The strong law of large numbers

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The strong lawof large numbers

• Artur Wasiak

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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.
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The strong law of large numbers (3)
Kolmogorovs theorem.Let (Xn) be a sequence of
independent random variables such that Var Xn lt 8
(n1,2,). Let (bn) be an increasing sequence of
positive real numbers disvergent to 8 and
thenIn particular, we can take ann.
almost surely
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The strong law of large numbers (4)
Kolmogorovs strong law of large numbers. If (Xn)
is a sequence of independent and identically
distributed random variables and EX1lt8,
thenalmost surely.
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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.
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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.