We consider the following family of random trigonometric polynomials form Sn(θ):=1 n ∑k=1nakcos(kθ)+bksin(kθ), where variables {ak,bk}k≥1 are i.i.d. and symmetric, defined on a common probability space (Ω,F,P). In seminal paper [SZ54], Salem Zygmund proved that for Rademacher coefficients, P−almost surely all t∈R limn→+∞ 1 2π∫02πeitSn(θ)dθ=e−t2∕2. To best our knowledge, natural question fluctua...

The problems of almost sure a.s. stability and a.s. stabilization are investigated for hybrid stochastic systems HSSs with time-varying delays. The different time-varying delays in the drift part and in the diffusion part are considered. Based on nonnegative semimartingale convergence theorem, Hölder’s inequality, Doob’s martingale inequality, and Chebyshev’s inequality, some sufficient conditi...

By the continuous and discrete nonnegative semimartingale convergence theorems, this paper investigates conditions under which the Euler–Maruyama (EM) approximations of stochastic functional differential equations (SFDEs) can share the almost sure exponential stability of the exact solution. Moreover, for sufficiently small stepsize, the decay rate as measured by the Lyapunov exponent can be re...