Let A(n) be a sequence of i.i.d. topical (i.e. isotone and additively
homogeneous) operators. Let x(n,x0) be defined by x(0,x0)=x0 and
x(n,x0)=A(n)x(n−1,x0). This can modelize a wide range of systems including,
task graphs, train networks, Job-Shop, timed digital circuits or parallel
processing systems. When A(n) has the memory loss property, we use the spectral
gap method to prove limit theorems for x(n,x0). Roughly speaking, we show
that x(n,x0) behaves like a sum of i.i.d. real variables. Precisely, we show
that with suitable additional conditions, it satisfies a central limit theorem
with rate, a local limit theorem, a renewal theorem and a large deviations
principle, and we give an algebraic condition to ensure the positivity of the
variance in the CLT. When A(n) are defined by matrices in the \mp semi-ring, we
give more effective statements and show that the additional conditions and the
positivity of the variance in the CLT are generic