[請益] Almost sure convergence and Converge …
看板Economics (經濟學)作者washburn (Just a game)時間18年前 (2008/01/31 17:45)推噓0(0推 0噓 3→)留言3則, 2人參與討論串1/2 (看更多)
Almost sure convergence 和 Convergence in probability 定義如下:
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 可分別以 Kolmogorov
strong law of large numbers 以及 Chebyshev weak law of large numbers 為例:
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.
我個人有兩個問題想請教:
第一, 雖然我了解兩種 convergence 及 l.l.n. 的定義, 但我想知道 Almost sure
convergence 和 Convergence in probability 有沒有比較直觀的解釋?
第二, {Zt} i.i.d. 的範例很容易找, 例如丟銅板就是最簡單的例子. 但是
E(Zt) = μ, var(Zt) = σ^2 < ∞ for all t and cov(Zt,Zs) = 0 for t ≠ s
的例子, 對我個人而言很難想像. 能不能提供一個簡單的範例?
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※ 編輯: washburn 來自: 211.77.241.3 (01/31 20:00)
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