作者washburn (Just a game)
看板Economics
标题[请益] Almost sure convergence and Convergence in Probability
时间Thu Jan 31 17:45:31 2008
Almost sure convergence 和 Convergence in probability 定义如下:
Let {bn(.)} be a sequence of real-value random variables,
and there exists a real number b.
Almost sure convergence:
bn(.) converges almost surely to b if P{w:bn(w)→b}=1 as n→∞.
Convergence in Probability:
bn(.) converges in probability to b if P(w:|bn(w)-b|<ε)→1
as n→∞ for every ε>0.
Almost sure convergence 和 Convergence in probability 可分别以 Kolmogorov
strong law of large numbers 以及 Chebyshev weak law of large numbers 为例:
Let bar(Zn) = (sum Zt)/n.
Kolmogorov strong law of large numbers:
bar(Zn) →{a.s.} μ as {Zt} i.i.d. with μ = E(Zt) < ∞.
Chebyshev weak law of large numbers:
bar(Zn) →{p} μ as E(Zt) = μ, var(Zt) = σ^2 < ∞ for all t
and cov(Zt,Zs) = 0 for t ≠ s.
我个人有两个问题想请教:
第一, 虽然我了解两种 convergence 及 l.l.n. 的定义, 但我想知道 Almost sure
convergence 和 Convergence in probability 有没有比较直观的解释?
第二, {Zt} i.i.d. 的范例很容易找, 例如丢铜板就是最简单的例子. 但是
E(Zt) = μ, var(Zt) = σ^2 < ∞ for all t and cov(Zt,Zs) = 0 for t ≠ s
的例子, 对我个人而言很难想像. 能不能提供一个简单的范例?
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