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Definition: | Definition: | ||
| − | A random sequence or a discrete-time random process is a sequence of random variables <math>\mathbf{X}_{1},\cdots,\mathbf{X}_{n},\cdots</math> defined on a probability space <math>\left(\mathcal{S},\mathcal{F},P\right)</math> . | + | A random sequence or a discrete-time random process is a sequence of random variables <math class="inline">\mathbf{X}_{1},\cdots,\mathbf{X}_{n},\cdots</math> defined on a probability space <math class="inline">\left(\mathcal{S},\mathcal{F},P\right)</math> . |
Note | Note | ||
| − | • We often write this random sequence as <math>\left\{ \mathbf{X}_{n}\right\}</math> or <math>\left\{ \mathbf{X}_{n}\right\} _{n\geq1}</math> or <math>\left\{ \mathbf{X}_{n}\right\} _{n\in\mathbf{N}}</math> . | + | • We often write this random sequence as <math class="inline">\left\{ \mathbf{X}_{n}\right\}</math> or <math class="inline">\left\{ \mathbf{X}_{n}\right\} _{n\geq1}</math> or <math class="inline">\left\{ \mathbf{X}_{n}\right\} _{n\in\mathbf{N}}</math> . |
| − | • For any specific <math>\omega_{0}\in\mathcal{S}</math> , <math>\mathbf{X}_{1}\left(\omega_{0}\right),\cdots,\mathbf{X}_{n}\left(\omega_{0}\right),\cdots</math> is a sequence of real numbers. | + | • For any specific <math class="inline">\omega_{0}\in\mathcal{S}</math> , <math class="inline">\mathbf{X}_{1}\left(\omega_{0}\right),\cdots,\mathbf{X}_{n}\left(\omega_{0}\right),\cdots</math> is a sequence of real numbers. |
---- | ---- | ||
*[[ECE 600 Convergence|Convergence]] | *[[ECE 600 Convergence|Convergence]] | ||
Latest revision as of 11:35, 30 November 2010
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
