# Point-wise scaling of stochastic processes on $\mathbb{R}$

Consider a $$n$$-dimensional random vector $$\boldsymbol{X}$$ with covariance matrix $$\mathbf{\Sigma} = (\sigma_{ij})$$. We may apply a element-wise scaling to $$\boldsymbol{X}$$ by multiplying it with a diagonal matrix $$\mathbf{A} = \operatorname{diag}(a_1, \dots, a_n)$$. Then the covariance matrix of the scaled random vector $$\mathbf{A} \boldsymbol{X}$$ is $$\operatorname{Cov}(\mathbf{A} \boldsymbol{X}) = \mathbf{A} \boldsymbol{X} \mathbf{A}^\top = (a_i a_j \sigma_{ij}).$$

My question is: Does this result generalize to stochastic processes on $$\mathbb{R}$$? In other words: Considering a stochastic process $$X(t)$$ with covariance function $$c_X(t, t')$$ and a (non-random) function $$f(t)$$, what is the covariance function of the stochastic process $$Y(t) = f(t) X(t)$$? Is it $$c_Y(t, t') = f(t) f(t') c_X(t, t')$$ or are things more complicated?

I am not very familiar with rigorous probability theory, but I appreciate any hints on the question or suggestions on where to read up on the topic. Thank you very much!

• The generalization is correct. This is because the covariance of a process depends only on its two-dimensional distributions. Also more simply, $\text{Cov}(f(t)X(t), f(t')X(t')) = f(t)f(t')\text{Cov}(X(t),X(t'))$ by bilinearity of the covariance. – Michh Dec 29 '18 at 17:44
• @Michh, thank you, that's really quite simple! Is it also true that a function of the form $f(t) f(t') c(t, t')$ is always a valid covariance function given that $c(t, t')$ is a valid covariance function? – bbrot Dec 29 '18 at 18:22
• Absolutely, by setting $Y(t) = f(t) \, X(t)$ where $X$ is a process with covariance $c$, $Y$ has the desired covariance function. – Michh Dec 29 '18 at 18:28