분포 수렴
{ X n } \{X_n\} { X n } X X X F X n F_{X_n} F X n F X F_X F X X n X_n X n X X X C ( F X ) C(F_X) C ( F X ) F X F_X F X
lim n → ∞ F X n ( x ) = F X ( x ) , For all x ∈ C ( F X ) \lim \limits_{n \to \infin} F_{X_n}(x) = F_X(x), \ \text{For all x } \in C(F_X) n → ∞ lim F X n ( x ) = F X ( x ) , For all x ∈ C ( F X ) 를 만족하면 X n X_n X n X X X X n → D X X_n \rightarrow^D X X n → D X
이 때 X X X X n X_n X n 극한 분포 라고 한다.
분포 수렴의 의미
분포 수렴은 F X F_X F X F X n F_{X_n} F X n F X F_X F X F X n F_{X_n} F X n F X F_X F X
T 분포의 극한 분포
Let T n have t-distribution with dof n F n ( t ) = ∫ − ∞ t Γ ( n + 1 2 ) π n Γ ( n 2 ) 1 ( 1 + y 2 n ) n + 1 2 d y lim n → ∞ F n ( t ) = lim n → ∞ ∫ − ∞ t f n ( y ) d y = ∫ − ∞ t lim n → ∞ f n ( y ) d y ⇒ lim n → ∞ F n ( t ) = ∫ − ∞ t 1 2 π e − y 2 2 d y ∴ T n → D N ( 0 , 1 ) \text{Let } T_n \text{ have t-distribution with dof }n \\ {} \\ F_n(t) = \int_{-\infin}^t \frac{\Gamma(\frac{n+1}{2})}{\sqrt{\pi n}\Gamma(\frac{n}{2})} \frac{1}{(1+\frac{y^2}{n})^{\frac{n+1}{2}}} dy \\ {} \\ \lim \limits_{n \to \infin} F_n(t) = \lim \limits_{n \to \infin} \int_{-\infin}^{t} f_n(y)dy = \int_{-\infin}^{t} \lim \limits_{n \to \infin} f_n(y)dy \\ {} \\ \Rightarrow \lim \limits_{n \to \infin} F_n(t) = \int_{-\infin}^t \frac{1}{\sqrt{2\pi}}e^{-\frac{y^2}{2}}dy \\ {} \\ \therefore T_n \rightarrow^D N(0, 1) Let T n have t-distribution with dof n F n ( t ) = ∫ − ∞ t π n Γ ( 2 n ) Γ ( 2 n + 1 ) ( 1 + n y 2 ) 2 n + 1 1 d y n → ∞ lim F n ( t ) = n → ∞ lim ∫ − ∞ t f n ( y ) d y = ∫ − ∞ t n → ∞ lim f n ( y ) d y ⇒ n → ∞ lim F n ( t ) = ∫ − ∞ t 2 π 1 e − 2 y 2 d y ∴ T n → D N ( 0 , 1 ) 균등 분포와 지수 분포
X 1 , X 2 , . . . , X n ∼ i i d U ( 0 , θ ) Y n = m a x { X 1 , X 2 , . . . , X n } , Z n = n ( θ − Y n ) F Z n ( t ) = P ( Z n ≤ t ) = P ( n ( θ − Y n ) ≤ t ) = P ( Y n ≥ θ − t n ) = 1 − P ( Y n ≤ θ − t n ) P ( Y n ≤ y ) = P ( m a x { X 1 , X 2 , . . . , X n } ≤ y ) = P ( X 1 ≤ y , X 2 ≤ y , . . . , X n ≤ y ) = ∏ i = 1 n F X i ( y ) = ( 1 θ y ) n ⇒ P ( Y n ≤ θ − t n ) = ( 1 θ ( θ − t n ) ) n = ( 1 − t / θ n ) n ⇒ F Z n ( t ) = 1 − ( 1 − t / θ n ) n lim n → ∞ F Z n ( t ) = 1 − e − t θ ∴ Z n → D e x p ( θ ) X_1, X_2, ..., X_n \sim^{iid} U(0, \theta) \\ {} \\ Y_n = max\{X_1, X_2, ..., X_n\}, \ Z_n = n(\theta - Y_n) \\ {} \\ F_{Z_n}(t) = P(Z_n \le t) = P(n(\theta - Y_n) \le t) = P(Y_n \ge \theta - \frac{t}{n}) = 1-P(Y_n \le \theta - \frac{t}{n}) \\ {} \\ P(Y_n \le y) = P(max\{X_1, X_2, ..., X_n\} \le y)= P(X_1 \le y, X_2 \le y, ..., X_n \le y) = \\ {} \\ \prod_{i=1}^n F_{X_i}(y) =\left(\frac{1}{\theta}y\right)^n \\ {} \\ \Rightarrow P(Y_n \le \theta - \frac{t}{n}) = \left(\frac{1}{\theta}(\theta - \frac{t}{n}) \right)^n = \left(1 - \frac{t/ \theta}{n} \right)^n \\ {} \\ \Rightarrow F_{Z_n}(t) = 1- \left(1 - \frac{t/ \theta}{n} \right)^n \\ {} \\ \lim \limits_{n \to \infin} F_{Z_n}(t) = 1-e^{-\frac{t}{\theta}} \\ {} \\ \therefore Z_n \rightarrow^D exp(\theta) X 1 , X 2 , . . . , X n ∼ i i d U ( 0 , θ ) Y n = m a x { X 1 , X 2 , . . . , X n } , Z n = n ( θ − Y n ) F Z n ( t ) = P ( Z n ≤ t ) = P ( n ( θ − Y n ) ≤ t ) = P ( Y n ≥ θ − n t ) = 1 − P ( Y n ≤ θ − n t ) P ( Y n ≤ y ) = P ( m a x { X 1 , X 2 , . . . , X n } ≤ y ) = P ( X 1 ≤ y , X 2 ≤ y , . . . , X n ≤ y ) = i = 1 ∏ n F X i ( y ) = ( θ 1 y ) n ⇒ P ( Y n ≤ θ − n t ) = ( θ 1 ( θ − n t ) ) n = ( 1 − n t / θ ) n ⇒ F Z n ( t ) = 1 − ( 1 − n t / θ ) n n → ∞ lim F Z n ( t ) = 1 − e − θ t ∴ Z n → D e x p ( θ ) 