為了表述之精確嚴謹，許多數學使用公設化形式系統，將如何看待它的利弊得失呢？大概最大弊端就是拗口難讀的吧！若是逆向操作，以概念和定義為提要，自己補足直觀解釋及範例，未必不是一種學習法乎？！比方說一個公正硬幣之

$\Omega = \{ H , T \}$

$F = \{ \Phi , \{ H \} , \{ T \} , \{ H , T \} \}$

$P(H) = \frac{1}{2} \ , \ P(T) = \frac{1}{2}$

假使擲公正硬幣 N 次，每次丟出頭 H 則得一元，丟出尾 T 輸一元。如是形成多個隨機變數 $X_i$ ， $X_i(H) = 1 \ , \ X_i(T) = -1$ 。

那麼可能的輸贏

$Y_n = \Sigma \limits_{k=1}^{n} X_k$

構成一個隨機程序， $S = \{-n , -n + 1 , \cdots , 0 , \cdots , n - 1 , n \}$ 。

Stochastic process

A stochastic process is defined as a collection of random variables defined on a common probability space $(\Omega ,{\cal {F}},P)$ , where $\Omega$ is a sample space, ${\cal {F}}$ is a $\sigma$ –algebra, and $P$ is a probability measure, and the random variables, indexed by some set $T$ , all take values in the same mathematical space $S$ , which must be measurable with respect to some $\sigma$ -algebra $\Sigma$ .^[28]

In other words, for a given probability space $(\Omega ,{\cal {F}},P)$ and a measurable space $(S,\Sigma )$ , a stochastic process is a collection of $S$ -valued random variables, which can be written as:^[60]

\{X(t):t\in T\}.

Historically, in many problems from the natural sciences a point

t\in T

had the meaning of time, so

X(t)

is random variable representing a value observed at time

t

.^[113] A stochastic process can also be written as

\{X(t,\omega ):t\in T\}

to reflect that it is actually a function of two variables,

t\in T

and

\omega \in \Omega

.^[28]^[114]

There are others ways to consider a stochastic process, with the above definition being considered the traditional one.^[115]^[116] For example, a stochastic process can be interpreted or defined as a $S^{T}$ -valued random variable, where $S^{T}$ is the space of all the possible $S$ -valued functions of $t\in T$ that map from the set $T$ into the space $S$ .^[27]^[115]

Index set

The set $T$ is called the index set^[4]^[51] or parameter set^[28]^[117] of the stochastic process. Often this set is some subset of the real line, such as the natural numbers or an interval, giving the set $T$ the interpretation of time.^[1] In addition to these sets, the index set $T$ can be other linearly ordered sets or more general mathematical sets,^[1]^[54] such as the Cartesian plane $R^{2}$ or $n$ -dimensional Euclidean space, where an element $t\in T$ can represent a point in space.^[48]^[118] But in general more results and theorems are possible for stochastic processes when the index set is ordered.^[119]

State space

The mathematical space $S$ is called the state space of the stochastic process. This mathematical space can be the integers, the real line, $n$ -dimensional Euclidean space, the complex plane or other mathematical spaces, which reflects the different values that the stochastic process can take.^[1]^[5]^[28]^[51]^[56]

Sample function

A sample function is a single outcome of a stochastic process, so it is formed by taking a single possible value of each random variable of the stochastic process.^[28]^[120] More precisely, if $\{X(t,\omega ):t\in T\}$ is a stochastic process, then for any point $\omega \in \Omega$ , the mapping

X(\cdot ,\omega ):T\rightarrow S,

is called a sample function, a realization, or, particularly when

T

is interpreted as time, a sample path of the stochastic process

\{X(t,\omega ):t\in T\}

.^[50] This means that for a fixed

\omega \in \Omega

, there exists a sample function that maps the index set

T

to the state space

S

.^[28] Other names for a sample function of a stochastic process include trajectory, path function^[121] or path.^[122]

Random variable

A random variable $X\colon \Omega \to E$ is a measurable function from the set of possible outcomes $\Omega$ to some set $E$ . The technical axiomatic definition requires $\Omega$ to be a probability space and $E$ to be a measurable space (see Measure-theoretic definition).

Note that although $X$ is usually a real-valued function ( $E=\mathbb {R}$ ), it does not return a probability. The probabilities of different outcomes or sets of outcomes (events) are already given by the probability measure $P$ with which $\Omega$ is equipped. Rather, $X$ describes some numerical property that outcomes in $\Omega$ may have — e.g., the number of heads in a random collection of coin flips, or the height of a random person. The probability that $X$ takes value $\leq 3$ is the probability of the set of outcomes $\{\omega \in \Omega :X(\omega )\leq 3\}$ , denoted $P(X\leq 3).$

Probability space

In short, a probability space is a measure space such that the measure of the whole space is equal to one.

