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From: Sayaka Fujisaki on 20 Apr 2010 20:11
Hello everyone,
I have a question about convergence of a function of some random
variables. In my problem, I have a function f(), which has the
property that

lim_{n goes to infinity} f(X_1, ..., X_n)/sqrt(n) = k

with probability 1, when the Xi's are i.i.d. samples drawn uniformly
from the unit square in the plane. Here f() can take any number of
arguments (in fact, f() is the length of a minimal travelling salesman
tour through the points X_i).

What I want to do now is consider a binomial random variable N ~
B(n,p) (with known parameters n and p), and I'd like to know if
there's any sense of convergence in which the statement

lim_{n goes to infinity} f(X_1, ..., X_N)/sqrt(np) = k

holds. Are there weaker notions of convergence for which this may
also be true?

Thank you! SF
From: Sayaka Fujisaki on 20 Apr 2010 20:34
After giving it a bit of thought, I believe my question can be reduced
to the following:

Suppose that X_1, ... , X_n is a sequence of random variables that
converges almost surely to some constant k. Also suppose that
Y_1, ..., Y_n is a sequence of random variables that converges almost
surely to 1. Is there a sense of convergence in which the sequence
{X_i * Y_i} must converge to k?

Thank you again! SF
From: Nate Eldredge on 21 Apr 2010 14:54
On Apr 20, 8:34 pm, Sayaka Fujisaki <psychoblad...(a)hotmail.com> wrote:
> After giving it a bit of thought, I believe my question can be reduced
> to the following:
>
> Suppose that X_1, ... , X_n is a sequence of random variables that
> converges almost surely to some constant k.  Also suppose that
> Y_1, ..., Y_n is a sequence of random variables that converges almost
> surely to 1.  Is there a sense of convergence in which the sequence
> {X_i * Y_i} must converge to k?

Um, is * ordinary multiplication? Then of course you get almost sure
convergence. (Multiplication is continuous...)
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