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