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Suppose we have a random walk where each step's distance $ X \in N$ is drawn from a uniform distribution:

$$X \sim \text{Uniform}\{1,\dots,n\}.$$

Each step takes exactly 1 second, and we denote by $ T$ the time (in seconds) it takes for the cumulative sum of steps to first exceed the boundary $n$. That is:

$$T = \min \left\{ t : \sum_{i=1}^t X_i > n \right\},$$

where $X_i$ are i.i.d. random variables drawn from $ \text{Uniform}\{1,\dots,n\}$.

My reasoning is as follows:

1)The expectation of a single step distance is: $$ \mathbb{E}[X] = \frac{n}{2}. $$
2) After $2$ steps, the expected cumulative distance is:

$$ \mathbb{E}\left[\sum_{i=1}^2 X_i\right] = 2 \cdot \frac{n}{2} = n. $$

Thus, it seems intuitive that the expected time $\mathbb{E}[T]$ to cross the boundary $n$ is: $$ \mathbb{E}[T] = 2. $$ Is this reasoning correct? If not, what is the correct value of $\mathbb{E}[T]$, and how can it be derived formally?

Mike Earnest
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Yarden Tziar
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  • Your reasoning is not quite correct. This is because $T$ and $X_i's$ are dependent random variables. The correct approach can be found here: https://math.stackexchange.com/questions/214399/summing-0-1-uniform-random-variables-up-to-1 – sudeep5221 Dec 31 '24 at 15:32
  • could the contrast between the uniform distribution over $(0,1)$ and $[0,n]$, such as implications of Kantor's theorem, influence the result? – Yarden Tziar Dec 31 '24 at 16:04
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    Yes, adding up discrete random variables makes it a different problem, but that different problem has also been asked before: https://math.stackexchange.com/q/3927657/ – Mike Earnest Dec 31 '24 at 17:55
  • Clarification question: when you say Uniform$(0,n)$, do you mean that $X$ is equally likely to be any number in the set ${1,\dots,n-1}$? Or did you intend the set ${1,\dots,n}$? – Mike Earnest Dec 31 '24 at 17:57
  • $X$ , the distance of a step, is equally likely to be any number in the set {1, …, n}, where n is a natural number representing the upper bound. thank you for the link above , it make sense. – Yarden Tziar Dec 31 '24 at 18:09

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