Approximate Fixed Point

In functional analysis and topology, the very important and very basic theorem is the fixed point theorem. It has been extensively used to guarantee the existence of the solution of some equation systems. Economists like Debreu and Nash borrow the idea of fixed point to shape the existence of the general equilibrium in economic theory. Along with assumptions about consumer preferences and production techlogoly, fixed point theory, such as Brower fixed point theorem and Kakutani fixed point theorem are utilized to prove the existence of the equilibrium. Nash applies the similar ideas in Game Theory to show the existence of Nash equilibrium.

You might feel overwhelmed by the statements above if you have never get to know these background economic nuisance. To state fixed point theorem simply, as cited from Wikipedia, "it says that any continuous function from the closed unit ball in n-dimensional Euclidean space to itself must have a fixed point." See http://en.wikipedia.org/wiki/Fixed_point_theorem
To give a simple example, when we put the World map on the desk, then there should be one point on the map that exactly coincides with the exact corresponding geographical real point.

However, albeit useful in expanding areas of scientific research today, it has its limitations. The fixed point theorem deals with deterministic functions. There has been a litterature trying to incorporate randomness into fixed point theorem. Bharucha-Reid (1976) has a great review of this litterature and Tan and Yuan (1993) explore into the relationship between the deterministic and random fixed point theorem. Random fixed point theorem introduce a stochastic sample space and make the function and variable all dependent on the realizatioin of the sample space. That is, instead of having f(x)=x, we have f(w,x(w))=x(w), equavalent to say that x(w) is a fixed point of f. Some conditions sufficient for the existence of random fixed point have been explored since 1950s.

Though as if it is complicated, with a realization of w, the random fixed point theorem degenerates to fixed point theorem in deterministic case. Another way of extending the fixed point theorem into stochastic situation is what called approximate fixed point. This is a new term I'd like to introduce to the litterature. What term means is that, instead of getting x via f(.) when input is x, we have f(x)->x in probability, as some index goes to infinity. This would be of great interest as many functions we have in Econometrics are random functions. It would be hard to get f(x)=x. While when f(x) approaches x in probability, some properties can be established. For an illustration, iteration via f(.), will give x when sample size is large enough, with an input of x. The idea is rather new and not much can be said at the moment. However, I believe that this will be an important point in probability, and especially in Econometrics, in a few years time.

When you can not get the exact solution to the problem anoying you, try to circumvent to an approximate one!