Basic Modelling in Quantitative Genetics
Trait z is represented by k genes: z1....zk (I am using k instead of l because the former is visually differentiable from 1 ).
For simplicity let's assume that there is only one mutable site in a gene. So a mutation can impart a change of $\pm \Delta z$. Starting from initial state of z at zero, the system will proceed to equilibrium where the rate of forward mutation would be same as that of backward mutation; since the rates are same the deterministic steady state of z should be zero. 

A mutation event can also be considered a binomially distributed RV. After n events, the mean number of forward mutations would be 0.5×n.   Mean fitness = mean forward mutation — mean backward mutations = 0 

You can also model the mutation events as a simple random walk. The mean of that is 0 and the variance is n. But what about large n; I am not sure. You should read about random walks or ask in CrossValidated. With $t\to \infty$, you would get a lot of variants but variants would be filtered out by selection (as the optimum is at 0). I think selection can be incorporated in the random walk with death rate being proportional to the distance from 0. But again I do not have much experience with such hybrid models. 

I tried a simulation with n=10000000 (using Monte Carlo); different colours denote different runs (or in another words different loci). Y-axis denotes the value of z.