metric spaces

In an arbitrary set, we are able to set some relations between the elements.

The order relationship in real number system is one of them.

[31] For each pair of real numbers $a$ and $b$, exactly one of the following is true:
       $a=b$,  $a<b$,  or  $b<a$
[32] If $a<b$ and $b<c$, then $a<c$
[33] If $a<b$, then $a+c<b+c$
[34] If $a<b$, then $ac<bc$, whenever $0<c$

Now we will introduce a new relation. For any elements $a, b$ in an arbitrary set $A$, there is a function $d(x,y)\in \mathbb{R}$ which satisfies following conditions.

(1)$d(a,b)\geq 0, \quad (d(a,b)=0 \quad \mbox{if and only if}  a=b)$
(2)$d(a,b)=d(b,a)$
(3)$d(a,b)+d(b,c)\geq d(a,c), \quad (\mbox{for any}   c\in A)$

Then, we call the set $A$ a metric space. (3) is the triangle inequality. The function $d$ is namely a distance function. The metric space is a set which has a quantifiable distance between elements.

For example, $d(a,b)=|b-a|, (a,b\in\mathbb{R})$ is a distance function. Therefore, $\mathbb{R}$ becomes a metric space. Please note that there are various kinds of sets, and every set may adopt multiple distance functions to determine the corresponding distance.

coffee break 5

I read "Winter Frost".

In the past, I have read "Frost at Christmas",
"Night Frost",
"A touch of Frost",
"Hard Frost",
 and short story "Early Morning Frost".
All works is very funny and interesting. This is what workaholic is.

It is a very sad fact that R. D. Wingfield has gone.
We can not be lost in the world of Frost and merry companions. 
Only "Killing Frost" is left for me.

weak law of large numbers

Let random variables$X_1,X_2,\cdots,X_n$ be $i.i.d.$ and any $E(X_i)=\mu<\infty$. If we put
\[ Y_n=\frac{1}{n}\sum_{i=1}^nX_i \]
then,
\[ P(\lim_{n\rightarrow\infty}Y_n=\mu)=1 \]
It is called strong law of large numbers.

On the other hand, weak law of large numbers has a weaker assertion in which $Y_n$ approches $\mu$ than strong law of large numbers. It says, for any $\epsilon>0$,
\[ P(\lim_{n\rightarrow\infty}|Y_n-\mu|>\epsilon )=0\quad  \]
Weak law of large numbers does not say that $Y_n$ approaches $\mu$ with probability 1.

On mathematical terms, strong law of large numbers is a almost everywhere convergence. Weak law of large numbers is a convergence in probability. Therefore, in the random variables, if strong law of large numbers is true, then weak law of large numbers is true. However the converse is not true.

epsilon-delta proofs 7

Let random variables$X_1,X_2,\cdots,X_n$ be i.i.d. and any $E(X_i)=\mu<\infty$. If we put
\[ Y_n=\frac{1}{n}\sum_{i=1}^nX_i \]
then law of large numbers says,
\[ P(\lim_{n\rightarrow\infty}Y_n=\mu)=1 \]
It means, for any $\epsilon>0$ there is a $N$ such that if $n>N$, $|Y_n-\mu|\leq \epsilon$ and it's probability becomes 1.

In the probability theory, as we are used to write elements $w$ of the sample space explicitly,
\[ P (w |\lim_{n\rightarrow\infty}Y_n(w)=\mu )=1. \]
Therefore, when $1/j,(j\in \mathbb{N})$  is used in place of $\epsilon$, if a set $A$ is defined by
\[ A=\left\{w |\forall j, \exists N, n\geq N, |Y_n-\mu|<1/j \right\} \]
, then $P(A)=1$  .  Furthermore, using symbols of the set theory, if a set $A$ is
\[ A=\bigcap_{j=1}^{\infty}\bigcup_{N=1}^{\infty}\bigcap_{n=N}^{\infty}  \left\{w  |  |Y_n-\mu|<1/j \right\} \]
, then $P(A)=1$  .

A epsilon-delta technique can be written in the form like this.

(strong) law of large numbers

Once more we will begin with the epsilon-delta technique.

In the probability theory there is a theorem called "(strong) law of large numbers". It is one of most famous theorems in the probability or statistics.

Random variables $X_1,X_2,\cdots ,X_n$ are independent identically distributed(i. i. d.). $E(X_i)=\mu<\infty$. Then,
\[ Y_n=\frac{1}{n}\sum_{i=1}^nX_i\rightarrow\mu\quad (n\rightarrow\infty)\quad a.s. \]
In other words,
\[  P\left\{ \lim_{n\rightarrow\infty}Y_n=\mu \right\}=1 \]

Law of large numbers means the average of sample data will converge to the expectation value of each random variable (which all will become same),  when sample data increase infinitely.
In general, law of large numbers indicates the strong law.

In next post, I will explain how applied the epsilon-delta technique is.  

During a long intermission

From June to September, I worked for a special lecture of the probability theory for finance. The themes of the lecture are as follow.

1. Introduction of the probability theory

2. Lebesgue integral theory

3. Normal distribution and Brownian motion

4. Conditional expectation and Martingales

5. Ito Calculus

The work was much overburdened for me. At 28/Sep it was completed.
So now I am easy, I am able to restart my blog. Hereafter, in this blog, I will induct the contents of the lecture in a different form.

Around the same time I read books, saw movies, and play games.
However, now I can not remember all of them. I will present them at every
opprtunity, too.