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Merging with respect to bounded uniformly continuous functions in terms of characteristic functions

1

1

$begingroup$

I would like to know if there are any results, where merging
of probability measures in $R^n$ with respect to bounded uniformly continuous functions is deduced from some conditions on characteristic functions?

In partucular, is it true that if for all t $f_n(t)-g_n(t)$ converges to zero, when $n$ goes to infinity, where $f_n(t)=int e^itx dP_n(x)$ and $g_n(t)=int e^itx dQ_n(x)$, then sequences of measures $P_n$ and $Q_n$ merge wrt bounded uniformly continuous functions?

(sequences $P_n$ and $Q_n$ of probability measures merge wrt bounded uniformly continuous functions if for every bounded uniformly continuous function $f$ $(int f dP_n)-(int f dQ_n)$ converges to zero when n goes to infinity)

probability weak-convergence characteristic-functions

share|cite|improve this question

edited Mar 18 at 11:14

ssss nnnn

asked Mar 18 at 9:59

ssss nnnnssss nnnn

62

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I don't understand the question. You have defined merging for sequences of probability measures and you are asking if sequences of sets merge.
  $endgroup$
  – Kavi Rama Murthy
  Mar 18 at 10:03
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Sorry for improperly formulating the question. I have edited it, is it now easier to understand?
  $endgroup$
  – ssss nnnn
  Mar 18 at 10:45
  

  
  
  
  

add a comment | 

1

1

$begingroup$

I would like to know if there are any results, where merging
of probability measures in $R^n$ with respect to bounded uniformly continuous functions is deduced from some conditions on characteristic functions?

In partucular, is it true that if for all t $f_n(t)-g_n(t)$ converges to zero, when $n$ goes to infinity, where $f_n(t)=int e^itx dP_n(x)$ and $g_n(t)=int e^itx dQ_n(x)$, then sequences of measures $P_n$ and $Q_n$ merge wrt bounded uniformly continuous functions?

(sequences $P_n$ and $Q_n$ of probability measures merge wrt bounded uniformly continuous functions if for every bounded uniformly continuous function $f$ $(int f dP_n)-(int f dQ_n)$ converges to zero when n goes to infinity)

probability weak-convergence characteristic-functions

share|cite|improve this question

edited Mar 18 at 11:14

ssss nnnn

asked Mar 18 at 9:59

ssss nnnnssss nnnn

62

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I don't understand the question. You have defined merging for sequences of probability measures and you are asking if sequences of sets merge.
  $endgroup$
  – Kavi Rama Murthy
  Mar 18 at 10:03
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Sorry for improperly formulating the question. I have edited it, is it now easier to understand?
  $endgroup$
  – ssss nnnn
  Mar 18 at 10:45
  

  
  
  
  

add a comment | 

$begingroup$

I would like to know if there are any results, where merging
of probability measures in $R^n$ with respect to bounded uniformly continuous functions is deduced from some conditions on characteristic functions?

In partucular, is it true that if for all t $f_n(t)-g_n(t)$ converges to zero, when $n$ goes to infinity, where $f_n(t)=int e^itx dP_n(x)$ and $g_n(t)=int e^itx dQ_n(x)$, then sequences of measures $P_n$ and $Q_n$ merge wrt bounded uniformly continuous functions?

(sequences $P_n$ and $Q_n$ of probability measures merge wrt bounded uniformly continuous functions if for every bounded uniformly continuous function $f$ $(int f dP_n)-(int f dQ_n)$ converges to zero when n goes to infinity)

probability weak-convergence characteristic-functions

share|cite|improve this question

edited Mar 18 at 11:14

ssss nnnn

asked Mar 18 at 9:59

ssss nnnnssss nnnn

62

$endgroup$

share|cite|improve this question

edited Mar 18 at 11:14

ssss nnnn

asked Mar 18 at 9:59

ssss nnnnssss nnnn

62

share|cite|improve this question

edited Mar 18 at 11:14

ssss nnnn

asked Mar 18 at 9:59

ssss nnnnssss nnnn

62

asked Mar 18 at 9:59

ssss nnnnssss nnnn

62

asked Mar 18 at 9:59

ssss nnnnssss nnnn

62