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:
Setting up M1 Mac for both TensorFlow and PyTorch
Macs with ARM64-based M1 chip, launched shortly after Apple’s initial announcement of their plan to migrate to Apple Silicon, got quite a lot of attention both from consumers and developers. It became headlines especially because of its outstanding performance, not in the ARM64-territory, but in all PC industry.
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4.8. Optional stopping theorem
In this section, we generalize the bounded version of optional stopping. After that as an example we will cover theorem regarding assymetric random walk.
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4.5. Square integrable martingales
In this section, we look into martingales with special property - square integrability. Square integrability gives martingale an upper bound for maximal expectation so that it can further be used to determine the convergence of the sequence.
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4.6.3. Riez's decomposition
We know that any submartingales can be decomposed into a martingale and a predictable sequence (Doob’s decomposition). Riez’s decomposition allows us to do the similar to uniformly integrable non-negative supermartingales.
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4.6.2. Levy's theorem
Consider a sequence of conditional expectations $E(X|\mathcal{F}_n)$ with fixed $X.$ By using the theorem from previous subsection we can determine convergence of this sequence as well.
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