But moment picking is tricky too.

This is the second part of the story that I started with the previous post. Wanting to build a quick counterexample, I wondered if I can create a random variable by solving the moment problem with only finite non-zero moments. In the language of the moment problem, my sequence was such that for all greater than some number . That is, only finite elements of … Continue reading But moment picking is tricky too.

Going moment-picking

Yesterday, while trying to build a counter-example for a research problem, I started playing with the idea of designing a random variable, not by picking a vanilla distribution, but by selecting its moments myself. That is, if I pick the moments myself, can I find a distribution that has them? More formally, if I provide , is there a : probability measure such that   … Continue reading Going moment-picking Solving a problem by adding 0 and multiplying by 1

Has it ever occurred to you that when solving a problem, a non-trivial step might make the problem trivial? Well, if you are a mathematician, it probably has. The only difference might be the number of steps needed to turn the problem into something easier. Take for example the following problem which I solved yesterday in a forum on which I have the privilege to … Continue reading Solving a problem by adding 0 and multiplying by 1

Kinds of convergence for sequences of random variables

This is a quick post on the different kinds of convergence that a sequence of random variable can have in a probability space. Let us suppose that we have a sequence of random variables , in a probability space .   is the probability space, is the σ-algebra of measurable sets and P is the probability measure. First of all, here’s a list of the … Continue reading Kinds of convergence for sequences of random variables

Adding past infinity : A mathematician’s point of view

In case you didn’t know, I’m a fan of the short youtube videos given by the user minutePhysics. The guy has a knack for exploring and explaining physics concepts in a simple way. But let’s cut to the chase. In one of his videos, he makes the following claim : You can see that video here. Of course, the internet does not forgive “errors” easily. … Continue reading Adding past infinity : A mathematician’s point of view

Neumann Series

So, I have a gut feeling this result might be important enough to remember. Let X be a Banach space, a space where every Cauchy sequence converges, and let L(X) be the set of all bounded operators from X to X. If we have an operator and suppose , then we can say a couple of interesting stuff about the operator, with my favourite being … Continue reading Neumann Series