强大数定律与弱大数定律在收敛意义上有何本质区别？
What is the essential difference between the strong and weak laws of large numbers in terms of convergence?
弱大数定律要求样本均值依概率收敛于期望，而强大数定律要求几乎处处（以概率 1）收敛。后者更强：几乎处处收敛蕴含依概率收敛，但反之不成立。典型证明借助 Borel-Cantelli 引理与三级数定理。
The weak law requires convergence in probability of the sample mean to the expectation, whereas the strong law requires almost-sure convergence (with probability one). The latter is strictly stronger; almost-sure convergence implies convergence in probability but not conversely. Proofs typically invoke the Borel-Cantelli lemma and the three-series theorem.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 来源待核验 · Source pending review
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出处备注：著录仅给书名，无版次与年份；同一书名常对应多个版次，不臆测具体版本。
美国数学会(AMS)《概率论与测度论》
American Mathematical Society (AMS), Probability and Measure
#强大数定律 #几乎处处收敛 #Borel-Cantelli
泊松过程为何常被用作建模稀有随机事件的计数过程？
Why is the Poisson process commonly used to model counting processes of rare random events?
泊松过程具有独立增量与平稳增量，且区间计数服从参数为 $\lambda t$ 的泊松分布，满足 $P(N(t)=k)=e^{-\lambda t}(\lambda t)^k/k!$。在事件发生率低且相互独立时，二项分布极限正好导出泊松分布，故它是稀有事件的自然模型。
The Poisson process has independent and stationary increments, and counts in an interval follow a Poisson distribution with parameter $\lambda t$, $P(N(t)=k)=e^{-\lambda t}(\lambda t)^k/k!$. When events are rare and independent, the binomial limit yields the Poisson law, making it the natural model for rare events.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-11-02
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美国数学会(AMS)《随机过程导论》
American Mathematical Society (AMS), Introduction to Stochastic Processes
独立核验依据 · Independent verification: 核验依据 1 ISBN 9787111315445（豆瓣书目条目: 随机过程导论）
#泊松过程 #独立增量 #泊松分布
什么是马尔可夫链的平稳分布，它与转移矩阵有何关系？
What is a stationary distribution of a Markov chain, and how is it related to the transition matrix?
平稳分布 $\pi$ 是一个概率向量，满足 $\pi P=\pi$，其中 $P$ 为一步转移矩阵。若链不可约且正常返，则存在唯一平稳分布，且从任意初态出发的长期分布都收敛到它。
A stationary distribution $\pi$ is a probability vector satisfying $\pi P=\pi$, where $P$ is the one-step transition matrix. If the chain is irreducible and positive recurrent, a unique stationary distribution exists, and the long-run distribution from any initial state converges to it.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-11-02
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美国数学会(AMS)《随机过程导论》
American Mathematical Society (AMS), Introduction to Stochastic Processes
独立核验依据 · Independent verification: 核验依据 1 ISBN 9787111315445（豆瓣书目条目: 随机过程导论）
#马尔可夫链 #平稳分布 #转移矩阵
随机游走的常返性为何在二维与三维表现完全不同？
Why does the recurrence of random walks differ so dramatically between two and three dimensions?
Polya 定理表明，对称简单随机游走在二维仍然常返（回到原点概率为 1），但在三维变为暂态（返回概率约 0.34，小于 1）。原因在于高维空间中可走的路径更多、原点被「稀释」，返回概率随维度升高而下降。
Polya's theorem states that the symmetric simple random walk is recurrent in two dimensions (return probability 1) but transient in three dimensions (return probability about 0.34, less than 1). In higher dimensions there are more paths and the origin is effectively diluted, so the return probability decreases with dimension.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 来源待核验 · Source pending review
来源 · Source
出处备注：著录仅给书名，无版次与年份；同一书名常对应多个版次，不臆测具体版本。
美国数学会(AMS)《概率论与测度论》
American Mathematical Society (AMS), Probability and Measure
#Polya定理 #常返 #暂态
如何利用吸收马尔可夫链计算从某状态到达吸收态的期望步数？
How can an absorbing Markov chain be used to compute the expected number of steps to reach an absorbing state?
