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Tried make roles of l and n more apparend to show what qoes to infinity. Also added another part ot the special case question.
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AlbertRapp
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From the standard literature it is well known that for sequences of random variables X1,nPX1 and X2,nPX2 as n it holds that (X1,n,X2,n)P(X1,X2) for n. Using a continuous mapping theorem argument this can be used to establish that X1,n+X2,nPX1+X2 for n.

Question in general case

Under which conditions can the arguments of the standard literature be extended from a simple sum of two sequences to a sum of sequences whose number of summands also increases to infinity as n goes to infinity. Specifically, given a sequences Xl,n, l=1,,nlN, with Xl,nPXl (n) for all l=1,,nlN under which conditions does it holds that limnnl=1Xl,nPl=1Xl as n?

Question in special case

In my particular use case something more specific would also suffice. Though, I think the more general question is also interesting. If we know that Xl,n, l=1,,nlN, are nonnegative random sequences, i.e. each sequence element is a nonnegative random variable, and Xl,nP0 (n) for all l=1,,nlN under which conditions does it holds that limnnl=1Xl,nP0 as n? Possibly, what would also be interesting and might be easier to answer is when the sample mean converges to zero, i.e., limn1nnl=1Xl,nP0 as n

From the standard literature it is well known that for sequences of random variables X1,nPX1 and X2,nPX2 as n it holds that (X1,n,X2,n)P(X1,X2) for n. Using a continuous mapping theorem argument this can be used to establish that X1,n+X2,nPX1+X2 for n.

Question in general case

Under which conditions can the arguments of the standard literature be extended from a simple sum of two sequences to a sum of sequences whose number of summands also increases to infinity as n goes to infinity. Specifically, given a sequences Xl,n, l=1,,n, with Xl,nPXl for all l=1,,n under which conditions does it holds that limnnl=1Xl,nPl=1Xl as n?

Question in special case

In my particular use case something more specific would also suffice. Though, I think the more general question is also interesting. If we know that Xl,n, l=1,,n, are nonnegative random sequences, i.e. each sequence element is a nonnegative random variable, and Xl,nP0 for all l=1,,n under which conditions does it holds that limnnl=1Xl,nP0 as n?

From the standard literature it is well known that for sequences of random variables X1,nPX1 and X2,nPX2 as n it holds that (X1,n,X2,n)P(X1,X2) for n. Using a continuous mapping theorem argument this can be used to establish that X1,n+X2,nPX1+X2 for n.

Question in general case

Under which conditions can the arguments of the standard literature be extended from a simple sum of two sequences to a sum of sequences whose number of summands also increases to infinity as n goes to infinity. Specifically, given a sequences Xl,n, lN, with Xl,nPXl (n) for all lN under which conditions does it holds that limnnl=1Xl,nPl=1Xl as n?

Question in special case

In my particular use case something more specific would also suffice. Though, I think the more general question is also interesting. If we know that Xl,n, lN, are nonnegative random sequences, i.e. each sequence element is a nonnegative random variable, and Xl,nP0 (n) for all lN under which conditions does it holds that limnnl=1Xl,nP0 as n? Possibly, what would also be interesting and might be easier to answer is when the sample mean converges to zero, i.e., limn1nnl=1Xl,nP0 as n

removed capitals, avoided abbreviation
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YCor
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Convergence in Probabilityprobability of series of random variables

From the standard literature it is well known that for sequences of random variables X1,nPX1 and X2,nPX2 as n it holds that (X1,n,X2,n)P(X1,X2) for n. Using a continuous mapping theorem argument this can be used to establish that X1,n+X2,nPX1+X2 for n.

Question in General Casegeneral case

Under which conditions can the arguments of the standard literature be extended from a simple sum of two sequences to a sum of sequences whose number of summands also increases to infinity as n goes to infinity. Specifically, given a sequences Xl,n, l=1,,n, with Xl,nPXl for all l=1,,n under which conditions does it holds that limnnl=1Xl,nPl=1Xl as n?

Question in Special Casespecial case

In my particular use case something more specific would also suffice. Though, I think the more general question is also interesting (imo). If we know that Xl,n, l=1,,n, are nonnegative random sequences, i.e. each sequence element is a nonnegative random variable, and Xl,nP0 for all l=1,,n under which conditions does it holds that limnnl=1Xl,nP0 as n?

Convergence in Probability of series of random variables

From the standard literature it is well known that for sequences of random variables X1,nPX1 and X2,nPX2 as n it holds that (X1,n,X2,n)P(X1,X2) for n. Using a continuous mapping theorem argument this can be used to establish that X1,n+X2,nPX1+X2 for n.

Question in General Case

Under which conditions can the arguments of the standard literature be extended from a simple sum of two sequences to a sum of sequences whose number of summands also increases to infinity as n goes to infinity. Specifically, given a sequences Xl,n, l=1,,n, with Xl,nPXl for all l=1,,n under which conditions does it holds that limnnl=1Xl,nPl=1Xl as n?

Question in Special Case

In my particular use case something more specific would also suffice. Though, the more general question is also interesting (imo). If we know that Xl,n, l=1,,n, are nonnegative random sequences, i.e. each sequence element is a nonnegative random variable, and Xl,nP0 for all l=1,,n under which conditions does it holds that limnnl=1Xl,nP0 as n?

Convergence in probability of series of random variables

From the standard literature it is well known that for sequences of random variables X1,nPX1 and X2,nPX2 as n it holds that (X1,n,X2,n)P(X1,X2) for n. Using a continuous mapping theorem argument this can be used to establish that X1,n+X2,nPX1+X2 for n.

Question in general case

Under which conditions can the arguments of the standard literature be extended from a simple sum of two sequences to a sum of sequences whose number of summands also increases to infinity as n goes to infinity. Specifically, given a sequences Xl,n, l=1,,n, with Xl,nPXl for all l=1,,n under which conditions does it holds that limnnl=1Xl,nPl=1Xl as n?

Question in special case

In my particular use case something more specific would also suffice. Though, I think the more general question is also interesting. If we know that Xl,n, l=1,,n, are nonnegative random sequences, i.e. each sequence element is a nonnegative random variable, and Xl,nP0 for all l=1,,n under which conditions does it holds that limnnl=1Xl,nP0 as n?

Source Link
AlbertRapp
  • 299
  • 1
  • 7

Convergence in Probability of series of random variables

From the standard literature it is well known that for sequences of random variables X1,nPX1 and X2,nPX2 as n it holds that (X1,n,X2,n)P(X1,X2) for n. Using a continuous mapping theorem argument this can be used to establish that X1,n+X2,nPX1+X2 for n.

Question in General Case

Under which conditions can the arguments of the standard literature be extended from a simple sum of two sequences to a sum of sequences whose number of summands also increases to infinity as n goes to infinity. Specifically, given a sequences Xl,n, l=1,,n, with Xl,nPXl for all l=1,,n under which conditions does it holds that limnnl=1Xl,nPl=1Xl as n?

Question in Special Case

In my particular use case something more specific would also suffice. Though, the more general question is also interesting (imo). If we know that Xl,n, l=1,,n, are nonnegative random sequences, i.e. each sequence element is a nonnegative random variable, and Xl,nP0 for all l=1,,n under which conditions does it holds that limnnl=1Xl,nP0 as n?