We consider the totally asymmetric simple exclusion process in a critical scaling parametrized by @math , which creates a shock in the particle density of order @math @math the observation time. When starting from step initial data, we provide bounds on the limiting law which in particular imply that in the double limit @math one recovers the product limit law and the degeneration of the correlation length observed at shocks of order @math . This result is shown to apply to a general last-passage percolation model. We also obtain bounds on the two-point functions of several @math processes.
A result we shall use repeatedly in this paper concerns the transversal fluctuations of the maximizer @math when @math are i.i.d. and @math there are bounds on the probability that the maximizer @math from @math to a point @math deviates more than @math from the straight line @math , @math . This is a result from @cite_11 and cited here as Theorem 3.2. Furthermore, for line-to-point problems, we have detailed control over the distribution of the (random) starting point of @math , see @cite_4 , Lemma 1.1, Lemma 1.2 and also (4.18) in @cite_1 .
@cite_1 @cite_4 @cite_11
2562674776 1724455341 2001663593 1540485574
We consider in this paper the problem of sampling a high-dimensional probability distribution @math having a density with respect to the Lebesgue measure on @math , known up to a normalization constant @math . Such problem naturally occurs for example in Bayesian inference and machine learning. Under the assumption that @math is continuously differentiable, @math is globally Lipschitz and @math is strongly convex, we obtain non-asymptotic bounds for the convergence to stationarity in Wasserstein distance of order @math and total variation distance of the sampling method based on the Euler discretization of the Langevin stochastic differential equation, for both constant and decreasing step sizes. The dependence on the dimension of the state space of these bounds is explicit. The convergence of an appropriately weighted empirical measure is also investigated and bounds for the mean square error and exponential deviation inequality are reported for functions which are measurable and bounded. An illustration to Bayesian inference for binary regression is presented to support our claims. Let @math be an i.i.d. sequence of random variables in @math , @math . We show that, for any function @math , under regularity conditions, [n^ 1 2 (n^ -1 i=1 ^n (X_i) f ^(X_i) - (x) ,dx ) P 0, ] where @math is the classical kernel estimator of the density of @math . This result is striking because it speeds up traditional rates, in root @math , derived from the central limit theorem when @math . Although this paper highlights some applications, we mainly address theoretical issues related to the later result. We derive upper bounds for the rate of convergence in probability. These bounds depend on the regularity of the functions @math and @math , the dimension @math and the bandwidth of the kernel estimator @math . Moreover, they are shown to be accurate since they are used as renormalizing sequences in two central limit theorems each reflecting different degrees of smoothness of @math . As an application to regression modelling with random design, we provide the asymptotic normality of the estimation of the linear functionals of a regression function. As a consequence of the above result, the asymptotic variance does not depend on the regression function. Finally, we debate the choice of the bandwidth for integral approximation and we highlight the good behavior of our procedure through simulations. par>We prove some non-approximability results for restrictions of basic combinatorial optimization problems to instances of bounded “degreeror bounded “width.” Specifically: We prove that the Max 3SAT problem on instances where each variable occurs in at most B clauses, is hard to approximate to within a factor @math , unless @math . H stad [18] proved that the problem is approximable to within a factor @math in polynomial time, and that is hard to approximate to within a factor @math . Our result uses a new randomized reduction from general instances of Max 3SAT to bounded-occurrences instances. The randomized reduction applies to other Max SNP problems as well. We observe that the Set Cover problem on instances where each set has size at most B is hard to approximate to within a factor @math unless @math . The result follows from an appropriate setting of parameters in Feige's reduction [11]. This is essentially tight in light of the existence of @math -approximate algorithms [20, 23, 9] We present a new PCP construction, based on applying parallel repetition to the inner verifier,'' and we provide a tight analysis for it. Using the new construction, and some modifications to known reductions from PCP to Hitting Set, we prove that Hitting Set with sets of size B is hard to approximate to within a factor @math . The problem can be approximated to within a factor B [19], and it is the Vertex Cover problem for B =2. The relationship between hardness of approximation and set size seems to have not been explored before. We observe that the Independent Set problem on graphs having degree at most B is hard to approximate to within a factor @math , unless P = NP . This follows from a comination of results by Clementi and Trevisan [28] and Reingold, Vadhan and Wigderson [27]. It had been observed that the problem is hard to approximate to within a factor @math unless P = NP [1]. An algorithm achieving factor @math is also known [21, 2, 30, 16 . The is the following problem: a coin is given, which lands on head with probability either @math or @math . We are given the outcome of @math independent tosses of this coin, and the goal is to guess which way the coin is biased, and to answer correctly with probability @math . When our computational model is unrestricted, the majority function is optimal, and succeeds when @math for a large enough constant @math . The coin problem is open and interesting in models that cannot compute the majority function. In this paper we study the coin problem in the model of . We prove that in order to succeed in this model, @math must be at least @math . For constant @math this is tight by considering the recursive tribes function, and for other values of @math this is nearly tight by considering other read-once AND-OR trees. We generalize this to a , where instead of independent tosses of a coin we are given independent tosses of one of two @math -sided dice. We prove that if the distributions are too close and the mass of each side of the dice is not too small, then the dice cannot be distinguished by small-width read-once branching programs. We suggest one application for this kind of theorems: we prove that Nisan's Generator fools width- @math read-once branching programs, using seed length @math . For @math , this seed length is @math . The coin theorem and its relatives might have other connections to PRGs. This application is related to the independent, but chronologically-earlier, work of Braver man, Rao, Raz and Yehudayoff BRRY .