1 definition

Convergence by distribution is defined as follows: Random variable sequence {Xn} n = 1 up \ {X_n \} _ {n = 1} ^ {\ infty} {Xn} n = 1 up, if their cumulative distribution functions the CDF sequence {F1} n = 1 up \ {F_1 \} _ {n = 1} ^ {\ infty}} {F1 up n = 1. And the CDF FFF of some random variable XXX, satisfies


lim ⁡ n – up F n ( x ) = F ( x ) \lim_{n\to\infty} F_n(x)=F(x)

It is true at any F(x)F(x)F(x) F(x) continuous point XXX. They are said to converge to the random variable XXX in distribution, denoted as Xn⟶DXX_n\stackrel{D}\longrightarrow XXn⟶DX.

In this definition, there are two easily overlooked but important points, one is that it must correspond to the CDF of some random variable, not any function, and the other is that the condition is only required to be true at the continuous point of F(x), F(x), F(x), F(x).

Next, let’s look at why we define it this way.

The limit function of 2 has to be CDF

Considering Xn ~ N (0, sigma n2) X_n \ sim N (0, \ sigma_n ^ 2) Xn ~ N (0, sigma n2), sigma and N + up \ sigma_n \ \ infty sigma to + N – > + up, we have


F n ( x ) = P ( x n sigma n Or less x sigma n ) = Φ ( x sigma n ) F_n(x)=P(\dfrac{x_n}{\sigma_n}\leq \dfrac{x}{\sigma_n})=\Phi(\dfrac{x}{\sigma_n})

Any place, XXX has Φ sigma n (x) – > Φ (0) = 12 \ Phi (\ dfrac {x} {\ sigma_n}) \ to \ Phi (0) = \ dfrac {1} {2} Φ (sigma nx) – > Φ (0) = 21, therefore, If F(x)=12F(x)=\dfrac{1}{2}F(x)=21, the limit condition in the definition is satisfied. But at this point, F(x), F(x), F(x) is not the CDF of any random variable, As random variables of CDF need to satisfy the lim ⁡ x – – up F (x) = 0, lim, limits_ {x \ – \ infty} F (x) = 0 x – – up limF (x) = 0 and lim ⁡ x – up F (x) = 1 \ lim \ limits_ {x \ the to \ infty} F (x) = 1 x – up limF (x) = 1.

How can this be corrected? We just let the sequence {Xn} n = 1 up \ {X_n \} _ {n = 1} ^ {\ infty} {Xn} n = 1 up is bounded in probability. In the definition, it is required that the limit form of the CDF function column must correspond to the CDF of a random variable.

3. Only the continuous points are considered

Recall another property of CDF: right continuous, that is, F(x)=F(x+)F(x)=F(x+)F(x)=F(x+). However, in the case of convergence according to distribution, it is likely to meet the situation, such as Xn=X+1nX_n=X+\dfrac1 nXn=X+ N1, easy to know


F n ( x ) = P ( X n Or less x ) = P ( X Or less x – 1 n ) = F ( x – 1 n ) F_n(x)=P(X_n\leq x)=P(X\leq x-\dfrac 1 n)=F(x-\dfrac{1}{n})

If the n – up n \ to \ inftyn – up, the Fn (x) and F (x) – F_n \ to F (x) (x) Fn (x) and F (x -). If FFF does not satisfy left continuity at XXX, then Fn(x)→F(x)F_n(x)\to F(x)Fn(x)→F(x) can not be satisfied. Therefore, the discontinuity of FFF should be excluded from the definition.

For example, Xn ~ U(0,1/n)X_n\sim U_{(0,1/n)}Xn ~ U(0,1/n), the distribution of XnX_nXn in the limit degenerates to X=1X=1X=1, and Fn(0)=0F_n(0)=0Fn(0)=0. But F (0) = 1, F (0) = 1, F (0) = 1, so for x = 0 to x = 0 x = 0 cannot meet the Fn (x) and F (x) F_n \ to F (x) (x) Fn (x) and F (x), but the x = 0 x = 0 x = 0 is F (x) F (x) F (x) of discrete points, so you can eliminate. Xn⟶DXX_n\stackrel{D}\longrightarrow XXn⟶DX.