Download
1 / 6
90 likes | 361 Views
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 X n to a constant. The strong law of large numbers (3).
E N D
The strong lawof 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 thenIn particular, we can take an=n. almost surely
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|<∞, thenalmost 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.
