Summarize the problem

Given A Stationary Random Processes (strict sense) $X_i$ I define two Stationary Ergodic Random Processes by

$$ \bar{X}_n = \frac{1}{n} \sum_{i=0}^{n-1} X_i \ \ \text{and} \ \ \bar{Y}_n = \frac{1}{n/\delta} \sum_{i=0}^{n/\delta-1} X_{i \cdot \delta} \\ $$

where the second one is sub-sampling the data with $\delta$. Does it always hold that

$$ \text{Var} \left ( \bar{X}_n \right ) \leq \text{Var} \left ( \bar{Y}_n \right ) $$

for any $\delta>1$ and how so?

Provide details and any research

If required you can assume that $\lim_{\tau \rightarrow \infty}\text{Cov}(X_i,X_{i+\tau}) \rightarrow 0 $ and assume that $n \rightarrow \infty$ and that $n$ is a multiple of $\delta$ and $E[X_i] = 0$.

This lecture I found discusses something similar but it does not include sub-sampling; http://isl.stanford.edu/~abbas/ee278/lect07.pdf

Describe what you’ve tried

Writing it out gives $$ \text{Var} \left ( \bar{X}_n \right ) \leq \text{Var} \left ( \bar{Y}_n \right ) $$ $$ \frac{1}{n^2} \sum_{i=0}^{n-1} \sum_{j=0}^{n-1} \text{Cov}(X_i,X_j) \leq \frac{\delta^2}{n^2} \sum_{i=0}^{n/\delta-1} \sum_{j=0}^{n/\delta-1} \text{Cov}(X_{i \cdot \delta},X_{j \cdot \delta}) $$ Using that $X_i$ is a Stationary Random Processes we can write it with the autocovariance function

$$ C_X(|i-j|) = \text{Cov}(X_i,X_j) $$

$$ \frac{1}{n^2} \sum_{i=0}^{n-1} \sum_{j=0}^{n-1} C_X(|i-j|) \leq \frac{\delta^2}{n^2} \sum_{i=0}^{n/\delta-1} \sum_{j=0}^{n/\delta-1} C_X(\delta |i-j|) $$

$$ \frac{1}{n^2} \left ( C_X(0) n + 2 \sum_{i=1}^{n-1} (n-i) C_X(i) \right ) \leq \frac{\delta^2}{n^2} \left ( C_X(0) n/\delta + 2 \sum_{i=1}^{n/\delta-1} (n/\delta-i) C_X(\delta i) \right ) $$

and with the autocorrolation

$$ R(i) = C_X(i)/C_X(0) $$

it gives

$$ n + 2 \sum_{i=1}^{n-1} (n-i) R(i) \leq \delta^2 \left ( n/\delta + 2 \sum_{i=1}^{n/\delta-1} (n/\delta-i) R(\delta i) \right ) $$

This is obviously true of $R(i) = 0$ for $i>0$ but in the general case, I'm unable to prove this expression and hence that is why I'm asking the question.

I have been checking it numerically using Mathematica and I could not find any counterexamples (I did find quality examples).

Any kind of help would be very much appreciated.



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