引用本书 · Citation
请引用 Haobai Wang，并注明三十一章公开阅读版及内容指纹；英文仍为完整翻译工作稿。
正式网页版本
Haobai Wang. 偶然的数学：概率论的历史、思想与数学结构. 2026. 正式网页第7版，前9章。 正式阅读网址.
@misc{wang2026chance,
author = {Wang, Haobai},
title = {{偶然的数学：概率论的历史、思想与数学结构}},
year = {2026},
url = {https://chance-mathematics-six-chapters.goldenmath7.chatgpt.site/},
note = {thirtyone-chapters-11e3e21; academic final review not implied; English is a working translation}
}引用当前公开阅读版
本页对应已获作者授权的三十一章公开阅读版；原稿快照与逐章指纹见版本页。
@misc{wang2026chance,
author = {Wang, Haobai},
title = {{偶然的数学：概率论的历史、思想与数学结构}},
year = {2026},
url = {https://chance-mathematics-six-chapters.goldenmath7.chatgpt.site/},
note = {thirtyone-chapters-11e3e21; academic final review not implied; English is a working translation}
}引用本章工作稿
第一章：一场没有结束的赌博
@misc{wang2026chance_ch1,
author = {Wang, Haobai},
title = {{第一章：一场没有结束的赌博（续）}},
year = {2026},
url = {https://chance-mathematics-six-chapters.goldenmath7.chatgpt.site/chapters/01-gambling.html},
note = {thirtyone-chapters-11e3e21; academic final review not implied; English is a working translation; SHA-256 13aef92ae4724b4b4c8b5aeea1462d17e345c72b8cb5c761a89edae63788243e}
}第二章：从机会到知识——我们如何认识未知的概率？
@misc{wang2026chance_ch2,
author = {Wang, Haobai},
title = {{第二章：从机会到知识——我们如何认识未知的概率？}},
year = {2026},
url = {https://chance-mathematics-six-chapters.goldenmath7.chatgpt.site/chapters/02-inference.html},
note = {thirtyone-chapters-11e3e21; academic final review not implied; English is a working translation; SHA-256 67f96226c5fc864eef686e9994db2d039c424444f16e66d240dee6017501596a}
}第三章：钟形曲线的诞生——为什么无数随机现象走向同一种分布？
@misc{wang2026chance_ch3,
author = {Wang, Haobai},
title = {{第三章：钟形曲线的诞生——为什么无数随机现象走向同一种分布？}},
year = {2026},
url = {https://chance-mathematics-six-chapters.goldenmath7.chatgpt.site/chapters/03-normal.html},
note = {thirtyone-chapters-11e3e21; academic final review not implied; English is a working translation; SHA-256 3ce001a8524b13a34da67c32bc07592a95d16377aff7f4ed3f78d5aa08773f5b}
}第四章：当随机性进入时间——从随机游走到 Brownian motion
@misc{wang2026chance_ch4,
author = {Wang, Haobai},
title = {{第四章：当随机性进入时间——从随机游走到 Brownian motion}},
year = {2026},
url = {https://chance-mathematics-six-chapters.goldenmath7.chatgpt.site/chapters/04-brownian.html},
note = {thirtyone-chapters-11e3e21; academic final review not implied; English is a working translation; SHA-256 07af2fb31042890a84e4f1fd33b39e4b2bc4073b6e8e49dd66dacc316d49309e}
}第五章：无限世界的代价——为什么概率需要测度论？
@misc{wang2026chance_ch5,
author = {Wang, Haobai},
title = {{第五章：无限世界的代价——为什么概率需要测度论？}},
year = {2026},
url = {https://chance-mathematics-six-chapters.goldenmath7.chatgpt.site/chapters/05-measure.html},
note = {thirtyone-chapters-11e3e21; academic final review not implied; English is a working translation; SHA-256 96ce49ea02f739700393d1febed32c353be47bcedc2f0cb0fb4f149020917782}
}第六章：平均值的重生——从加权求和到 Lebesgue 积分
@misc{wang2026chance_ch6,
author = {Wang, Haobai},
title = {{第六章：平均值的重生——从加权求和到 Lebesgue 积分}},
year = {2026},
url = {https://chance-mathematics-six-chapters.goldenmath7.chatgpt.site/chapters/06-integral.html},
note = {thirtyone-chapters-11e3e21; academic final review not implied; English is a working translation; SHA-256 2663ea1806fc26d134cb82bf112dad89a38d7d5d4fb3e9ad087fae745a15dc33}
}第七章：收敛究竟意味着什么？——随机世界中的四种极限
@misc{wang2026chance_ch7,
author = {Wang, Haobai},
title = {{第七章：收敛究竟意味着什么？——随机世界中的四种极限}},
year = {2026},
url = {https://chance-mathematics-six-chapters.goldenmath7.chatgpt.site/chapters/07-convergence.html},
