<t>By the central limit theorem your process is Brownian motion in the limit t goes to infinity. Therefore the conditional probability distribution for the position is Gaussian with variance t. The probability that the particle's position is between a and b will therefore go to 0 as as t goes to inf...

<t>It is trivial to show that for a positive definite matrix, A, and for a real scalar, a>0, aA is also positive definite. It is also easy to show that a sum of two positive definite matrices is also positive definite. Therefore, if 0<y<1 your C matrix will be positive definite.The reverse does not ...