Intro to Probability - YAQ Week 8

Question

What is the meaning of the notations $o()$ (little o) and $O()$ (big o)?

The symbols are used to indicate order of magnitudes of quantities. Take for instance a sequence $(x_n)_{n\in \mathbb{N}}$ of non-negative real numbers. Then, for $k\in \mathbb{Z}$, $$ x_n=O(n^k) \text{ as }n\to\infty $$ means that there is a constant $C$ independent of $n\in \mathbb{N}$ such that $$ x_n\leq C n^k\text{ for }n \text{ large.} $$ Fromally $$ x_n=O(n^k) \text{ as }n\to\infty\text{ iff there is} N\in \mathbb{N} \text{ s.t. } x_n\leq C n^k\text{ for }n\geq N. $$ This intuitively means that $x_n$ grows (decays if $k<0$) at most (least if $k<0$) as fast as $n^k$ does when $n\to\infty$. In a similar fashion one says that $$ x_n=o(n^k) \text{ as }n\to\infty\text{ iff } \lim_{n\to\infty}\frac{x_n}{n^k}=0, $$ which means that $x_n$ grows slower (decays faster if $k<0$) than $n^k$.

Similarly, given a function $f:\mathbb{R}\to\mathbb{R}$ and $k\in \mathbb{N}$, we can define \begin{align*} f=O(h^k)\text{ as }h\to 0+&\text{ iff there is} C>0, h_0>0\text{ s.t. }|f(h)|\leq C\, h^k \text{ for }0\leq h\leq h_0\text{, and} \\ f=o(h^k)\text{ as }h\to 0+&\text{ iff }\lim_{h\to 0+}\frac{f(h)}{h^k}=0, \end{align*} with a similar interpretation as above. The following are examples $$ n\log(n)=O(n^2)\text{ as }n\to\infty,\text{ }\sin(n)=O(1)=O(n^0)\text{ as }n\to\infty, \text{ } log(n)=o(n)\text{ as }n\to\infty,\text{ }h^{1.5}=o(h)\text{ as }h\to 0+ $$

Question

Why is the cumulative distribution function of a random variable not left-continuous in general?

It is enough to produce a counterexample. Let $S=\{ H,T\}$ be the sample space corresponding to a fair coin toss and let the random variable $X$ be given by $$ X(\omega)=\begin{cases} 0,& \omega=T,\\ 1,&\omega=H.\end{cases} $$ Using the definition $F_X(x)=P(X\leq x)$ for $x\in \mathbb{R}$, it is easily seen that $$ F_X(x)=\begin{cases} 0,& x<0,\\ 0.5,&0\leq x<1,\\ 1,&x\geq 1,\end{cases}, $$ which is clearly not left-continuous at $x=0$ and $x=1$.