Dear all,
I am searching a result concerning the weak convergence of signed measures defined on the Borel algebra of a given metric space. The result that i find so far is established for weak convergence of probability measures (please see Portmanteau theorem page 11; book attached).
Does the same result apply for signed measures? If not, what could be the possible differences in the equivalent conditions?
(providing me with good references would be appreciated)
Sincerely,
Nathalie
Dear Nathalie,
I had a look at Theorem 2.1 on page 11 of the book attached to your message. Some of items do not look possible to generalize because if you multiply probability measures by -1, all the inequalities reverse. Less evident is the situation with the item (v), but there is an easy example demonstrating that (v) does not follow from (i). Namely, consider S=[0,1], and define xn < 1/2 and yn>1/2 in such a way that both xn and yn converge to 1/2 as n goes to infinity. Consider the sign measure mn which is the difference of the unit mass at xn and the unit mass at yn. Then mn goes weakly to P=0, but (v) is not valid (take A = (0, 1/2)).
All the best, Vladimir
Dear Nathalie,
I had a look at Theorem 2.1 on page 11 of the book attached to your message. Some of items do not look possible to generalize because if you multiply probability measures by -1, all the inequalities reverse. Less evident is the situation with the item (v), but there is an easy example demonstrating that (v) does not follow from (i). Namely, consider S=[0,1], and define xn < 1/2 and yn>1/2 in such a way that both xn and yn converge to 1/2 as n goes to infinity. Consider the sign measure mn which is the difference of the unit mass at xn and the unit mass at yn. Then mn goes weakly to P=0, but (v) is not valid (take A = (0, 1/2)).
All the best, Vladimir
Dear Followers of this question,
Let me add, that for signed finite measures on the special space, which is the real line R, a necessary condition for the weak convergence is aditionally the point-wise convergence of their characteristic functions (ch.f.-s).
Nota bene: Vladimir's answer provides an interesting example: if $x_n \to 0+$ and $y_n \to 0-$, then the characteristic functions (ch.f.-s) of the measures $\delta_{x_n} - \delta_{y_n}$ are equal to $\exp(i x_n t ) - $\exp(i y_n t ) $ and approach 0 , as $n\to \infty$.
Question: is the pointwise convergence of the characteristic functions of uniformly bounded signed measures on R to a characteristic function of a signed (finite) measure sufficient for their weak convergence? Positive answer would extend the corresponding inverse continuity theorem for probability measures (p.m.-s), which can be formulated as follows (c.f. Patric Billingsley's book: Probability and measure):
If the pointwise limit function f of the characteristic functions f_{P_n) of probability measures $P_n$ is a ch.f. of a p.m.Q, then Q is the weak limit of the sequence of p.m.-s $P_n$
Another question: is there some extension to signed measures of the inverse continuity theorem of the following form?
If the pointwise limit function $ f$ of the characteristic functions $f_{P_n)$ of probability measures $P_n$ is continuous, then $f$ is a ch.f of a p.m., say $Q$, and $Q$ is the weak limit of the sequence of probability measures $P_n$.
Regards
3 Recommendations
2nd Feb, 2017
Nathalie T. Khalil
University of Porto
It is exactly what I am searching for. Thank you all for your helpful answers.
Sincerely,
Nathalie
1 Recommendation
2nd Feb, 2017
Joachim Domsta
The State University of Applied Sciences in Elbląg,
Dear All Followers of this question,
let me give answer to my subproblem by presenting a counterexample, where $\delta(a)$ stands for the p.d. concentrated at $a$, for $a \in R$:
For $n \in Z_0=\{...,-2,-1, 1,2,3, . . . \}$ let the coefficients be
$b_n : = sign(n) \sqrt{ \pi |n|}$ and let the signed measures be
$Q_n := ( -\delta(b_{-2 n-1} +\delta(b_{-2 n} + \delta(b_{2 n} - \delta(b_{2 n+1} $.
Their characteristic functions equal
$ f_n(t) = 2 [ \cos(t \sqrt{2n\pi} - \cos(t \sqrt{(2n+1)\pi} ] $
By elementary trigonometry, $f_n(t) \to 0$, for all $t \in R$.
On the other hand the integrals of the obviously continuous and bounded function $\cos(x^2), x \in \R$ with respect to $Q_n$ are all equal to
$ \int_R \, cos(x^2) \, dQ_n(x) = 2 [ \cos ((b_{2n})^2) - \cos ((b_{2n+1})^2) $,
which equals 4, for every $n \in N$. Thus the measures $Q_n$ are not weakly convergent to the zero measure, although absolute variations of the measures are bounded (equal 4).
Thus, thank you Natalie for your question! Without it we could miss this special feature of signed measures.
Joachim
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