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Convergence by probability

• 1. background
• 2. Definition
• 3. Theorem
  • 3.1 Definition
  • 3.2 prove
• 4. reference

1. background

Suppose we check whether the products are qualified one by one on the production line . remember For the first time The quantity of unqualified products in this inspection , It can only take 0 and 1, And , among Is the unqualified rate of the product . This is a Bernoulli experiment sequence , It corresponds to an independent identically distributed （ The binomial distribution ） A sequence of random variables ：, Write it down as . If you consider before The number of nonconforming products in each inspection Be situated between And What is the probability between ？ namely

Of course , from This probability can be calculated , But when The larger the （ Such as or ） It is difficult to calculate . Can we find a simple random variable , Use its distribution （ stay large ） The approximate value of the above probability can be easily calculated , namely

So under what conditions and in what sense , A sequence of random variables It can converge to random variables

Another example is in the above Bernoulli experiment sequence , front Frequency of nonconforming products in this inspection Rate of nonconforming products The deviation of Can it be arbitrarily small ？ When I'm sure I can't when I'm young ; But when What happens when it's very big ？ So we will study random sequences Limit state of .

2. Definition

Definition ： set up Is a sequence of random variables , Is a random variable , If to any , Yes

It's called sequence Converges in probability to , Write it down as .

The meaning of probability convergence is ： Yes The probability that the absolute deviation of is not less than any given quantity will increase with Increase and decrease . Or say , Absolute deviation The probability of being less than any given quantity will increase with Increase and become closer to 1, That is to say (1) Equivalent to

Special When the distribution is deterministic , namely , It's called sequence Converges in probability to , namely

3. Theorem

3.1 Definition

Definition ： set up Is a sequence of two random variables , It's two constants . If

Then there are

3.2 prove

prove ： （1） because

therefore

namely

It can be obtained. . Similar verifiable

（2） In order to prove , We do it in a few steps ：

i) if , Then there are This is because for any Yes

ii) if , Then there are This is because in the when , Yes

And when when , The conclusion is clear .

iii) if , Then there are . This is because there are a series of conclusions ：,,,, namely iv) from iii) And （1） know ,, Thus there are

(3) In order to prove , Let's prove it first ：. This is because for any , Yes

This proves , And again combination , utilize （2） Immediate . From this theorem we can see , The limit of random variable sequence in the sense of probability （ That is, it converges to a constant in probability a） It still holds under four operations .

4. reference

[1] Mao Shisong , Cheng Yiming , Pu Xiaolong . Probability theory and mathematical statistics course （ The second edition ）[M]. Higher Education Press , 2019.