Convergence of indicator functions for L1 r.v.

Thread starter jk_zhengli
Start date Sep 28, 2012
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Convergence Functions

In summary, the convergence of indicator functions for L1 r.v. refers to the idea that as the number of observations or trials increases, the value of the indicator function for a random variable approaches a certain limit or value. It is a type of weak convergence, different from other types of convergence such as almost sure convergence or convergence in distribution. The study of convergence of indicator functions for L1 r.v. is important in understanding the behavior and properties of random variables and has various real-world applications. For convergence to occur, the sequence of random variables must be independent and identically distributed, and must also be bounded in L1 norm.

Sep 28, 2012

jk_zhengli

Hi all,

I wonder if the following are equivalent.

1) E(|X|) < infinity

2) I(|X| > n) goes to 0 as n goes to infinity (I is the indicator function)

3) P(|X| > n) goes to 0 as n goes to infinity.

1) => 2) and 2) => 3) are easy to see, please help me to show 2) => 1) and 3) => 2) if they are indeed true. Thanks.

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Sep 28, 2012

micromass

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Does this hold for the Cauchy distribution? http://en.wikipedia.org/wiki/Cauchy_distribution

Related to Convergence of indicator functions for L1 r.v.

1. What is the definition of convergence of indicator functions for L1 r.v.?

The convergence of indicator functions for L1 r.v. refers to the idea that as the number of observations or trials increases, the value of the indicator function for a random variable approaches a certain limit or value. This is also known as convergence in probability.

2. How is the convergence of indicator functions for L1 r.v. different from other types of convergence?

The convergence of indicator functions for L1 r.v. is a type of weak convergence, meaning it does not require the random variable to converge to a specific value, but rather to converge in probability. This is different from other types of convergence, such as almost sure convergence or convergence in distribution.

3. What is the importance of studying convergence of indicator functions for L1 r.v.?

The study of convergence of indicator functions for L1 r.v. is important in understanding the behavior and properties of random variables. It allows us to make predictions about the behavior of a large number of observations or trials, and is crucial in many areas of statistics and probability theory.

4. What are the main conditions for convergence of indicator functions for L1 r.v. to occur?

In order for convergence of indicator functions for L1 r.v. to occur, there are two main conditions that must be satisfied. First, the sequence of random variables must be independent and identically distributed. Secondly, the sequence must also be bounded in L1 norm, meaning the expected value of the random variable is finite.

5. Are there any real-world applications of convergence of indicator functions for L1 r.v.?

Yes, there are many real-world applications of convergence of indicator functions for L1 r.v. For example, in finance and economics, it is often used to model the behavior of stock prices and other financial variables. It is also used in various fields of science, such as biology and physics, to analyze experimental data and make predictions about future outcomes.