Let X1, X2, ... Xn be a random variable then.
Weakest
1) lim Fxn(x) = Fx(x) as n -> infiniti (Converges in distribution)
2) lim Pr(|Xn - Xn|>= e) = 0 as n -> infiniti for every e>0. (Converges in probability)
3) Pr(lim Xn = X) = 1 as n -> inifiniti (Converges almost surely)
4) lim Xn(w) = X(w) as n -> infiniti for all w in Omega where Omega is the sample space of the underlying probability space over which the random variable is defined. (Converges Surely)
Strongest
Prove:
If 4 is true then 3, 2 and 1 are true.
If 3 is true then 2 and 1 are true.
If 2 is true then 1 is true.
Weakest
1) lim Fxn(x) = Fx(x) as n -> infiniti (Converges in distribution)
2) lim Pr(|Xn - Xn|>= e) = 0 as n -> infiniti for every e>0. (Converges in probability)
3) Pr(lim Xn = X) = 1 as n -> inifiniti (Converges almost surely)
4) lim Xn(w) = X(w) as n -> infiniti for all w in Omega where Omega is the sample space of the underlying probability space over which the random variable is defined. (Converges Surely)
Strongest
Prove:
If 4 is true then 3, 2 and 1 are true.
If 3 is true then 2 and 1 are true.
If 2 is true then 1 is true.
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