymptotic oracle inequality.

To obtain non-asymptotic oracle inequalities, we have to control the centered empirical process nn,T fj. We apply Theorem 3. Then, with probability larger than 1 - Ae-x. Since its introduction by Gordin , the martingale method has attracted number of probabilists. It was originally designed for the Central Limit Theorem CLT but it has been successfully used for almost any limit theorem the Weak Invariance Principle WIP and its quenched version, the Law of the Iterated Logarithm and its functional versions, the Marcinkiewicz-Zygmund strong law of large numbers Those characterizations are of theoretical interest but proved to be useful in the applications as well see for instance Gordin and Peligrad The martingale method applies to "different" situations : stationary processes adapted or not , non-invertible dynamical systems, functionals of Markov chain.

In this note, for the sake of clarity, we shall only be concerned with adapted stationary processes. As we recall below, this case is actually equivalent to considering functionals of Markov chains. The case of non-adapted processes may be treated similarly see for instance Volny or Cuny a.

The adaptation of the results mentioned below to the setting of non-invertible dynamical systems needs more care since what we really obtain in that case is a"reverse" martingale approximation. We shall consider only real-valued processes, but some results extend to Hilbert space-valued processes. We use the same notations as in the introduction. X0 is Fo-measurable. For simplicity, we assume T to be ergodic.

It turns out that this operator allows to see our process Xn neZ as a functional of a Markov chain, see Cuny and Volny Another advantage of this operator is that it allows to translate projective conditions in terms of Q, hence to make use of classical facts from ergodic theory of operators. For instance, the martingale-coboundary decomposition of Gordin is easily characterized as follows, which has been observed by Volny the non-adapted case is also considered there.

Our goal now will be to explain that, in several situations, the proof of a limit theorem by means of a martingale approximation may be split into two steps: the first step is to prove the desired result when one has a martingale-coboundary decomposition as above.

The second step consists of obtaining a maximal inequality adapted to the limit theorem under consideration. This approach has been used explicitly in Cuny a and Cuny b and implicitly, for instance, in Jiang and Wu , Cuny and Volny or Cuny and Merlevede The notation ASIP stands for the almost sure invariance principle. The above martingale approximations are unique.

If Xo admits a martingale approximation of one of the above types then it satisfies the corresponding limit theorem. We would like to study the validity of some limit theorems under the Hannan condition 0. In terms of the operator Q those conditions read. It is not hard to prove that those spaces are Banach spaces and that Q induces a contraction of H2 and of MW2.

With little effort one can also prove that, for every. By the mean ergodic theorem see e. Theorems 1. We first have to find D0. According to Gordin and Peligrad , if there exists a martingale approximation of type CLT , necessarily.

Then, D is an unbounded operator on L2 H, P. As one may expect from the proof of Proposition 4. The operator D is well-defined i. For some problems, it may be useful to have an other form of D, more adapted to the conditions 4. One can prove that if the representation 4. Before proving the martingale approximation properties, we shall need the corresponding estimate with a big "O" instead of a little "o".

For this we need some maximal inequalities. We start with the martingale case. We have. Proposition 4. The first estimate is nothing but Doob's maximal inequality. Both inequalities hold without ergodicity.

The maximal inequality 4. In the iid case, it has been proved by Pisier To emphasize the usefulness of such inequality we mention, that thanks to 4. In the next proposition, X stands either for H2 or for MW2. The estimate 4. For the other estimates, at first, notice that to prove 4. For the martingale part both estimates follow from Proposition 4. It follows from 4. Similarly, by 4.

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Show 3 more comments. Active Oldest Votes. It remains to give Proof of Lemma 1: Suppose the highlighted statement is true. Improve this answer. Iosif Pinelis Iosif Pinelis I went through the paper that you have cited, but I couldn't find Theorem 3.

Perhaps you meant Theorem 3? I have a couple of more questions : 1. I had given a wrong link. This is now corrected. On the other hand, Azuma's inequality does not assume the conditional symmetry. Also, Azuma's inequality is due entirely to Hoeffding '63 see the last paragraph of Section 2 of Hoeffding's paper.

Add a comment. Yuval Peres Yuval Peres 6, 1 1 gold badge 18 18 silver badges 28 28 bronze badges. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name. Email Required, but never shown. Featured on Meta. Opt-in alpha test for a new Stacks editor. Visual design changes to the review queues. Related 8. Question feed. The basic result on the convergence of martingale is Doob's theorem: If is a submartingale andthen with probability 1, exists martingale.

A natural generalization of a martingale is the concept of a local martingale, that is, a stochastic process for which there is a sequence of martingale stopping times with probability 1, such that for each the "stopped" processes.

In the case of discrete time each local martingale is a martingale transform, that is, can be represented in the formwhere is a predictable process and is a martingale. Each submartingale has, moreover, a unique Martingale decomposition martingale, where is a local martingale and is martingale predictable non-decreasing process. In particular, if is a square-integrable martingale, then its square martingale a submartingale in whose Doob—Meyer decomposition the process martingale called the quadratic characteristic of the martingale.

For read article square-integrable martingale and martingale process such that with probability 1, it is possible to define the stochastic integral. In the case of continuous time martingale Doob, Burkholder and Davis inequalities are still true for right-continuous processes having left limits. Stopping times are also called optimal times, or, in the older literature, Markov times or Markov moments, cf.

Markov moment. The optimal sampling theorem martingale also called the stopping theorem or Doob's stopping theorem. The notion of a martingale is one of the most important concepts martingale modern probability theory. It is martingale in the theories of Markov processes and stochastic integrals, martingale is useful in many parts of analysis convergence theorems in ergodic theory, derivatives and lifting in measure theory, inequalities martingale the theory of singular integrals, etc.

Log in. Namespaces Page Discussion. Views View View source History. Martingale to: navigationsearch. Martingale notions are stochastic processes which form a submartingale, if or a supermartingale, if Example martingale. Let be a martingale submartingalea predictable sequence that is, is not only -measurable but also -measurable, and let Then, if the variables are integrable, the stochastic process forms a martingale submartingale.

In particular, if is a sequence martingale independent random variables corresponding to a Bernoulli scheme and 2 then is a martingale. Markov momentif and if 3 martingale with probability 1where As a particular case of this the Wald identity follows: Among martingale basic results of the theory of martingales is Doob's inequality: If is a non-negative submartingale, then 4 5 6 If is a martingale, then for the Burkholder inequalities hold generalizations of the inequalities of Khinchin and Marcinkiewicz—Zygmund for sums of martingale random variables : 7 martingale and are certain universal constants not depending on orfor which one can take and Taking 5 and 7 into account, it follows that 8 where When inequality 8 can be generalized.

Namely, Davis' inequality martingale There are universal martingale and such that In the proof of a different kind of theorem on the convergence of submartingales with probability 1, a key role is played by Doob's inequality for the mathematical expectation of the number martingale upcrossings,of the interval by the submartingale in steps; namely 9 The basic result on the convergence of submartingales is Doob's theorem: If is a submartingale andthen with martingale 1, exists and.

A natural generalization flamingo casino a martingale is the concept of a martingale martingale, that is, a stochastic process for which there is a sequence of finite stopping times with probability 1martingale, such that for each the "stopped" processes are martingales. For each square-integrable martingale and predictable process such that with probability 1, it is possible to define the stochastic integral which is a local martingale.

Doob, "Classical potential theory and its probabilistic counterpart"Springer pp. Heyde, "Martingale limit theory and its application"Acad. Encyclopedia of Mathematics. This article was adapted from an original article by A. See original article. Categories : Probability and statistics Probability theory and stochastic processes Stochastic processes.

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