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A particle moves randomly over the integer points of the real line. Jumps ofthe particle outside the membrane (a fixed "locally perturbating set") arei.i.d., have zero mean and finite variance, whereas jumps of the particle fromthe membrane have other distributions with finite means which may be differentfor different points of the membrane; furthermore, these jumps are mutuallyindependent and independent of the jumps outside the membrane. Assuming thatthe particle cannot jump over the membrane we prove that the weak scaling limitof the particle position is a skew Brownian motion with parameter $\gamma\in[-1,1]$. The path of a skew Brownian motion is obtained by taking eachexcursion of a reflected Brownian motion, independently of the others, positivewith probability $2^{-1}(1+\gamma)$ and negative with probability$2^{-1}(1-\gamma)$. To prove the weak convergence result we offer a newapproach which is based on the martingale characterization of a skew Brownianmotion. Among others, this enables us to provide the explicit formula for theparameter $\gamma$. In the previous articles the explicit formulae for theparameter have only been obtained under the assumption that outside themembrane the particle performs unit jumps.

A functional limit theorem for locally perturbed random walks