arXiv subjects
Publications and source records attributed to Xiequan Fan.
We establish nonuniform Berry-Esseen bounds for martingales under the conditional Bernstein condition. These bounds imply Cramér type large deviations for moderate $x$'s, and are of exponential decay rate as de la Peña's inequality when $x\rightarrow \infty$. Statistical applications associated with linear regressions and self-normalized large deviations are also provided.
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.
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).
Using changes of probability measure developed by \mbox{Grama} and Haeusler (Stochastic Process.\ Appl., 2000), we obtain two generalizations of the deviation inequalities of Lanzinger and Stadtmüller (Stochastic Process.\ Appl., 2000) and Fuk and Nagaev (Theory Probab. Appl., 1971) to the case of martingales. Our inequalities recover the best possible decaying rate of independent case. Applications to linear regressions and weak invariance principles for martingales are provided.
Multistable Lévy motions are extensions of Lévy motions where the stability index is allowed to vary in time. Several constructions of these processes have been introduced recently, based on Poisson and Ferguson-Klass-LePage series representations and on multistable measures. In this work, we prove a functional central limit theorem for the independent-increments multistable Lévy motion, as well as of integrals with respect to these processes, using weighted sums of independent random variables. This allows us to construct continuous approximations of multistable Lévy motions. In particular, we prove that multistable Lévy motions are stochastic Hölder continuous and strongly localisable.
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).
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.
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.
We consider a Markov chain X_1, X_2, ..., X_n belonging to a class of iterated random functions, which is "one-step contracting" with respect to some distance d. If f is any separately Lipschitz function with respect to d, we use a well known decomposition of S_n=f(X_1, ..., X_n) -E[f(X_1, ..., X_n)]$ into a sum of martingale differences d_k with respect to the natural filtration F_k. We show that each difference d_k is bounded by a random variable eta_k independent of F_{k-1}. Using this very strong property, we obtain a large variety of deviation inequalities for S_n, which are governed by the distribution of the eta_k's. Finally, we give an application of these inequalities to the Wasserstein distance between the empirical measure and the invariant distribution of the chain.
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.
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.
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.