note = {thirtyone-chapters-11e3e21; academic final review not implied; English is a working translation; SHA-256 56bd85f8860ce7b69c33031df04f03567e20311e968ec43de4da900908b5e664}
}第八章：如何构造无限随机实验？
@misc{wang2026chance_ch8,
author = {Wang, Haobai},
title = {{第八章：如何构造无限随机实验？}},
year = {2026},
url = {https://chance-mathematics-six-chapters.goldenmath7.chatgpt.site/chapters/08-infinite-experiments.html},
note = {thirtyone-chapters-11e3e21; academic final review not implied; English is a working translation; SHA-256 a01d3147929e96aaff254ac32092fcb81d6b022f5d08657c4d0f9d0dad86a536}
}第九章：看见一部分世界之后——条件期望与信息
@misc{wang2026chance_ch9,
author = {Wang, Haobai},
title = {{第九章：看见一部分世界之后——条件期望与信息}},
year = {2026},
url = {https://chance-mathematics-six-chapters.goldenmath7.chatgpt.site/chapters/09-conditional-expectation.html},
note = {thirtyone-chapters-11e3e21; academic final review not implied; English is a working translation; SHA-256 23a3efff0d5b607b2fd9807e6654f0a46eb1e707be4f7f2a92fde08db33c9278}
}第十章：一份概率怎样分解成另一份概率？——密度、条件分布与概率核
@misc{wang2026chance_ch10,
author = {Wang, Haobai},
title = {{第十章：一份概率怎样分解成另一份概率？——密度、条件分布与概率核}},
year = {2026},
url = {https://chance-mathematics-six-chapters.goldenmath7.chatgpt.site/chapters/10-densities-and-kernels.html},
note = {thirtyone-chapters-11e3e21; academic final review not implied; English is a working translation; SHA-256 d7936210e53916250665e658e25f35a74a875accabc7594e54665481f660deb3}
}第十一章：分布怎样被函数记录？——特征函数、独立和与 Gaussian 极限
@misc{wang2026chance_ch11,
author = {Wang, Haobai},
title = {{第十一章：分布怎样被函数记录？——特征函数、独立和与 Gaussian 极限}},
year = {2026},
url = {https://chance-mathematics-six-chapters.goldenmath7.chatgpt.site/chapters/11-characteristic-functions.html},
note = {thirtyone-chapters-11e3e21; academic final review not implied; English is a working translation; SHA-256 b2b099fd191928a19f8ec7a9b2743b5e1a1f236e20d4f0a65258851ce0029782}
}第十二章：无穷多次相加，何时仍有答案？
@misc{wang2026chance_ch12,
author = {Wang, Haobai},
title = {{第十二章：无穷多次相加，何时仍有答案？}},
year = {2026},
url = {https://chance-mathematics-six-chapters.goldenmath7.chatgpt.site/chapters/12-infinite-sums.html},
note = {thirtyone-chapters-11e3e21; academic final review not implied; English is a working translation; SHA-256 08a787e5184f58c7ff205099719c6ee18effc835cd212560ccc416c22ce0d62e}
}第十三章：罕见事件为什么会产生新的稳定规律？
@misc{wang2026chance_ch13,
author = {Wang, Haobai},
title = {{第十三章：罕见事件为什么会产生新的稳定规律？}},
year = {2026},
url = {https://chance-mathematics-six-chapters.goldenmath7.chatgpt.site/chapters/13-rare-events.html},
note = {thirtyone-chapters-11e3e21; academic final review not implied; English is a working translation; SHA-256 8c6c8bb2598cf86b83678cd30c1978d300add82f77aaca43c252a0059531be0c}
}第十四章：不是所有极限都是 Gaussian——三角阵列、稳定分布与无限可分性
@misc{wang2026chance_ch14,
author = {Wang, Haobai},
title = {{第十四章：不是所有极限都是 Gaussian——三角阵列、稳定分布与无限可分性}},
year = {2026},
url = {https://chance-mathematics-six-chapters.goldenmath7.chatgpt.site/chapters/14-arrays-and-stability.html},
note = {thirtyone-chapters-11e3e21; academic final review not implied; English is a working translation; SHA-256 e822e84341ed8fb6e8811e606114197eda4a7ebf6162be22ebb12e7f32311eca}
}第十五章：一场永远公平的游戏，最终会怎样？
@misc{wang2026chance_ch15,
author = {Wang, Haobai},
title = {{第十五章：一场永远公平的游戏，最终会怎样？}},
year = {2026},
url = {https://chance-mathematics-six-chapters.goldenmath7.chatgpt.site/chapters/15-martingales.html},
note = {thirtyone-chapters-11e3e21; academic final review not implied; English is a working translation; SHA-256 b34eeca7a839a3921898725bb7431f22b37896bd9562860bc733eed69d185f4f}
