Assuming a symmetric random walk; 1st part: I think they are uncorrelated, let's prove by induction, corr(X0 Y0) = 0 obv. Let Xn, Yn be coordinates of a random walk after time n; then, E(Xn) = 0, E(Yn) = 0, so covariance will be E(Xn Yn) = sum_{x, y} x y p_n(x, y) = sum_{x, y} x y (1/4 p_{n-1}(x, y + 1) + 1/4 p_{n-1}(x, y -1) + 1/4 p_{n-1}(x - 1, y) + 1/4 p_{n-1}(x + 1, y)), where the sum is over all possible x, y (even unreachable ones, p_n(x, y) = 0 then); sum_{x, y} x y p_{n-1}(x, y + 1) = sum_{x, y} x (y + 1) p_{n-1}(x, y + 1) - sum_{x, y} x p_{n-1}(x, y + 1) = E(X_{n-1}Y_{n-1}) - sum_{x} x p_{n-1}(x); doing the same for other sums, assembling it all back, we get E(X_n Y_n) = 0. 2nd part idk but probably correlated, one can see this for t = 1 easily