Suppose we have a symmetric simple random walk and the step is 1 and -1. Obviously, if we take N steps, the PMF of position k for N steps is given by $${{N}\choose{\frac{k+N}{2}}} \left(\frac{1}{2}\right)^N$$ And if N is large, this follows the normal distribution with $\mu = 0$ and $\sigma = \sqrt{N}$. But how can I obtain the distribution of position for a certain random walk.
For example, we ran a 10000 steps random walk simulation, and the hist is as follows for each position.
But it was not a normal distribution, and it was thinner than normal distribution.
Is there a way to use a distribution function to generate the approximate hist without doing random walk simulation？ In this way, it is more efficient. And we could obtain the approximate maximum and minimum without doing simulation.
I checked many papers, but no answer exists.
Thank you in advance.