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My aim is to find a few examples of positive definite covariance matrices $\pmb{R} = \{R(s,t)\}_{s,t=1}^n$ that satisfy $$R(s,t) = R(s+T, t+T),~~~1\leq s,t \leq n-T,$$ where $T$, $1\leq T\leq n-1$, is a fixed positive integer.

I have tried to construct some examples, in an arbitrary manner, but they appeared to be positive semi-definite only. For example, I have tried

  1. $R(s,t) = \exp\left\{-\alpha|(s ~\text{mod}~ T) - (t ~\text{mod}~ T)|\right\}$,
  2. $R(s,t) = \exp\left\{-\alpha|\sin(2\pi s/T) - \sin(2\pi t/T)|\right\}$,
  3. $R(s,t) = \min \{s ~\text{mod}~ T, t ~\text{mod}~ T\}$, and a few other structures. Below, I provide one simple reproducible R code where I constructed many such examples (with $T=24$) and checked for their positive definiteness:
n = 500; t = 1:n


R0 = function(s, t, alpha = 0.3) exp(-alpha*abs(s%%24 - t%%24))

#R0 = function(s, t, alpha = 0.3) exp(-alpha*(s%%24 - t%%24)^2) 
#R0 = function(s, t, alpha = 0.3) exp(-alpha*abs(sin(pi*s/12) - sin(pi*t/12))) 
#R0 = function(s, t, alpha = 0.3) exp(-alpha*abs(cos(pi*s/12) - cos(pi*t/12))) 
#R0 = function(s, t) min(s%%24, t%%24) 
#R0 = function(s, t) (1+abs(s%%24-t%%24))*exp(-abs(s%%24-t%%24)) 
#R0 = function(s, t) (1+abs(sin(pi*s/12)-sin(pi*t/12)))*
#                             exp(-abs(sin(pi*s/12)-sin(pi*t/12))) 
#R0 = function(s, t, sig=0.2, nu=0.5, beta=0.5) 
#  rSPDE::matern.covariance((s%%24 - t%%24), kappa = 1/beta, nu = nu, sigma = sig)
#R0 = function(s, t, sig=0.2, nu=0.5, beta=0.5)
#  rSPDE::matern.covariance((cos(pi*s/12)-cos(pi*t/12)), kappa = 1/beta, nu = nu, sigma = sig)
#R0 = function(s, t, lambda=0.2) exp(-lambda*(sin(s%%24-t%%24))^2)


R = matrix(nrow = n, ncol = n)
for (i in 1:n) {
  for (j in 1:n) {
    R[i,j] = R0(t[i], t[j])
  }
}
matrixcalc::is.positive.definite(R)
matrixcalc::is.positive.semi.definite(R)

Can anyone please suggest one or more non-trivial examples of such covariance matrices?

OR, if such periodic covariance matrices can never be positive definite, can you please provide a proof (or sketch of a proof) supporting this statement? Any help will be greatly appreciated!

EDIT: I am looking for a non-trivial (that is, non-diagonal) example of such matrices.

Joy
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    Could you please be clearer about what you mean by a periodic matrix? Would it be that there is a fixed $T,$ $1\le T\le n-1,$ such that for all $1\le s,t\le n-T,$ $R(s,t)=R(s+T,t+T)$? Or perhaps you're taking $s+T$ and $t+T$ modulo $n$? Or something else? Regardless, one obvious choice would be a diagonal matrix with positive entries that are periodic in whatever sense you mean. – whuber Jun 26 '20 at 18:28

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