#StackBounty: #time-series #autocorrelation #covariance #stochastic-processes #brownian Time-series Auto-Covariance vs. Stochastic Proc…

Bounty: 50

My background is more on the Stochastic processes side, and I am new to Time series analysis. I would like to ask about estimating a time-series auto-covariance:

$$ lambda(u):=frac{1}{t}sum_{t}(Y_{t+u}-bar{Y})(Y_{t}-bar{Y}) $$

When I think of the covariance of Standard Brownian motion $W(t)$ with itself, i.e. $Cov(W_s,W_t)=min(s,t)$, the way I interpret the covariance is as follows: Since $mathbb{E}[W_s|W_0]=mathbb{E}[W_t|W_0]=0$, the Covariance is a measure of how "often" one would "expect" a specific Brownian motion path at time $s$ to be on the same side of the x-axis as as the same Brownian motion path at time t.

It’s perhaps easier to think of correlation rather than covariance, since $Corr(W_s,W_t)=frac{min(s,t)}{sqrt(s) sqrt(t)}$: with the correlation, one can see that the closer $s$ and $t$ are together, the closer the Corr should get to 1, as indeed one would expect intuitively.

The main point here is that at each time $s$ and $t$, the Brownian motion will have a distribution of paths: so if I were to "estimate" the covariance from sampling, I’d want to simulate many paths (or observe many paths), and then I would fix $t$ and $s=t-h$ ($h$ can be negative), and I would compute:

$$ lambda(s,t):=frac{1}{i}sum_{i}(W_{i,t}-bar{W_i})(W_{i,t-h}-bar{W_i}) $$

For each Brownian path $i$.

With the time-series approach, it seems to be the case that we "generate" just one path (or observe just one path) and then estimate the auto-covariance from just that one path by shifting throught time.

Hopefully I am making my point clear: my question is on the intuitive interpretation of the estimation methods.

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#StackBounty: #distributions #brownian #diffusion What is the distribution of the peak time of the first hitting time process

Bounty: 50

I need to find the distribution of the random variable $T_{peak}$ where $T_{peak}$ represents the peak time of the first hitting time process.

Detailed Explanation of the System:
There are emitted molecules from a specific point in 3D environment. Molecules diffuse in the environment according to the followings:

$$ r[t] = r[t-1] + (Delta r_1, Delta r_2, Delta r_3)$$
$$ Delta r_i sim mathcal{N}(0,, 2DDelta t)$$
where $r[t]$, $r_i$, $D$, and $Delta t$ are the location vector at time $t$, $i$-th component of the location vector, diffusion coefficient, and the time step, respectively.

If there is an absorbing spherical trap at a distance $d$, the mean number of arriving/hitting molecules until time $t$ is:

$$ E[N^{absorb}(t)] = frac{r_{trap}}{d+r_{trap}} , text{erfc}
right) = frac{r_{trap}}{d+r_{trap}} , 2Phi

where $r_{trap}$ is the radius of the absorbing spherical trap.

When you plot $N^{absorb}(t)$ in small intervals, you get something like the following figure (Scaled Inverse Gaussian distribution)
enter image description here

And the expected value of the peak time of hitting time histogram is
$$ E[T_{peak}] = frac{d^2}{6D}$$

When simulating this diffusion process and the focusing on the absorbing times, $T_{peak}$ differs from simulation instance to instance, $T_{peak}$ is a random variable and I need to find the distribution of $T_{peak}$.

P.S. These absorbed molecules are considered as the received signal in molecular communications and the peak time distribution of the received signal is important for many applications.

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