# Maximum possible number of random variables with the same correlation?

A set contains random variables where any two random variables in the set have the same correlation $$\rho$$. Then what is the maximum number of the elements of this set?

The answer depends on what $$\rho$$ is. For $$-1 \leq \rho < -\frac 12$$, the answer is two random variables. More generally, the maximum number of random variables that can have common correlation $$\rho$$ is $$n$$ for $$\rho$$ in the range $$\left[-\frac{1}{n-1}, -\frac{1}{n}\right)$$. For $$\rho \geq 0$$, the number of random variables is unbounded. See the answers to this question for some results.
• @Acccumulation I don't think that what you are saying can be true. Consider $n$ independent discrete random variables $X_1i$ each taking on values $\pm 1$ with equal probability $\frac 12$, and define $Y_i = X_i - \bar{X}$ where $\bar{X}$ is the sample mean $\frac 1n \sum_i$X_i$. Then, as shown in this answer to the question cited in my answer above, the$Y_i$are zero-mean random variables with common correlation$-\frac{1}{n-1}\$ exactly as claimed above. – Dilip Sarwate Jan 28 at 22:18
To supplement Dilip Sarwate's answer, if you take $$I_0 \sim Bernoulli(p_0)$$ and any number of independent $$I_i \sim Bernoulli(p)$$, all independent then $$cor(I_0 + I_i, I_0 + I_j) = \frac 1 {1 + \frac {p (1-p)} {p_0(1-p_0)}},$$ so you can choose $$p_0$$ and $$p$$ to get any $$\rho$$ in the interval $$(0,1)$$.