Why random walk sample path seen as if it is continuous-time stochastic process?

Question

Random walk is a discrete-time stochastic process. In many references, instead of using dots to draw its sample path, why does random walk use line-styled graph as if it is a continuous-time stochastic process?

Answer 1

It's easier on the eyes and gives a clearer picture of what's going on. It's true that it's discrete, so the connected line graph is not entirely correct. On the other hand, under the right conditions a random walk can be rescaled to be an approximately continuous process such as brownian motion. In other words, if your step size is small, and your jump size is small, one can make a continuous approximation which becomes exact (in a tricky sense) when you take the step size to zero.