icon: 📑

📑
Note for Probability Theory

Main References

Richard Durrett. (1996). "Probability Theory and Examples", second edition, Duxbury Press, Washington.
Kim, Seung Hyun. (2024). "Measure Theory for Applications to Probability and Economics", Work in progress, https://seunghyun-97.github.io/
Alessandro Rinaldo. (2018 sprig). Lecture Notes on 36-752 Advanced Probability Overview. Carnegie Mellon University.

Probability Theory

Theorem 32 (measurability of function).

Suppose a function , for a measurable space . Let be a collection of subsets in such that . Then is -measurable if and only if for all .

Theorem 7 (properties of distribution function).

Let be any distribution function. Then, we have the following properties:

Theorem 9 (sufficient condition for distribution function).

If a function satisfies the following conditions,

then is a distribution of some random variable.

Theorem 19 (properties of expectation).

Let be random variables on , and suppose or . Then we have

Theorem 28 (change of the variable formula).

Let be a random element and be a measurable function. Denote as a distribution of on , i.e. . If or (), then

Theorem 40 (independence of generated sigma-field).

Suppose are independent and each is a system. Then are independent.

Corollary 41 (sufficient condition for independence).

The sufficient condition for the independence of random variables is

Theorem 45 (distribution of random vector).

Suppose independent random variables for with distribution distribution (i.e. , ). Then the random vector has distribution , i.e. , .

Theorem 49 (convolution).

Let and be independent random variables and their distributions be . Then the convolution of and is the distribution of and derived as

Remark 27 (common form of Chebyshev).

Proposition 2 (lim and aslim implies plim).

Lemma 22 (second condition for truncation convergence).

For a random variable , if as , then we have as .

Lemma 24 (weak law of large numbers).

Let be i.i.d. random variables with as for any . Now put and . Then