Consider the following two random walks:

The first random walk $\{S_n\}$ has i.i.d. step size $$ X_i\sim\mathcal{N}(1,1) $$ The second random walk $\{S'_n\}$ has i.i.d. step size $$ Y_i\sim\mathcal{N}(4,4) $$ And let $t_1,t_2$ be the minimizer of the two random walks: $$ t_1=\arg\min_n S_n $$ $$ t_2=\arg\min_n S'_n $$

I'm now interested in the distribution of $t_1$ and $t_2$, are they kind of a rescale of each other? It makes some sense since a step in random walk $2$ is the same as four steps in random walk $2$.

However, I don't know how to prove or disapprove this, any help would be appreciated :-)