}第十六章：如果玩家自己决定何时停止？
@misc{wang2026chance_ch16,
author = {Wang, Haobai},
title = {{第十六章：如果玩家自己决定何时停止？}},
year = {2026},
url = {https://chance-mathematics-six-chapters.goldenmath7.chatgpt.site/chapters/16-stopping-times.html},
note = {thirtyone-chapters-11e3e21; academic final review not implied; English is a working translation; SHA-256 6edf50d9952c9e3a41dd4f39de491aeedb54ad3a66c5cd2bcf56fc33bb63dca3}
}第十七章：只记住现在，就足以决定未来吗？
@misc{wang2026chance_ch17,
author = {Wang, Haobai},
title = {{第十七章：只记住现在，就足以决定未来吗？}},
year = {2026},
url = {https://chance-mathematics-six-chapters.goldenmath7.chatgpt.site/chapters/17-markov-chains.html},
note = {thirtyone-chapters-11e3e21; academic final review not implied; English is a working translation; SHA-256 0669eafadd5864319dbd782e0d5fb97825998244c7c280f450c1045e8f671ee3}
}第十八章：随机游走为什么有时回来，有时消失？
@misc{wang2026chance_ch18,
author = {Wang, Haobai},
title = {{第十八章：随机游走为什么有时回来，有时消失？}},
year = {2026},
url = {https://chance-mathematics-six-chapters.goldenmath7.chatgpt.site/chapters/18-return-and-escape.html},
note = {thirtyone-chapters-11e3e21; academic final review not implied; English is a working translation; SHA-256 c3c7eee165bc3391a02cdcfa0359f7f25e08c64e6059d4e7576dbfd6175c2f7f}
}第十九章：等待下一次开始
@misc{wang2026chance_ch19,
author = {Wang, Haobai},
title = {{第十九章：等待下一次开始}},
year = {2026},
url = {https://chance-mathematics-six-chapters.goldenmath7.chatgpt.site/chapters/19-renewal.html},
note = {thirtyone-chapters-11e3e21; academic final review not implied; English is a working translation; SHA-256 e2afde8965d0d43b9b610fe2a8eb65f9106a28f6398dbe0a8b9ae6a8b1e03bf2}
}第二十章：一个祖先怎样变成一个随机家族？
@misc{wang2026chance_ch20,
author = {Wang, Haobai},
title = {{第二十章：一个祖先怎样变成一个随机家族？}},
year = {2026},
url = {https://chance-mathematics-six-chapters.goldenmath7.chatgpt.site/chapters/20-branching.html},
note = {thirtyone-chapters-11e3e21; academic final review not implied; English is a working translation; SHA-256 f947d405d92d9d88f3e41bda551600033a0c470dc890f86880582321283529a7}
}第二十一章：没有记忆的等待时间
@misc{wang2026chance_ch21,
author = {Wang, Haobai},
title = {{第二十一章：没有记忆的等待时间}},
year = {2026},
url = {https://chance-mathematics-six-chapters.goldenmath7.chatgpt.site/chapters/21-poisson-process.html},
note = {thirtyone-chapters-11e3e21; academic final review not implied; English is a working translation; SHA-256 2bae45e59bbb09776c4409da45f682dda70ffe32ad2cd1e88bca25eefe1d57ab}
}第二十二章：随机过程怎样在瞬间发生跳跃？——Continuous-time Markov chains
@misc{wang2026chance_ch22,
author = {Wang, Haobai},
title = {{第二十二章：随机过程怎样在瞬间发生跳跃？——Continuous-time Markov chains}},
year = {2026},
url = {https://chance-mathematics-six-chapters.goldenmath7.chatgpt.site/chapters/22-continuous-time-chains.html},
note = {thirtyone-chapters-11e3e21; academic final review not implied; English is a working translation; SHA-256 a7d772615df579c70fb36d5a5273fe9679c586cf9e7516db188e35ca7173dab6}
}第二十三章：Gaussian 快照怎样真正拼成连续路径？——连续修改、Gaussian processes、Wiener measure
@misc{wang2026chance_ch23,
author = {Wang, Haobai},
title = {{第二十三章：Gaussian 快照怎样真正拼成连续路径？——连续修改、Gaussian processes、Wiener measure}},
year = {2026},
url = {https://chance-mathematics-six-chapters.goldenmath7.chatgpt.site/chapters/23-gaussian-paths.html},
note = {thirtyone-chapters-11e3e21; academic final review not implied; English is a working translation; SHA-256 3e6c76db2e52f24786d5c3872d3cce9866cdc559a799df786256c4aa515e3029}
}第二十四章：一整条随机路径，究竟怎样收敛？——Portmanteau、Prokhorov与函数空间拓扑
@misc{wang2026chance_ch24,
author = {Wang, Haobai},