The expanded definition is the following: a probability space is a triple $(\Omega ,{\mathcal {F}},P)$ consisting of:

the sample space $\Omega$ — an arbitrary non-empty set,
the σ-algebra ${\mathcal {F}}\subseteq 2^{\Omega }$ (also called σ-field) — a set of subsets of $\Omega$ , called events, such that:
- ${\mathcal {F}}$ contains the sample space: $\Omega \in {\mathcal {F}}$ ,
- ${\mathcal {F}}$ is closed under complements: if $A\in\mathcal{F}$ , then also $(\Omega \setminus A)\in {\mathcal {F}}$ ,
- ${\mathcal {F}}$ is closed under countable unions: if $A_{i}\in {\mathcal {F}}$ for $i=1,2,\dots$ , then also $\textstyle (\bigcup _{i=1}^{\infty }A_{i})\in {\mathcal {F}}$
  - The corollary from the previous two properties and De Morgan’s law is that ${\mathcal {F}}$ is also closed under countable intersections: if $A_{i}\in {\mathcal {F}}$ for $i=1,2,\dots$ , then also $\textstyle (\bigcap _{i=1}^{\infty }A_{i})\in {\mathcal {F}}$
the probability measure $P:{\mathcal {F}}\to [0,1]$ — a function on ${\mathcal {F}}$ such that:
- P is countably additive: if $\{A_{i}\}_{i=1}^{\infty }\subseteq {\mathcal {F}}$ is a countable collection of pairwise disjoint sets, then $\textstyle P(\bigcup _{i=1}^{\infty }A_{i})=\sum _{i=1}^{\infty }P(A_{i})$ ,
- the measure of entire sample space is equal to one: $P(\Omega )=1$ .

Measure (mathematics)

Let $X$ be a set and $Σ$ a $σ$ -algebra over $X$ . A function $μ$ from $Σ$ to the extended real number line is called a measure if it satisfies the following properties:

Non-negativity: For all $E$ in $Σ$ : $μ (E) \geq 0$ .
Null empty set: $μ (\emptyset) = 0$ .
Countable additivity (or $σ$ -additivity): For all countable collections $\{E_{i}\}_{i=1}^{\infty }$ of pairwise disjoint sets in $Σ$ :

\mu \left(\bigcup _{k=1}^{\infty }E_{k}\right)=\sum _{k=1}^{\infty }\mu (E_{k})

One may require that at least one set $E$ has finite measure. Then the empty set automatically has measure zero because of countable additivity, because

\mu (E)=\mu (E\cup \varnothing \cup \varnothing \cup \dots )=\mu (E)+\mu (\varnothing )+\mu (\varnothing )+\dots ,

which implies (since the sum on the right thus converges to a finite value) that $\mu (\varnothing )=0$ .

If only the second and third conditions of the definition of measure above are met, and $μ$ takes on at most one of the values $\pm\infty$ , then $μ$ is called a signed measure.

The pair $(X, Σ)$ is called a measurable space, the members of $Σ$ are called measurable sets. If $\left(X,\Sigma _{X}\right)$ and $\left(Y,\Sigma _{Y}\right)$ are two measurable spaces, then a function $f:X\to Y$ is called measurable if for every $Y$ -measurable set $B\in \Sigma _{Y}$ , the inverse image is $X$ -measurable – i.e.: $f^{(-1)}(B)\in \Sigma _{X}$ . The composition of measurable functions is measurable, making the measurable spaces and measurable functions a category, with the measurable spaces as objects and the set of measurable functions as arrows.

A triple $(X, Σ, μ)$ is called a measure space. A probability measure is a measure with total measure one – i.e. $μ (X) = 1$ . A probability space is a measure space with a probability measure.

For measure spaces that are also topological spaces various compatibility conditions can be placed for the measure and the topology. Most measures met in practice in analysis (and in many cases also in probability theory) are Radon measures. Radon measures have an alternative definition in terms of linear functionals on the locally convex space of continuous functions with compact support. This approach is taken by Bourbaki (2004) and a number of other sources. For more details, see the article on Radon measures.

Sigma-algebra

Let X be some set, and let 2^X represent its power set. Then a subset Σ ⊂ 2^X is called a σ-algebra if it satisfies the following three properties:^[3]

X is in Σ, and X is considered to be the universal set in the following context.
Σ is closed under complementation: If A is in Σ, then so is its complement, $X\A$ .
Σ is closed under countable unions: If A₁, A₂, A₃, … are in Σ, then so is A = A₁ ∪ A₂ ∪ A₃ ∪ … .

From these properties, it follows that the σ-algebra is also closed under countable intersections (by applying De Morgan’s laws).

It also follows that the empty set ∅ is in Σ, since by (1) X is in Σ and (2) asserts that its complement, the empty set, is also in Σ. Moreover, since ${X, \emptyset$ } satisfies condition (3) as well, it follows that ${X, \emptyset$ } is the smallest possible σ-algebra on X. The largest possible σ-algebra on X is 2^X.

Elements of the σ-algebra are called measurable sets. An ordered pair $(X, Σ)$ , where X is a set and Σ is a σ-algebra over X, is called a measurable space. A function between two measurable spaces is called a measurable function if the preimage of every measurable set is measurable. The collection of measurable spaces forms a category, with the measurable functions as morphisms. Measures are defined as certain types of functions from a σ-algebra to [0, ∞].

A σ-algebra is both a π-system and a Dynkin system (λ-system). The converse is true as well, by Dynkin’s theorem (below).

如是當閱讀少用術語的書時也不會覺得怪怪的耶！？

Time Series Analysis and Its Applications: With R Examples

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