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Kesten et al.( 1975) proved the stable law for the transient RWRE (here werefer it as the $\kappa$-transient RWRE). After that, some similar interestingproperties have also been revealed for its continuous counterpart, thediffusion proces in a Brownian environment with drift $\kappa$. In the presentpaper we will investigate the connections between these two kind of models,i.e., we will construct a sequence of the $\kappa$-transient RWREs and prove itconvergence to the diffusion proces in a Brownian environment with drift$\kappa$ by proper scaling. To this end, we need a counterpart convergence forthe $\kappa$-transient random walk in non-random environment, which isinteresting itself.

Scaling limit theorems for the $\kappa$-transient random walk in random and non-random environment