# How to simulate a sequence of partial sums $(X)_n(w) = \sum\limits_{i=1}^n (Y_i(w)-Y_{i-1}(w)),$ given some properties.

I want to generate/simulate a sequence of partial sums.

$$(X)_n(w) = \sum\limits_{i=1}^n (Y_i(w)-Y_{i-1}(w)),\text{ for }1 \leq n \leq 100$$

Let $W$ be a random variable such that: $W \thicksim N(0,1)$.

I know that: $\forall i, (Y_i(w)-Y_{i-1}(w))$ has the same distribution as $W$.

However, I don't know anything about how the individual $Y_i$ random variables are distributed, other than to say that they are, together with W, defined on the same probability space.

I'm really confused by a number of aspects about this question. In order to generate the partial sums, is it sufficient to take n independent draws out of a standard normal and then calculate the partial sums of these as my sequence $(X)_n(w)$?

But then, from the definition of the partial sums above, it is clear that $w$ is fixed throughout the partial sum. What does that mean for me?

My intuitive response is to ignore the fixed $w$. While I can draw on my knowledge of the distribution of $Y_i - Y_{i-1}$ to draw randomly from that distribution, I don't actually know anything about what their underlying functions look like. And so, for different values of $i$, $(Y_i(w) - Y_{i-1}(w)$ may map the same $w$ to vastly different values.

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