In this paper, a novel statistical application of large deviation principle (LDP) to the robot trajectory tracking problem is presented. The exit probability of the trajectory from stability zone is evaluated, in the presence of small-amplitude Gaussian and Poisson noise. Afterward, the limit of the partition function for the average tracking error energy is derived by solving a fourth-order system of Euler–Lagrange equations. Stability and computational complexity of the proposed approach is investigated to show the superiority over the Lyapunov method. Finally, the proposed algorithm is validated by Monte Carlo simulations and on the commercially available Omni bundleTM robot.

Let W = {Wn: n ∈ N} be a sequence of random vectors in Rd, d ≥ 1. In this paper we consider the logarithmic asymptotics of the extremes of W, that is, for any vector q > 0 in Rd, we find that logP(there exists n ∈ N: Wnuq) as u → ∞. We follow the approach of the restricted large deviation principle introduced in Duffy (2003). That is, we assume that, for every q ≥ 0, and some scalings {an}, {vn}, (1 / vn)logP(Wn / an ≥ uq) has a, continuous in q, limit JW(q). We allow the scalings {an} and {vn} to be regularly varying with a positive index. This approach is general enough to incorporate sequences W, such that the probability law of Wn / an satisfies the large deviation principle with continuous, not necessarily convex, rate functions. The equations for these asymptotics are in agreement with the literature.

Consider the random variable Ln defined as the length of a longest common subsequence of two random strings of length n and whose random characters are independent and identically distributed over a finite alphabet. Chvátal and Sankoff showed that the limit γ=limn→∞E[Ln]/n is well defined. The exact value of this constant is not known, but various methods for the computation of upper and lower bounds have been discussed in the literature. Even so, high-precision bounds are hard to come by. In this paper we discuss how large deviation theory can be used to derive a consistent sequence of upper bounds, (qm)m∈ℕ, on γ, and how Monte Carlo simulation can be used in theory to compute estimates, q̂m, of the qm such that, for given Ξ > 0 and Λ ∈ (0,1), we have P[γ < q̂ < γ + Ξ] ≥ Λ. In other words, with high probability the result is an upper bound that approximates γ to high precision. We establish O((1 − Λ)−1Ξ−(4+ε)) as a theoretical upper bound on the complexity of computing q̂m to the given level of accuracy and confidence. Finally, we discuss a practical heuristic based on our theoretical approach and discuss its empirical behavior.

The third-generation (3G) mobile communication system uses a technique called code division multiple access (CDMA), in which multiple users use the same frequency and time domain. The data signals of the users are distinguished using codes. When there are many users, interference deteriorates the quality of the system. For more efficient use of resources, we wish to allow more users to transmit simultaneously, by using algorithms that utilize the structure of the CDMA system more effectively than the simple matched filter (MF) system used in the proposed 3G systems. In this paper, we investigate an advanced algorithm called hard-decision parallel interference cancellation (HD-PIC), in which estimates of the interfering signals are used to improve the quality of the signal of the desired user. We compare HD-PIC with MF in a simple case, where the only two parameters are the number of users and the length of the coding sequences. We focus on the exponential rate for the probability of a bit-error, explain the relevance of this parameter, and investigate how it scales when the number of users grows large. We also review extensions of our results, proved elsewhere, showing that in HD-PIC, more users can transmit without errors than in the MF system.

The theme of this paper is that the simple concept of risk sensitivity raises ideas, shows associations, and forces clarification of issues that one would have thought quite unrelated.

We investigate the optimal allocation of effort to a collection of n projects. The projects are ‘restless' in that the state of a project evolves in time, whether or not it is allocated effort. The evolution of the state of each project follows a Markov rule, but transitions and rewards depend on whether or not the project receives effort. The objective is to maximize the expected time-average reward under a constraint that exactly m of the n projects receive effort at any one time. We show that as m and n tend to ∞ with m/n fixed, the per-project reward of the optimal policy is asymptotically the same as that achieved by a policy which operates under the relaxed constraint that an average of m projects be active. The relaxed constraint was considered by Whittle (1988) who described how to use a Lagrangian multiplier approach to assign indices to the projects. He conjectured that the policy of allocating effort to the m projects of greatest index is asymptotically optimal as m and n tend to∞. We show that the conjecture is true if the differential equation describing the fluid approximation to the index policy has a globally stable equilibrium point. This need not be the case, and we present an example for which the index policy is not asymptotically optimal. However, numerical work suggests that such counterexamples are extremely rare and that the size of the suboptimality which one might expect is minuscule.