# Bounds on distribution of sum

1. Mar 2, 2009

### bpet

Take any distribution function F(x) where the n-fold convolution F_n(x) is unknown or difficult to calculate. Here

$$F_{k+1}(x) = \int_{-\infty}^{\infty}F_k(x-t)dF(t).$$

Are there any good techniques for estimating bounds on F_n(x), given F(x) ?

Suppose the distribution does not have finite moments?

2. Mar 8, 2009

### bpet

Ok so one approach to derive bounds on F_n(x) is to break up (-inf,inf) into (-inf,0), (0,x), and (x,inf). Using the fact that F_n(x) is non-decreasing, we get

$$F_k(x)F(0) + F_k(0)(F(x)-F(0)) \le F_{k+1}(x) \le F(0)+F_k(x)(F(x)-F(0))+F_k(0)(1-F(x))$$

for x>=0 and thus

$$F(0)^{n-1}(n(F(x)-F(0))+F(0)) \le F_n(x) \le 1-(1-F(0))^n+(F(x)-F(0))^n$$

however these bounds rapidly tend to 0 and 1 respectively as n increases. Are any tighter bounds known?