Recent content by Epsilon36819

Hi everyone. I was going through a proof of Wald's equation, where it was claimed that if {S_n} is a sequence defined as S_n = \sum_1^{n} Y_i where the Y_i are iid with finite mean \mu, then Z_n = S_n - n \mu is a martingale. But I don't see why... at all! Help!

Here goes: If F is a probability distribution function and /phi is its integrable characteristic function. If the mean of F exists, why can we say that there exists u>0 st int[abs(1- /phi(t))/t] < infinity, where the integral is over the set of all t st abs(t)<u ? (abs = absolute value)...

Hi everyone, I am currently writing my personal statement for the application to grad school in pure mathematics. I am aware that grad school comitees usually want to hear about the specifics of your research experience: the project, your role in it, etc. However, my only research...

Thanks mathwonk, I'll have a look at this book.

Mathwonk, I have a question for you. There is this graduate class given next term which is a second course in topology. The first class was given this term and I unfortunately couldn't take it, as it overlapped a core course for my degree. This first course covered the basic of topology...

It can be a subset of a vector space.

Can't we use an epsilon/3 argument, just as in the proof that uniform convergence of continuous functions => continuity of the limit? In this case, the given delta valid for fn will be the one used for f, so f will be part of the equicontinuous family. Now that I think of it, though, I don't...

I am having trouble accepting two well known results of analysis as non contradictory. First, given a vector space equipped with norm ||.||, the unit ball is compact iff the space is finite dimensional. Second, the Arzela-Ascoli theorem asserts that given a compact set X, a set S contained...

Hi Calleigh, It may help to visualize the function as a surface in 3D, where every point (x,y) of the x-y plane is mapped to a function z=f(x,y). If you fix y, say to 3, you can "cut the plane" at that particular y and obtain a function z=f(x,3), ie a function of one variable which can be...

Ah! Yes, of course. Good one!

It's not obvious to me why this needs to be so.

Right. I knew how to prove the cases d(f(x),f(y))<d(x,y) (as in post 5) as well as d(f(x),f(y))<= c*d(x,y) (as in the first part of HallsofIvy's post), but I was looking for other proofs. Actually, I'll keep in mind your method and I'll get back to it once I have the proper math background.

The (wrong) "proof" I initially gave was for the case d(f(x),f(y))<d(x,y). But it doesn't hold for the reason I gave in post 2 (and 13). For the case c<1, HallsofIvy's outline does it.

Since X is compact, every sequence has a convergent subsequence. But nothing tells us that any particular convergent subsequence satisfies f(x_{n})= x_{n+1}.

Mmm... that's all a little over my head right now. I'll meditate on it.