What is the difference between as convergence and convergence in probability?
Answers
Answered by
1
One of the key differences is that convergence in probability tells you something about random variables being close in a pointwise sense (with a high probability) whereas convergence in distribution says only something about the closeness of the distributions.
Let me illustrate this with an example: Take to identically distributed random variables XX and YY and define
Xn:=Yfor all n≥1.Xn:=Yfor all n≥1.
Since all the random variables have the same distribution, we clearly have Xn→XXn→X in distribution. On the other hand, we can, in general not, expect that XX is close to YY in a pointwise sense. For instance, if X∼N(0,1)X∼N(0,1) and we set Y=Xn:=−XY=Xn:=−X, then
P(|X−Xn|>δ)=P(|X|>δ/2)>0P(|X−Xn|>δ)=P(|X|>δ/2)>0
which means that XnXn does not convergence in probability to XX.
Convergence in distribution does not even require that the random variables are all defined on the same probability space, i.e. each random variable XnXn may be defined on some probability space (Ωn,An,Pn)(Ωn,An,Pn). In particular, it doesn't even make sense to ask whether the random variables XnXn are being close to each other since we can't calculate the probability of the set
{ω∈??;|Xn(ω)−Xm(ω)|>δ}{ω∈??;|Xn(ω)−Xm(ω)|>δ}
(probability with respect to which measure? PnPn or PmPm...? In fact, we can't even write down the set properly since the ωω's are elements in different probability spaces.)
Let me illustrate this with an example: Take to identically distributed random variables XX and YY and define
Xn:=Yfor all n≥1.Xn:=Yfor all n≥1.
Since all the random variables have the same distribution, we clearly have Xn→XXn→X in distribution. On the other hand, we can, in general not, expect that XX is close to YY in a pointwise sense. For instance, if X∼N(0,1)X∼N(0,1) and we set Y=Xn:=−XY=Xn:=−X, then
P(|X−Xn|>δ)=P(|X|>δ/2)>0P(|X−Xn|>δ)=P(|X|>δ/2)>0
which means that XnXn does not convergence in probability to XX.
Convergence in distribution does not even require that the random variables are all defined on the same probability space, i.e. each random variable XnXn may be defined on some probability space (Ωn,An,Pn)(Ωn,An,Pn). In particular, it doesn't even make sense to ask whether the random variables XnXn are being close to each other since we can't calculate the probability of the set
{ω∈??;|Xn(ω)−Xm(ω)|>δ}{ω∈??;|Xn(ω)−Xm(ω)|>δ}
(probability with respect to which measure? PnPn or PmPm...? In fact, we can't even write down the set properly since the ωω's are elements in different probability spaces.)
Similar questions