p-dimensional random vectorX and a sequence of random vectors {Xn} are said to converge in probability to n→∞ as Xn if they satisfy the following. It is denoted by Xn→PX.
∀ε>0,n→∞limP[∥Xn−X∥<ε]=1
∥⋅∥ is defined as the Euclidean norm, defined by ∥(x1,⋯,xn)∥=x12+⋯+xn2.
Theorem
Let us represent the p-dimensional random vector as X=(X1,⋯,Xp). Then
Xn→PX⟺Xnk→PXk,∀k=1,⋯,p
Proof
(⇒)
Let Xn→PX. According to the definition of the Euclidean norm for ε>0,
ε≤∣Xnk−Xk∣≤∥Xnk−Xk∥
therefore,
n→∞limsupP[∣Xnk−Xk∣≥ε]≤n→∞limsupP[∥Xnk−Xk∥≥ε]=0
(⇐)
Let Xnk→PXk,∀k=1,⋯,p. According to the definition of the Euclidean norm for ε>0,
ε≤∥Xn−X∥≤k=1∑p∣Xnk−Xk∣
therefore,
≤≤=n→∞limsupP[∥Xn−X∥≥ε]n→∞limsupP[∣Xnk−Xk∣≥ε]k=1∑pn→∞limsupP[∣Xnk−Xk∣≥ε]0
