Distribution of positive and negative values in a Brownian bridge Recently there was a question about the occurrence of a large discrepancy in the differences between two ordered sequences of random numbers. The difference between these two sequences can be related to a process that approaches a scaled zero mean Brownian bridge and it is due to the correlation that there is a large variance in the number of negative and positive values.
I wonder, can we make an approximation of the distribution of positive and negative values?

A more formal description of the question:
Say we compute $T+1$ values $X_t$, with $t = 0,1,2,\dots,T$, as a multivariate Gaussian distribution with zero mean and the following covariance structure $$\Sigma_{s,t} = \text{min}(s,t) - \frac{st}{T}$$ That is, it has the same mean and covariance of a Brownian bridge. Then what is the distribution of the following?
$$S = \sum_{t=0}^T \text{sign}(X_t)$$

Example of a single simulated Brownian bridge:

set.seed(1)
x = c(0,rnorm(1000))
t = 0:1000
y = cumsum(x)-sum(x)*t/1000

S = sum(sign(y))


plot(t,y, main = "Example of Brownian bridge \n with 754 positive values and 245 negative values \n sum sign(y) = 509", cex= 0.4)

lines(t,t*0)


The distribution is uniform.
A more well known related relationship is Lévy's arcsine law: the distribution of time that a random walk is positive follows an arcsine distribution (or Beta 1/2,1/2).
On mathematics the same question was asked and answered Distribution of time spent above 0 by a Brownian Bridge.
We should be able to derive the uniform distribution of the Brownian bridge by using the arcsine laws for the Wiener process. Schematically it looks like below.

The Wiener process can be viewed as the combination of a scaled Brownian bridge at times below $t$, and a final piece that is entirely below or above zero at times above $t$.
Then the fraction of time that the Wiener process is above zero $f_W$ is equal to
$$f_W = t \cdot f_B + (1-t) \cdot X$$
where $f_B$ is the fraction of time that the Brownian bridge is above zero, $t$ is the last time since the random walk hits zero, and $X$ is a Bernoulli variable (the last part is fifty-fifty either all positive or all negative).
With the arcsine laws we know that $f_W,t \sim B(0.5,0.5)$ with this we should be able to derive $f_B$ (I still have to work that part out, but this seems to be the strategy to get from the arcsine laws for the Brownian motion to the laws for the Brownian bridge).
