We establish some limit theorems for the elephant random walk (ERW), including Berry–Esseen’s bounds, Cramér moderate deviations and local limit theorems. These limit theorems can be regarded as refinements of the central limit theorem for the ERW. Moreover, by these limit theorems, we conclude that the convergence rate of normal approximations and the domain of attraction of normal distribution mainly depend on a memory parameter p which lies between 0 and 3/4.

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Recently, the elephant random walk has attracted much attention. A wide range of literature studies are available for the asymptotic behavior of the process, such as central limit theorems, functional limit theorems, and the law of the iterated logarithm. However, there is no result concerning the Wasserstein-1 distance for normal approximations. In this paper, we show that the Wasserstein-1 distance in the central limit theorem is totally different when a memory parameter p belongs to one of the three cases 0 < p < 1/2, 1/2 < p < 3/4, and p = 3/4.

Personal sound zone systems aim at creating multiple listening zones within a shared space with minimum interference between zones, but the performance is often poorer than simulations predict and effects of nonlinear distortion are sometimes audible. This paper assesses the impact of nonlinear distortion on sound zones through simulations and measurements performed under anechoic conditions. Two sound zones, one bright and one dark, are created with acoustic contrast control using two loudspeaker arrays driven at 250 Hz. Nonlinear distortion is modelled using second or third order nonlinearities. Simulations show that nonlinear distortion degrades the acoustic contrast, which is confirmed by experimental measurements. The harmonic distortion is audible in the dark zone. Frequency resolved measurements reveal that harmonic distortion contributes to contrast loss, but nonlinear effects on the fundamental component are the main cause. Nonlinear distortion can be controlled through regularization of the loudspeaker control effort. Simulations and experiments show an optimum regularization range where contrast is optimized by balancing linear performance and nonlinear distortion.

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