### 2.4. Strong law of large numbers

probability Durrett

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).

### 2.3. Borel-Cantelli lemmas

probability Durrett

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.

### 2.2. Weak laws of large numbers

probability Durrett

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
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.