It is straightforward to verify that for two random variables $X$ and $Y$ with variances $\sigma^2_X \neq \sigma^2_Y$, we have that

$$\Big|{\rm Cov}(X, Y)\Big| \leq \max\{\sigma^2_X,\, \sigma^2_Y\}$$

On the other hand, is is not true in general that $\Big|{\rm Cov}(X, Y)\Big| \leq \min\{\sigma^2_X,\, \sigma^2_Y\}$

Assume $\sigma^2_X$ is the smaller variance. Then if we have

$$\sigma^2_X \leq \Big|{\rm Cov}(X, Y)\Big| \leq \sigma^2_Y \Rightarrow \sigma^2_X \leq |\rho| \sigma_X \sigma_Y \Rightarrow \frac {\sigma_X}{\sigma_Y}\leq |\rho|$$

Theoretically, this is perfectly feasible, the bi-variate normal case being the easiest example. But somehow it doesn't feel very "likely in practice" to observe the ratio of standard deviations being smaller than the (absolute) correlation coefficient -but I may be wrong.

So I am looking for any kind of real-world-data-sets experience on such a phenomenon, in an attempt to informally assess whether it can be considered "likely to observe in practice" or not.

This relates to research in the following way: I am planning to use an approximating function which is piecewise, taking different functional forms depending on whether the argument of the approximated function is negative or not (which in turns depends on the above relations). And I need to obtain theoretical results prior to any empirical implementation, so it is not just a matter of "letting the data speak". Moreover this is not tied to any specific "part of the real world". In the interests of economy, it would be good to do all the theoretical work based on one hypothesis rather than derive all different scenarios from the beginning. And then it would be better if this hypothesis could be said to be the one "expected to hold" (informally).

It is straightforward to verify that for two random variables $X$ and $Y$ with variances $\sigma^2_X \neq \sigma^2_Y$, we have that

$$\Big|{\rm Cov}(X, Y)\Big| \leq \max\{\sigma^2_X,\, \sigma^2_Y\}$$

On the other hand, is is not true in general that $\Big|{\rm Cov}(X, Y)\Big| \leq \min\{\sigma^2_X,\, \sigma^2_Y\}$

Assume $\sigma^2_X$ is the smaller variance. Then if we have

$$\sigma^2_X \leq \Big|{\rm Cov}(X, Y)\Big| \leq \sigma^2_Y \Rightarrow \sigma^2_X \leq |\rho| \sigma_X \sigma_Y \Rightarrow \frac {\sigma_X}{\sigma_Y}\leq |\rho|$$

Theoretically, this is perfectly feasible, the bi-variate normal case being the easiest example. But somehow it doesn't feel very "likely in practice", when $X$ and $Y$ are measured in the same units, to observe the ratio of standard deviations being smaller than the (absolute) correlation coefficient -but I may be wrong.

So I am looking for any kind of real-world-data-sets experience on such a phenomenon, in an attempt to informally assess whether it can be considered "likely to observe in practice" or not.

This relates to research in the following way: I am planning to use an approximating function which is piecewise, taking different functional forms depending on whether the argument of the approximated function is negative or not (which in turns depends on the above relations). And I need to obtain theoretical results prior to any empirical implementation, so it is not just a matter of "letting the data speak". Moreover this is not tied to any specific "part of the real world". In the interests of economy, it would be good to do all the theoretical work based on one hypothesis rather than derive all different scenarios from the beginning. And then it would be better if this hypothesis could be said to be the one "expected to hold" (informally).