Portmanteau Theorem

Jaime Frade

Department of Statistics Florida State University

Florentina Bunea STA5447

April 25, 2009

Outline

1 Show 7 ⇔ 9
2 Show 7 ← 9
3 Show 9 → 12
4 Show 9 → 11
5 Show 11 → 12
6 Show 12 → 11
7 Show 11 and 12 → 13
8 Show 13 → 7

Show 7 ⇔ 9

F_{n} \underrightarrow{}^{d} F ⇒ E[g_{n}(x_{n})] = ∫ gdF_{n} → ∫ gdF = E[g(x)]

By Hally-Bray Theorem, consider some (Ω, A, P). Suppose that g is bdd and is continuous a.s. F . (g ∈ C_{b}), then

E[g_{n}(x_{n})] = ∫ gdF_{n} → ∫ gdF = E[g(x)]  (1)

By converse, since E[g_{n}(x_{n})] → E[g(x)] ⇒ F_{n} → F

Show 7 ← 9

F_{n} \underrightarrow{}^{d} F ⇐ E[g_{n}(x_{n})] = ∫ gdF_{n} → ∫ gdF = E[g(x)]

←: Notice that for any δ(B) = (C_{b})^{c}. For any B with P(x ∈ δB) = 0 by continuous mapping theorem, E(1_{B}(x_{n})) → E(1_{B}(x)) as n → ∞ F_{n}(x) = P(x_{n} ∈ (−∞, x]), with δ(−∞, x] = {x} so provided P(X = x) = 0, then P(x_{n} ∈ (−∞, x]) → P(x ∈ (−∞, x]). x ∈ B ↔ 1_{B}. x → 1_{B}(x) is not continuous at δB. Otherwise, h(x) = 1_{B}(x) is continuous and bounded ∀ ∉ δB

x_{n} → x ⇒ h(x_{n}^{#}) \underrightarrow{}^{a.s} h(x^{#}) and E[h(x_{n}^{#})] \underrightarrow{}^{DCT} h(x^{#}). So E[h(x_{n})] → h(x) E[h(x_{n})] = E[1_{B}(x_{n})] = P(x_{n}) → E[h(x)] = P(x ∈ B)

F_{n} \underrightarrow{}^{d} F ⇒ E[g_{n}(x_{n})] = ∫ gdF_{n} → ∫ gdF = E[g(x)]

Show 9 → 12

→: Take B = (−∞, x], then δB = {x} P_{n}(B) → P(B) with P(δB) = 0 F_{n}(x) → F (x) ∀x, P({x, y}) = 0, ∀x, F has no jumps at x, all x ∈ C_{b} Show that E[g_{n}(x_{n})] = ∫ gdF_{n} → ∫ gdF = E[g(x)] ∀ g ∈ C_{b} ⇒ lim inf P_{n}(B) ≥ P(B) ∀ open sets B

Define a function, g, such that satisfies first part, then will show because of the way the function is setup, the function will imply the second part.

Let B be an open set and F = B^{c}. Consider a sequence of functions, g_{n} ∈ C_{b},

g_{n}(x) = min(1, nd(x, F ))  (2)

Show that g_{n}(x) ↑ 1(x)

a. When x ∉ F ⇒ let ϵ = d(x, F ) > 0, then

g_{n}(x) = min(1, nϵ), where nϵ ≥ 1. For n large, ⇒ g_{n}(x) = 1 b. When x ∈ F , then d(x, F ) = 0 ⇒ g_{n}(x) = 0

Therefore,

g_{n}(x) = \{ \begin{matrix}0 & x ∉ F = B^{c} \\ 1 & x ∈ F = B^{c} \end{matrix}

as n → ∞ By 9, P_{n}(x) ≥ ∫ g_{k}dF_{n} → ∫ g_{k}dF as n → ∞ and lim inf P_{n}(B) ≥ ∫ g_{k}dF by letting k → ∞ and MCT,

P_{n}(B) ≥ ∫ g_{k}dF_{n} \underrightarrow{}^{9} ∫ g_{k} ↑ P(B), n → ∞
lim inf_{n} P_{n}(B) ≥ lim_{k→∞}∫ g_{k}dF \overbrace{=}^{MCT} ∫ lim_{k→∞}g_{k}dF = ∫ gdF = P(B)
lim inf_{n} P_{n}(B) ≥ ∫ 1_{B}dF = P(B)  (3)

Show that E[g_{n}(x_{n})] = ∫ gdF_{n} → ∫ gdF = E[g(x)] ∀ g ∈ C_{b} ⇒ lim sup P_{n}(B) ≤ P(B)∀ closed sets B

P(B) = ∫ 1_{B}dF_{n} ≤ ∫ g_{k}dF_{n} \underrightarrow{}^{9} ∫ g_{k}dF ↓ P(B^{c})
lim sup_{n}P_{n}(B) ≤ lim_{k→∞}∫ g_{k}dF_{n} \overbrace{=}^{MCT} ∫ lim_{k→∞}g_{k}dF = ∫ gDF = P(B)

lim sup P_{n}(B) ≤ P(B)∀ closed sets B → lim inf P_{n}(B) ≥ P(B)∀ open sets B

lim sup P_{n}(B) ≤ P(B)∀ closed sets B
1 − lim sup P_{n}(B) ≥ P(B^{c})
1 + lim inf(−P_{n}(B)) ≥ P(B^{c})
lim inf(1 − P_{n}(B)) ≥ P(B^{c})
lim inf(P_{n}(B^{c})) ≥ P(B^{C})  (4)

lim inf P_{n}(B) ≥ P(B)∀ open sets B → lim sup P_{n}(B) ≤ P(B)∀ closed sets B

lim inf P_{n}(B) ≥ P(B)∀ open sets B
1 − lim inf P_{n}(B) ≤ P(B^{c})
1 + lim sup(−P_{n}(B)) ≤ P(B^{c})
lim sup(1 − P_{n}(B)) ≤ P(B^{c})
lim sup P_{n}(B^{c}) ≤ P(B^{c})  (5)

Show 11 and 12 → 13

lim inf P_{n}(B) ≥ P(B)∀ open sets B and lim sup P_{n}(B) ≤ P(B)∀ closed sets B → lim P_{n}(B) = P(B)

Define the following:

a. B^{0} is the interior of B, where B^{0} = ∪{U ⊂ B, s.t. U is open}, (Union of all open subsets of B.

b. B̄ is the closure of B, where B̄ = ⋂{B ⊂ C, s.t. C is closed}, (Union of all open subsets of B.

P(B) ≤ lim inf P_{n}(B) ≤ lim sup P_{n}(B) ≤ P(B)

∀ P−continuity sets B, (P(δB) = 0), then the boundary B, δB = B̄/B^{0} Since P(δB) = 0, then P(B̄) = P(B^{0}) = P(B) → lim P_{n}(B) = P(B)

lim P_{n}(B) = P(B) → F_{n} \underrightarrow{}^{d} F

For lim P_{n}(B) = P(B) ∀ P− continuity sets B, , (P(δB) = 0), where B ⊂ Ω = ℝ. Let B = (−∞, x] → δ(−∞, x] = {x}. Then P(−∞, x]) = 0 ∀ continuity points x ∈ Ω, which implies

F_{n} \underrightarrow{}^{d} F ⇒ F_{n}((−∞, x]) → F ((−∞, x])