# When can we use fixed design regression results for the random design setting? [closed]

Suppose I have an independent vector $$X$$ and a dependent scalar random variable $$Y$$ and I wish to construct a regression model to predict $$Y$$ using $$X$$ given data $$\{(x_i,y_i)\}_{i=1}^{n}$$. For concreteness, suppose the true relationship is linear with $$Y = {\theta^*}^{\text{T}} X + \varepsilon$$, where $$\varepsilon$$ are errors independent of $$X$$ with $$\mathbb{E}[\varepsilon] = 0$$, and my least squares multiple linear regression yields an estimate $$\hat{\theta}_n$$ of $$\theta^*$$.

Background: As I understand it from these lecture notes, there are typically two different ways of analyzing the performance of the regression estimates depending on whether we are in the fixed design ($$X$$ is deterministic and $$\{x_i\}_{i=1}^{n}$$ are chosen by us) or random design ($$X$$ is a random variable and $$\{x_i\}_{i=1}^{n}$$ are random draws from the distribution of $$X$$) setting. In the fixed design setting, we care about the mean-squared estimation error at the training data points $$\dfrac{1}{n} \displaystyle\sum_{i=1}^{n} \left\lvert {\theta^*}^{\text{T}} x_i - \hat{\theta}^{\text{T}}_n x_i \right\rvert^2$$, whereas in the random design setting, we probably care about the average mean-squared prediction error averaged over the distribution of $$X$$, i.e. $$\mathbb{E}_X\left[ \left({\theta^* - \hat{\theta}_n}^{\text{T}}\right) X + \varepsilon \right]^2$$.

Main question: Suppose we have a result for the fixed design setting, but the result does not place any restrictions on the choice of the design points $$\{x_i\}_{i=1}^{n}$$. To be concrete, consider Theorem 2.2 in these lecture notes, which concludes that for fixed design least squares regression, we have without any restriction on the design points $$\{x_i\}_{i=1}^{n}$$: $$\mathbb{E}\left[\dfrac{1}{n} \displaystyle\sum_{i=1}^{n} \left\lvert {\theta^*}^{\text{T}} x_i - \hat{\theta}^{\text{T}}_n x_i \right\rvert^2\right] \leq C \dfrac{\sigma^2 r}{n},$$

$$\mathbb{P}\left( \dfrac{1}{n} \displaystyle\sum_{i=1}^{n} \left\lvert {\theta^*}^{\text{T}} x_i - \hat{\theta}^{\text{T}}_n x_i \right\rvert^2 > C\sigma^2\dfrac{r + \log\left( \frac{1}{\delta} \right)}{n} \right) \leq \delta ,$$

when the errors $$\varepsilon$$ are subGaussian with variance proxy $$\sigma^2$$, where $$r$$ is the rank of the design matrix $$[x_1 \: x_2 \: \cdots \: x_n]$$, $$C > 0$$ is a constant, and $$\delta \in (0,1)$$ is a reliability level.

Can I simply use the second probabilistic inequality above for the random design case since it holds conditionally for every random draw of the points $$\{x_i\}_{i=1}^{n}$$? Does the first inequality on the expected error hold as well by the law of iterated expectation? Do I need to assume that the draws $$\{x_i\}_{i=1}^{n}$$ are i.i.d., or do these hold for non-i.i.d. samples as well?

PS: My question is complementary to this and this.