Sentence examples for mean square convergence from inspiring English sources

Mean square convergence of the random Fourier series are discussed.

In particular, we propose the asymptotic and per-step (mean square) convergence factors as measures of the convergence speed and derive the exact value for the per-step (mean square) convergence factor.

Then we study their performances in terms of the mean square convergence and the convergence in probability.

The mean square convergence result of the method in temporal direction is tested numerically, and numerical comparisons with finite difference method are also investigated.

Note that the mean square convergence results stated in [12] relies on some strict assumptions.

A random mean value theorem is established and the mean square convergence of these methods is proved.

The strong and mean-square convergence rates are obtained.

The strong and mean-square convergence rates are explicitly obtained.

A two-step algorithm is developed, which demonstrates desired convergence properties including strong convergence, mean-square convergence, convergence rates, asymptotic normality, and asymptotical efficiency in terms of the Cramér Rao lower bound.

To this end, the Karhunen Loève (K L) expansion, also known as principal component analysis (PCA), is the most popular model reduction method due to its uniform mean-square convergence.

Fig. 7 Mean-square convergence of the splitting method.