[按：我有写邮件笔记的习惯，之前注意到有关博客文章*，今略关切。]

Recently, a bloger has pointed out in a few posts about certain basic problem in the textbooks of stochastic process.

---- It is argued that the function of time X(t) is replaced by random variables as a logic mistake.

---- The starting point is a continues function X(t).

---- By writing X(t) as ΔX1(t) + ΔX2(t) ... + ΔXn(t), it is assumed { ΔXi(t) } to be i.i.d.

---- So, X(t) obeys the normal distribution as a ramdom variable, according to some well-known theorem.

(I checked on Wikipedia that X(t) is just an "abuse of function notion", link omitted.)

I think it is no need to define the stochastic process as above. I would not comment further as I skipped the class totally... 

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Reference: "一图看懂《随机过程》如何偷换概念".

(chatGPT-3.5 的回应)

The blogger's observation raises a critical point about certain fundamental issues in the treatment of stochastic processes within textbooks. They argue that replacing the function of time X(t) with random variables constitutes a logical mistake. The traditional starting point involves a continuous function X(t). By expressing X(t) as ΔX1(t) + ΔX2(t) ... + ΔXn(t), it is assumed that { ΔXi(t) } are independent and identically distributed (i.i.d.). Consequently, this assumption leads to the conclusion that X(t) follows a normal distribution as a random variable, based on certain well-known theorems. This criticism points out potential flaws in how stochastic processes are presented and reasoned about in educational materials, shedding light on key considerations when working with such frameworks.

show me the definition of stochastic process X(t)

The stochastic process X(t) can be defined as a collection of random variables, where t represents time. This process is characterized by the fact that for any sequence of time instants t1 < t2 < ... < tn, the values X(t1), X(t2), ..., X(tn) are random variables. Stochastic processes are used to model systems or phenomena that involve randomness and evolve over time. The behavior of the process at any given time is not entirely predictable, making it a key tool in fields such as probability theory, statistics, finance, and various branches of science and engineering.