Resources
This space serves as a repository for reference materials, study notes, cheat sheets, and technical guides compiled throughout my studies and research.
Posts below are arranged chronologically—spanning probability theory, machine learning, software tools, and computational notes. You can also use the Archive to browse all articles organized by tags.
Selected notes have also been compiled into standalone PDF documents:
4.6.1. Uniform integrability and convergence in $L^1$
In section 4.4, we covered the condition where martingales converges in $L^p.$ We only covered the case where $p>1.$ In this section, the notions of uniform integrability is introduced to compensate convergence in $p=1$ case.
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4.4. Martingale inequalities and convergence in $L^p$
In this section we look into the condition that makes a martingale converges in $L^p$, $p>1$ in detail. We start by proving Doob’s inequality. By using this result we prove martingale inequalities which will then be used to prove Doob’s $L^p$ maximal inequality. $L^p$ convergence is direct from them. Lastly, as an extension of Doob’s inequality, I will briefly state a version of optional stopping.
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4.3. Application of martingales
For applications of martingales, I would like to cover the case of martingales with bounded increments and the branching process.
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4.2. Martingales and a.s. convergence
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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4.1. Conditional expectation
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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