  In this article, we first give a comprehensive description of random walk
(RW) problem focusing on self-similarity, dynamic scaling and its connection to
diffusion phenomena. One of the main goals of our work is to check how robust
the RW problem is under various different choices of the step size. We show
that RW with random step size or uniformly shrinking step size is exactly the
same as for RW with fixed step size. Krapivsky and Redner in 2004 showed that
RW with geometric shrinking step size, such that the size of the $n$th step is
given by $S_n=\lambda^n$ with a fixed $\lambda<1$ value, exhibits some
interesting features which are different from the RW with fixed step size.
Motivated by this, we investigate what if $\lambda$ is not a fixed number
rather it depends on the step number $n$? To this end, we first generate $N$
random numbers for RW of $t=N$ which are then arranged in a descending order so
that the size of the $n$th step is $\lambda_n^n$. We have shown, both
numerically and analytically, that $\lambda_n=(1-n/N)$, the root mean square
displacement increases as $t^{1/4}$ which are different from all the known
results on RW problems.