把转移矩阵分块为吸收态块 $I$ 与瞬态块，瞬态间转移矩阵记为 $Q$，则基本矩阵 $N=(I-Q)^{-1}$ 的元素 $N_{ij}$ 表示从状态 $i$ 出发、到达吸收态前经过状态 $j$ 的期望次数，各行和给出期望吸收步数。
Block the transition matrix with absorbing states forming $I$ and transient states forming $Q$; the fundamental matrix $N=(I-Q)^{-1}$ has entries $N_{ij}$ equal to the expected number of visits to state $j$ starting from $i$ before absorption, and row sums give the expected time to absorption.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-10-04
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Grinstead 与 Snell《Introduction to Probability》（AMS 出版）吸收马尔可夫链章
Grinstead and Snell, Introduction to Probability, AMS (absorbing Markov chains chapter)
独立核验依据 · Independent verification: 核验依据 1 ISBN 9780821807491 (1997)
#吸收马尔可夫链 #基本矩阵 #期望步数
在一维对称简单随机游走中，回到原点的概率是多少？
In a one-dimensional symmetric simple random walk, what is the probability of ever returning to the origin?
一维对称随机游走是常返的：从原点出发，回到原点的概率为 $1$。但首次返回时间的期望是无穷大，意味着回到原点虽然必然，却可能等待极久。
The one-dimensional symmetric random walk is recurrent: starting at the origin, the probability of ever returning is $1$. However, the expected return time is infinite, meaning a return is certain but may take arbitrarily long.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-10-04 · 3 条独立证据
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Feller《An Introduction to Probability Theory and Its Applications》第 1 卷（Wiley）随机游走章
Feller, An Introduction to Probability Theory and Its Applications, Vol. I, Wiley (random walk chapter)
独立核验依据 · Independent verification: 核验依据 1 ISBN 9789994311071 (1968) · 核验依据 2 ISBN 9780821807491 (1997) · 核验依据 3 ISBN 9789994311071 (1991)
#随机游走 #常返性 #首达时间
弱大数定律说明了独立同分布随机变量样本均值的什么行为？
What behavior of the sample mean of i.i.d. random variables does the weak law of large numbers describe?
设 $X_1,X_2,\dots$ 为独立同分布且期望 $\mu$ 存在的随机变量，则样本均值 $\bar X_n=(X_1+\cdots+X_n)/n$ 依概率收敛于 $\mu$。也就是说，当样本量增大时，样本均值偏离真实均值的概率趋于零。
If $X_1,X_2,\dots$ are i.i.d. random variables with mean $\mu$, the sample mean $\bar X_n=(X_1+\cdots+X_n)/n$ converges in probability to $\mu$. Thus as the sample size grows, the probability that the sample mean deviates from the true mean goes to zero.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-11-02
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Durrett《Probability: Theory and Examples》（第 5 版，Cambridge University Press）
Durrett, Probability: Theory and Examples, 5th ed., Cambridge University Press
独立核验依据 · Independent verification: 核验依据 1 OPENLIBRARY 标识 /works/OL15209593W：Probability: Theory an
#大数定律 #依概率收敛 #样本均值
中心极限定理对独立同分布变量的样本和给出了怎样的渐近分布？
What asymptotic distribution does the central limit theorem give for sums of i.i.d. variables?
设 $X_i$ 独立同分布，均值 $\mu$、方差 $\sigma^2>0$，则标准化和 $(\sum_{i=1}^n X_i-n\mu)/(\sigma\sqrt n)$ 依分布收敛于标准正态分布 $N(0,1)$。这是统计推断中大样本理论的根基。
For i.i.d. $X_i$ with mean $\mu$ and variance $\sigma^2>0$, the normalized sum $(\sum_{i=1}^n X_i-n\mu)/(\sigma\sqrt n)$ converges in distribution to the standard normal $N(0,1)$. This underpins large-sample theory in statistical inference.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-11-02
来源 · Source
Durrett《Probability: Theory and Examples》（第 5 版，Cambridge University Press）
Durrett, Probability: Theory and Examples, 5th ed., Cambridge University Press
独立核验依据 · Independent verification: 核验依据 1 OPENLIBRARY 标识 /works/OL15209593W：Probability: Theory an
#中心极限定理 #正态分布 #依分布收敛