The Law Of Large Number And Central Limit Theorem Of Independent Random Sequence And Their Applications

This paper concerns on the convergence and relation of the series of random variable. By means of the studying on laws of large numbers and central-limit theorems under the condition either of independent and identically distributions or independent but different-distributions comprehensively, we illuminate that the mean value is stable which is the essential property for the stochastic event. And we enplaned the theoretical rules that the sum of independent random variables submited the normal distribution according to central-limit theorems. At the same time, we analyzed and stated the Markov theorem under the different terms and conditions, So its deductive theorems, i.e. Chebyshev theorem, Bernoulli theorem, Poisson theorem including Khintchine theorem based on the identically distribution. Moreover, we have got the strong laws of large numbers in the sense of its probability with 1 by generalization. Central-limit theorems were studied in the same way.In addition, we showed the relations between various laws of large numbers, various central-limit theorems and themselves mutually. The more interesting is that plenty of correlative examples including positive and negative are illustrated to emphasize the importance of identifying the conditions of these theorems in all sorts of their applications.At last, we brought forth the value of laws of large numbers and central-limit theorems in some subjects such as mathematical statistics, administrative decision-making, approximate calculation, insurance and so on.