“convergence in probability”的概念、定义、翻译、参考文献-科学参考

convergence in probability

Mathematics

A sequence of random variables X_n converges in probability to a random variable X if for all ε‎ > 0, Pr(|X_n−X|>ε‎) tends to 0 as n tends to infinity. This is written $X_{n} \overset{P}{\to} X$ . Convergence in probability implies convergence in distribution. See weak law of large numbers.

Economics

A sequence of random variables x₁,…, x_n,… converges in probability to a random variable x if for every positive number ε‎ the probability of the (Euclidean) distance between x_n and x exceeding ε‎ converges to zero as n tends to infinity, so

$x = p lim x_{n} \Leftrightarrow x_{n} \overset{p}{\to} x \Leftrightarrow lim_{n \to \infty} P [| x_{n} - x | \geq ε] = 0 \Leftrightarrow lim_{n \to \infty} P [| x_{n} - x | \leq ε] = 1$

for every ε‎ > 0. This means that, if we consider a sequence of probabilities, P_n = P[|x_n − x| ≥ ε‎], then starting from some n₀ each probability in this sequence is arbitrarily small. In particular, x can be a constant. Convergence in probability implies convergence in distribution (the converse does not hold, in general).

单词	convergence in probability
释义	convergence in probability Mathematics A sequence of random variables X_n converges in probability to a random variable X if for all ε‎ > 0, Pr(\|X_n−X\|>ε‎) tends to 0 as n tends to infinity. This is written $X_{n} \overset{P}{\to} X$ . Convergence in probability implies convergence in distribution. See weak law of large numbers. Economics A sequence of random variables x₁,…, x_n,… converges in probability to a random variable x if for every positive number ε‎ the probability of the (Euclidean) distance between x_n and x exceeding ε‎ converges to zero as n tends to infinity, so $x = p lim x_{n} \Leftrightarrow x_{n} \overset{p}{\to} x \Leftrightarrow lim_{n \to \infty} P [\| x_{n} - x \| \geq ε] = 0 \Leftrightarrow lim_{n \to \infty} P [\| x_{n} - x \| \leq ε] = 1$ for every ε‎ > 0. This means that, if we consider a sequence of probabilities, P_n = P[\|x_n − x\| ≥ ε‎], then starting from some n₀ each probability in this sequence is arbitrarily small. In particular, x can be a constant. Convergence in probability implies convergence in distribution (the converse does not hold, in general).
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