# is it possible to make changes to use the affine property of Normal random variables, rather than the Central Limit Theorem?

I have proven the distribution of a discrete time model, evolving over a uniform mesh with $$\delta t = T/L$$ is given by

$$S(t_{i+1}) = S(t_i) + \mu \delta t S(t_i) + \sigma\sqrt{\delta t}S(t_i)Y_i,$$

for $$i = 0, . . . , M − 1$$, where $$Y_i$$ is an i.i.d. N(0, 1) sequence, converges to that of a log-normal random variable as $$\delta\to0$$(and hence as $$L\to \infty$$). This required an application of the Central Limit Theorem.

So now if I let $$X_1, X_2, . . . , X_n$$ be a set of independent Normal random variables, with means $$\mu_i$$ and variances $$θ^2_i$$ for $$i = 1, . . . n$$ respectively is it possible to make changes to use the affine property of Normal random variables, rather than the Central Limit Theorem?