Figure 1.13: Arbitrary snapshot of the particle under going random walk

In case we simplify the above notations then we consider the case where at the starting of the process, i.e., , the particle is at any given point say, , such that . At time  the particle undergoes a jump of quantum Z₁ (which is a random variable with a particular distribution). Furthermore at time  the same particle undergoes another jump Z₂ where Z₂ is independent of Z₁ but has the same distribution as Z₁ . Thus the particle undergoes jumps in a manner that after end of the first time period, it is at the position , after the second time period,  it is at the position . Thus for ,  is a sequence of mutually independent and identically distributed random variables.

More generally ,  and in case , or  or , then what is of interest to
us is ,  and , such that . There may be instances where , in which case we should have