Asymptotic Meaning

[Abhishek...]

I have come across several instances of the word asymptotic and I have Googled it only to get more confused, I am putting three terms that I have encountered please could anybody explain.
Asymptotic Dependence, Asymptotic Independence, Asymptotic Distribution

I understand that a higher level of knowledge is required but for exam purposes how much should I know also for later if somebody could name some books that go beyond the mathematics of FRM.


ps. the definition at http://mathworld.wolfram.com/Asymptotic.html only confused me more.

 

[Alex_1]

Hi Abhishek, I am normally not a huge fan of referencing Wikipedia, but in this case I found there a helpful pdf document:

http://rowdy.msudenver.edu/~talmanl/PDFs/APCalculus/OnAsymptotes.pdf

from which I post an interesting quote:

<<an asymptote is “A line such that a point, tracing a given curve and simultaneously receding to an infinite distance from the origin, approaches indefinitely near to the line; a line such that the perpendicular distance from a moving point on a curve to the line approaches zero as the point moves off an infinite distance from the origin. [...]

An asymptote is a tangent at infinity, i.e., a line tangent to (touching) the curve at an ideal point.”>>


The basic concept is (in my eyes) that, for example an asymptotic distribution (or the graphical depiction hereof) approaches up to a certain pre-defined level another distribution.

But I am not a mathematician and am basing my statements only on my high school calculus knowledge and on the almighty google.

I hope it helps.

 

[joacogimeno]

I don't think you should worry about it at all.

An asymptote is a straight line that a curve approaches but never crosses. An example is f(n)= 1/n. As n becomes very large, this function approaches but never touches "0". Or you may say that it only touches 0 at infinity, but this might be confusing. Y=0 is the asymptote for this function.

Asymptotic analysis deals with this kind of limiting behaviour for infinite functions. Two common examples are the Central Limit Theorem and Taylor's expansion (or any power series). For the latter, as you may see throughout the FRM curriculum, in the "real world" we truncate a function after the second term (Duration and Convexity, in the case of fixed income products), as a simpler approximation of the behaviour of the real series.

Imagine you have the function f(x)= x^3 + 2x. As x becomes large we can "ignore" the second term and say that g(x) = x^3 is asymptotically equivalent to f(x), or in other words, g(x) is the asymptotic distribution to which f(x) converges, so when we want to deal with the tail of the distribution (as "x" becomes large", we may as well use the second, simpler, function.

Anyway, don't worry about it for the test.

 

[joacogimeno]

By the way (@Alex_1), I am a Mathematician

 

[cdbsmith]

I like illustrations...especially as they relate to mathematical/statistical concepts.

http://www.mathsisfun.com/algebra/asymptote.html