Download The strong law of large numbers

The strong law
of large numbers
Artur Wasiak
The strong law of large numbers (2)
The strong law of large numbers is
every theorem about convergence
almost surely the arithmetical mean Xn
to a constant.
The strong law of large numbers (3)
Kolmogorov’s theorem.
Let (Xn) be a sequence of independent
random variables such that Var Xn < ∞
(n=1,2,…). Let (bn) be an increasing
sequence of positive real numbers
disvergent to +∞ and
then
almost surely
In particular, we can take an=n.
The strong law of large numbers (4)
Kolmogorov’s strong law of large
numbers. If (Xn) is a sequence of
independent and identically distributed
random variables and E|X1|<∞, then
almost surely.
The strong law of large numbers (5)
Conclusions.
1) Frequency definition of probability is
correct.
2) Intuitive and theoretical definitions
of expexted value are corresponding.
The strong law of large numbers (6)
Applications.
1) Verification if the probability space has
been chosen correctly.
2) Empirical distribution function.
3) Monte Carlo method.