Sequences of Random Variables
From the course notes of Sangchun Han, ECE PhD student.
Definition:
A random sequence or a discrete-time random process is a sequence of random variables $ \mathbf{X}_{1},\cdots,\mathbf{X}_{n},\cdots $ defined on a probability space $ \left(\mathcal{S},\mathcal{F},P\right) $ .
Note
• We often write this random sequence as $ \left\{ \mathbf{X}_{n}\right\} $ or $ \left\{ \mathbf{X}_{n}\right\} _{n\geq1} $ or $ \left\{ \mathbf{X}_{n}\right\} _{n\in\mathbf{N}} $ .
• For any specific $ \omega_{0}\in\mathcal{S} $ , $ \mathbf{X}_{1}\left(\omega_{0}\right),\cdots,\mathbf{X}_{n}\left(\omega_{0}\right),\cdots $ is a sequence of real numbers.
- Convergence
- Chebyshev Inequality
- Weak law of large numbers
- Strong law of large numbers (Borel)
- Central Limit Theorem
- Random Sum
