[請益] Almost sure convergence and Convergence in Probability

[washburn (Just a game)]

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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