title = {{第二十四章：一整条随机路径，究竟怎样收敛？——Portmanteau、Prokhorov与函数空间拓扑}},
year = {2026},
url = {https://chance-mathematics-six-chapters.goldenmath7.chatgpt.site/chapters/24-path-convergence.html},
note = {thirtyone-chapters-11e3e21; academic final review not implied; English is a working translation; SHA-256 fc5d382e93f6e43f9a972cc009f587afc6e0923d415a26da6d9295d2bc5d2904}
}第二十五章：随机游走什么时候变成 Brownian motion？——Donsker 不变性原理
@misc{wang2026chance_ch25,
author = {Wang, Haobai},
title = {{第二十五章：随机游走什么时候变成 Brownian motion？——Donsker 不变性原理}},
year = {2026},
url = {https://chance-mathematics-six-chapters.goldenmath7.chatgpt.site/chapters/25-donsker.html},
note = {thirtyone-chapters-11e3e21; academic final review not implied; English is a working translation; SHA-256 4885cbf2f8df28546b9fff054e8708e1531e11169d881af696a92059e862daf6}
}第二十六章：连续的路径为什么如此粗糙？
@misc{wang2026chance_ch26,
author = {Wang, Haobai},
title = {{第二十六章：连续的路径为什么如此粗糙？}},
year = {2026},
url = {https://chance-mathematics-six-chapters.goldenmath7.chatgpt.site/chapters/26-brownian-properties.html},
note = {thirtyone-chapters-11e3e21; academic final review not implied; English is a working translation; SHA-256 f1011c154e0e3a46691341b276cb04a588d88b15c41fdde989d200d1bbdf7e6e}
}第二十七章：如果随机运动会跳跃而不只是扩散？——Lévy processes 与 Lévy–Khintchine 理论
@misc{wang2026chance_ch27,
author = {Wang, Haobai},
title = {{第二十七章：如果随机运动会跳跃而不只是扩散？——Lévy processes 与 Lévy–Khintchine 理论}},
year = {2026},
url = {https://chance-mathematics-six-chapters.goldenmath7.chatgpt.site/chapters/27-levy-processes.html},
note = {thirtyone-chapters-11e3e21; academic final review not implied; English is a working translation; SHA-256 1153ba148830f9d1fa4e416451b0aa7d3a7a091ee1b34e33350f4374fac0f636}
}第二十八章：随机运动为什么能产生确定性方程？——半群、生成元与 resolvent
@misc{wang2026chance_ch28,
author = {Wang, Haobai},
title = {{第二十八章：随机运动为什么能产生确定性方程？——半群、生成元与 resolvent}},
year = {2026},
url = {https://chance-mathematics-six-chapters.goldenmath7.chatgpt.site/chapters/28-semigroups.html},
note = {thirtyone-chapters-11e3e21; academic final review not implied; English is a working translation; SHA-256 3944dfdc0be734d4a51c74019fcd3e94c9c6cf9b97b521f9b93a768261d55ba2}
}第二十九章：当小概率变成指数级的小概率——Large deviations
@misc{wang2026chance_ch29,
author = {Wang, Haobai},
title = {{第二十九章：当小概率变成指数级的小概率——Large deviations}},
year = {2026},
url = {https://chance-mathematics-six-chapters.goldenmath7.chatgpt.site/chapters/29-large-deviations.html},
note = {thirtyone-chapters-11e3e21; academic final review not implied; English is a working translation; SHA-256 31a0ebba5c26390716007a4dada53183fb2bda7382e8dcf658ab2c7b03d47580}
}第三十章：什么叫在事情发生之前就知道？
@misc{wang2026chance_ch30,
author = {Wang, Haobai},
title = {{第三十章：什么叫在事情发生之前就知道？}},
year = {2026},
url = {https://chance-mathematics-six-chapters.goldenmath7.chatgpt.site/chapters/30-predictability.html},
note = {thirtyone-chapters-11e3e21; academic final review not implied; English is a working translation; SHA-256 c2c0dcf998f9788a113fc6f930bbe5df4e5569cc08dafff166cac54c5e66d4a1}
}第三十一章：对一条不可导的随机路径怎样积分？——Itô integral、quadratic variation
@misc{wang2026chance_ch31,
author = {Wang, Haobai},
title = {{第三十一章：对一条不可导的随机路径怎样积分？——Itô integral、quadratic variation}},
year = {2026},
url = {https://chance-mathematics-six-chapters.goldenmath7.chatgpt.site/chapters/31-ito-integral.html},
note = {thirtyone-chapters-11e3e21; academic final review not implied; English is a working translation; SHA-256 56c35be4418f3eb70b2423dc9ebbebc4c30d89a76a4774935f6707e178944ef0}
}引用格式是学术署名工具，不等于转载、改编或商业再出版许可。