4.2. Martingales and a.s. convergence

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Remaining sections in chapter 4 is about martingales and convergence of it. Regarding martingales, our first topic will be convergence in almost sure sense. Next we will look into convergence in $L^p,$ with $p>1$ and $p=1$ separately. In the meantime the theory of optional stopping will be covered.

4.1. Conditional expectation

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In this chapter we study convergence of a sequence of random variables with dependency. To be specific, I will cover theory of martingales. The first subsection is about conditional expectation which is essential for defining martingales.

3.9. Infinitely divisible distributions

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A certain kind of well behaving distributions has characteristic functions that can be represented in canonical form. In this section we cover conditions that such distributions have and its canonical representation.

3.10. Limit theorems in ℝᵈ

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This part covers limit theorems regarding random vectors $\mathbf{X} = (X_1,\cdots,X_d)’\in\mathbb{R}^d.$

3.6. Poisson convergence

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I would like to finish reviewing Probability theory I by briefly mentioning the Poisson convergence (section 3.6) and limit theorems in $\mathbb{R}^d$ (3.10).