“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).
随便看	Geosynchronous Satellite Launch Vehicle geosynchronous transfer orbit geosyncline geotagging GEOTAIL Geotail geotail geotargeting geotaxis geotechnical map geotherm geothermal brine geothermal energy geothermal field geothermal flux geothermal gradient geothermic survey geothermometer geotropism geovisualization geo-web gerade geraniol geranium Gerard, Alexander (1728–95)