Lecture 14/ limiting distribution/2017 2018 /

Limiting Distribution

The weak law of large numbers, applied to a sequence Xi ? {0, 1} of coin tosses,
says that Sn/n must lie in an arbitrarily small interval around 1/2 with high
probability (arbitrarily close to 1) if n is taken big enough. A stronger statement
would be to say that, with probability one, a sequence of coin tosses yields a
sum Sn such that Sn/n actually converges to 1/2.
To explain the meaning of the stronger claim, let us be more explicit and
view the random variables as functions Xi : S ? R on the same probability
space (S, F, P). Then, for each s ? S we can consider the sample sequence
X1(s), X2(s), . . . , as well as the arithmetic averages Sn(s)/n, and ask whether
Sn(s)/n (an ordinary sequence of numbers) actually converges to 1/2. The
strong law of large numbers states that the set of s for which this holds is an
event of probability 1. This is a much more subtle result than the weak law,
and we will be content with simply stating the general theorem.
The Central Limit Theorem

The reason why the normal distribution arises so often is the central limit theo-
rem. We state this theorem here without proof, although experimental evidence
for its validity will be given in a number of examples.
Let (S, F, P) be a probability space and X1, X2, . . . be independent random
variables de?ned on S. Assume that the Xi have a common distribution with
?nite expectation m and ?nite nonzero variance ?2. De?ne the sum
Sn = X1 + X2 + ••• + X
Limiting Distribution
hese two terms are limiting distribution
and asymptotic distribution.
A distribution is called a limiting distribution function if it is the limit distribution function of a sequence of distribution
functions. Equation (31) provides us with an example; (z) is the limiting
distribution function of the sequence of distribution functions F (z_1 ),…,F(z_n ).
Also ? (z) is called the limiting distribution of the sequence of random
variables Z_1,Z_2 ,..,Z_n .
On the other hand, an asymptotic distribution of a random variable, say Y in a sequence of random variables Y_1,Y_2 ,…,Y_n is any distribution that is approximately equal to the actual distribution of Y,for large n.
Yet the two terms are closely related since it was precisely the fact
that the sequence Z_1,Z_2 ,..,Z_n had limiting standard normal distribu-
tion that allowed us to say that X ?,. had an asymptotic normal distribution with mean ? and variance ?^2/n. The idea is that if the distribution of Z,. is converging
to ? (z),then for large n the distribution of Z,. must be approximately distributed N(0, I).
But if Z_n=((X ?_n-?))/(???n) is approximately distributed N(0, I),
then X ?_nis approximately distributed N(?,?^2?(n)).