Law of Great Numbers and Central Limit Theorem of Series of Correlated Random Variables;

Note on the almost sure central limit theorem for ρ~--mixing sequences.;

The present paper firstly represents the model about random walks in time-random environments on the right line, then the studies about recurrence-transience criteria and limit theorem by using some relative theories of Markov chains, and finally a center limit theorem of this random walks in the non-recurrence case.

In this article,a Pólya Urn Model is discussed,We obtain a strong law of large numbers and a center limit theorem.

Then we further study the limit properties and derive a strong law of large numbers and a center limit theorem of this random walks in i.

The method is mainly based on the central limit theory, and can avoid the state explosion problem.

In this paper, we will use the theory of stochastic systems fluctuations, random walk, central limit theory to discuss the weak convergence properties of the two random paths and study the weak convergence limit of the model .

Note on the almost sure central limit theorem for ρ~--mixing sequences.;

The almost sure central limit theorem for the maximum and minimum of stationary Gaussian sequences of d-mensional random vectors was considered as max1≤p≠q≤d supn≥0 |rn(p,q)|<1 and ρnlog n(log log n)1+ε=O(1).

Under the con- ditions of D′(u_n)and D_2({u_k,u_n}),an almost sure central limit theorem for the maximum of weakly de- pendent sequences is obtained.

In order to improve the invisibility and the robustness of semi-fragile watermark,this paper brings up characteristic and representation in DWT transform domain based on visual features model,and puts forward the quantized central limit theorem which applies to adjust the coefficients in general transform domain.