Interesting Problem: Probability Theory Limit

[csopi]

Hi everybody,

On a probability theory lecture I met an interesting problem:

Let f : [0;1] -> R a bounded, 3 times continously differentiable function. Calculate the following limit

n * \int_0^1 \int_0^1 ... \int_0^1 f(\frac{x_1+...+x_n}n)-f(0.5) dx_1...dx_n while n -> infinity

Any ideas?

 

[statdad]

As a suggestion: treat the first term as a function of x_1, x_2, ..., x_n, and use the fact that f is three times differentiable to expand the integrand in a multi-variable Taylor polynomial centered at 0.5. Simplify, use the fact that f is bounded to help with the error term, and try to get an expression for the multiple integral at n fixed. IF you can do that, multiply by n and take the limit.

Disclaimer: I haven't tried this, so I cannot give any insight on its correctness or lack thereof, nor of its level of difficulty. It seems like the reasonable approach (thinking back to my probability/math stat courses), but (as my father used to say) it may be "as useless as a hog on ice". Let us know what you find.

 

[mXSCNT]

Looks like the change of variables y = (x1 + x2 + ... + xn)/n might be a good first step. You integrate like:
\int_0^1 M(y)(f(y) - f(0.5)) dy
The tricky part is figuring out the multiplier M(y). For n=2 the multiplier is 2 sqrt(2) min(y, 1-y). For n=3 it is a little more complicated. M(y) is the "area" of the hyperplane portion (x_1 + x_2 + ... + x_n)/n = y, where 0 <= x_i <= 1.

Assuming that the limit as n -> infinity is 0, perhaps all you need to prove about M(y) is that it bulges in the middle more and more as n increases, so eventually nearly all of the integral ends up being near y=0.5.