What random variable is this?

Question

I have a sequence of reals $S = s_1,s_2,\dots,s_n$ such that $s_i-s_{i-1}$ is a Gaussian distribution. From histogram of sequence $S$ (10000 elements) it appears that it is uniform distribution. Is it true ? If yes, can we prove it ? If no, what can we say about $s_i$ ?

Answer 1

Define $d_n = s_n - s_{n-1}$. Then $$\begin{align} s_n &= s_{n-1} + d_n\\ &= s_{n-2} + d_{n-1} + d_n\\ &\cdots\\ &=s_0 + (d_1 + \dots + d_n) \end{align} $$

For definiteness, suppose $s_0$ is negligible. Now if the $d_i$ are jointly Gaussian, then their sum (and hence $s_n$) is also Gaussian, with parameters easily calculated from those of the joint distribution:

$$\operatorname{E}\left(s_n\right) = \operatorname{E}\left(\sum_{i=1}^n d_i\right) = \sum_{i=1}^n \operatorname{E}(d_i)$$ $$\operatorname{Var}\left(s_n\right) = \operatorname{Var}\left(\sum_{i=1}^n d_i\right) = \sum_{i=1}^n \operatorname{Var}(d_i) + 2\sum_{1\le i}\sum_{<j\le n}\operatorname{Cov}(d_i,d_j).$$

Answer 2

The marginal distribution of $S$ cannot be uniform: If it were, the distribution of $s_i-s_{i-1}$ would have finite support. But the Gaussian distribution has infinite support, hence $S$ cannot have a uniform marginal.

If I should guess, I would say that $S$ is an autoregressive process.