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enter image description here

Picture is from Ross' Introduction to Probability Models, 11th ed.

I understand the definition of $[t/\Delta t]$, I just don't see how it connects to the position at time $t$ (eq. 10.1). For $\Delta t$ small, won't $t/ \Delta t$ be itself much larger than $t$? How does that coincide with the position at time $t$? Sorry for my confusion.

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Note that $X_n$ denotes the direction you move on the $n$th step, not at time $n$. So the fact that the sum goes up to $[t/\Delta t]$ means that you take $[t/\Delta t]$ steps. Why does this make sense? Well, you take a step every $\Delta t$ units of time. So by time $t$, $[t/\Delta t]$ is exactly the number of steps you will have taken.

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