Limiting Moment Generating Function

Theorem 2
If the sequence of random variables, X1, X2, . . ., converges in probability to a random variable X , the sequence also converges in distribution to X .
Theorem 3
The sequence of random variables, X1, X2, . . ., converges in probability to a constant µ if and only if the sequence also converges in distribution to µ. That is, the statement
P (|Xn ? µ| > ?) ? 0 for every ? > 0
is equivalent to
P (Xn ? x)? {?(0&x < µ@1&x > µ)?
Theorem 4 (Central limit theorem(
Let X1, X2,…., be a sequence of iid random variables whose mgfs exist in a neighborhood of 0 (,that is, MXi(t) exists for
|t| < h, for some positive h). Let E(Xi) = µ and Var(Xi )= ?2 .Both µ and ?2 are finite since the mgf exists.) Define X ?_n=?_(i=1)^n?X_i/n. Let Gn(x) denote the cdf of?n( Xn ? µ)/?. Then, for any x, ?? < xlimn? ?Gn(x)=F(x) the df of standard normal distribution, that is?n( Xn ? µ)/? has a limiting standard normal distribution.

The sequence of random variables, X1,…,Xn, is said to converge in probability to the constant c, if for every ?>0, ?lim??(n??)??P(|X_n-c|??)=1?
Weak Law of Large Numbers (WLLN): Let X1,…,Xn be iid random variables with E(Xi)=m and V(Xi)=S2 < ?. Then the sample mean converges in probability to m:

?lim??(n??)??P(|X ?_n-c|??)=1? or equivalent to
?lim??(n??)??P(|X ?_n-c|>?)=0?

Where X ?_n=?_(i=1)^n?X_i/n
Limiting Moment Generating Function
Theorem
The limiting distribution of a Poisson(?) distribution as ???
is normal.

By using the moment generating function of a Poisson random variable and expanding the exponential function as a series. This can be recognized as the moment generating function of a standard normal random variable. This implies that the associated unstandardized
random variable Xn has a limiting distribution that is normal with mean n and variance.
This result is the basis for the “normal approximation to the Poisson distribution.”