### 2.5. Convergence of random series

As the last section in chapter 2, we cover convergence of random series. Especially, since I already explained what tail $\sigma$-fields and tail events are, our focus will be on *Kolmogorov’s maximal inequality* and the *three series theorem*.

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### 2.4. Strong law of large numbers

Putting together all the topics we have covered so far, we now move on to one of the most impactful theorem in probability theory: the strong law of large numbers (SLLN).

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### 2.3. Borel-Cantelli lemmas

In this section I would like to cover the *Borel-Cantelli lemmas*, or B-C lemmas for short. Borel-Cantelli lemmas are quintessential tools for analysis of *tail events* and deriving almost sure convergence from $P$-convergence.

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### 2.2. Weak laws of large numbers

We say a random variable $X_n$ **converges in probability** (or $P$-converges) to another random variable $X$ and write $X_n \overset{P}{\to} X$ if $\lim_n P(|X_n - X| > \epsilon) = 0$ for all $\epsilon >0.$ We can also define convergence in probability to a constant by letting $X = c\in\mathbb{R}$. It is easy yet useful to know that $X_n \overset{L^p}{\to} X$ with $p > 0$ implies $X_n \overset{P}{\to} X$ by Markov-Chebyshev inequality with $\varphi(x) = |x|^p$.^{1}

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### 2.1. Independence

First chapter was about the essential of measure theory. We especially focused on important result for finite measure or at least $\sigma$-finite measures. We defined a probability space as a measure space and a random variable as a measurable function in it.

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