The two correlations you are handed do not pin the third one down, but they fence it into exactly [−1/50, 1], and that interval dips below zero. The fence falls out of a 3×3 determinant read as a quadratic in the unknown, and out of a picture: 0.7 is an angle of 45.573°, both stocks live on a cone of that half-angle around the index, and putting them on opposite sides opens 91.146° between them. Also here: why the real tipping point is ab ≥ 0 together with a² + b² ≥ 1 rather than "both above 0.707", why 0.9 and 0.5 force a positive answer while 0.9 and −0.9 allow −1, why standing on the floor costs a rank, and why three Bernoulli(0.5) indicators with the same two correlations are confined to [0.40, 1] instead.
Aussi posé sous le nom decorrelation transitivity · is correlation transitive · transitivity of correlation · bounds on a third correlation · correlation triangle inequality · if A and B are both correlated with C
A plain t sitting next to the t squared in a Gaussian exponent looks like a new function and is only a shift. Completing the square turns the integral of e to the minus a t squared over two plus b t, from x to infinity, into e to the b squared over 2a times the root of 2 pi over a times the standard normal at a rescaled and shifted argument, never at x itself. The worked case comes out as exactly half a bell, e root pi over two or 2.40901455, but only because its lower limit happens to land on the centre b over a.
Daily, weekly and monthly returns give per-day variance estimates of 1.0000, 1.3225 and 1.4000, and the reflex is to average them into 1.2408, a figure no horizon produced. The variance ratio is a weighted sum of autocorrelations, so a forty percent overshoot at twenty periods measures dependence rather than noise, and the coefficient that reproduces it is 0.17554. With twenty years of daily data that ratio sits 4.6 standard errors above one and with five years only 2.3, which is why the number means nothing without the sample size attached.
Two weights that add to 0.98 give the variance forecast a half-life of 34.31 days; delete the second one and the half-life is 0.2744 days, gone before the next open. The same recursion turns strictly normal daily draws into a year with kurtosis exactly 297/67, and one shuffle of those same numbers separates the fat tail from the clustering. It also has a condition nobody quotes: stationarity is not enough for that kurtosis to be finite.