In Stein and Shakarchi's functional analysis,
We recall the random walk in $\mathbb{R}^d$...is given by a sequence $\{s_n\}_{n=1}^\infty$ where $$s_n = s_n(x) = \sum_{k=1}^n \tau_k(x),$$ with $s_n(x) \in \mathbb{Z}^d$ for each $x$ in the probability space $\mathbb{Z}_{2d}^\infty$. This probability space carries the probability measure $m$ which is the product measure on $\mathbb{Z}_{2d}^\infty$... Next we consider the rectilinear paths obtained by joining these successive points, and then rescale both time and distance, so that between two consecutive steps the elapsed time is $1/N$ and the traversed distance is $1/N^{1/2}$, all in accordance with our experience with the central limit theorem. That is, for each $N$ we consider $$S_t^{(N)}(x) = {1 \over N^{1/2}} \sum_{1 \le k \le [Nt]} \tau_k(x) + {(Nt) - [Nt] \over N^{1/2}} \tau_{[Nt] + 1} (x).$$
Here, $x \in X := \mathbb{Z}_{2d}^\infty$ and $\tau \in \mathbb{Z}_{2d} := \{(1, 0, \dots, 0), (-1, 0, \dots, 0), \dots, (0, \dots, 1), (0, \dots, -1)\}$ which is basically the set of unit steps you can take in a random walk on $d$ dimensional euclidean space. $\tau_k$ denotes the step taken at time $k$.
The first term in $S_t^{(N)}$ makes sense: I am scaling it by ${1 \over N^{1/2}}$ and by adding upto $[Nt]$, I'm scaling the time by ${1/N}$. I'm having trouble understanding the second term in the definition of $S_t^{(N)}$. In fact, it seems like the first term already satisfies what we want.
I also tried plotting the above form and got something quite weird:
I'd like to know why and what the second term is doing. Thanks in advance.