# Multivariate normal distribution logic

From Introduction to Probability here (PDF page 348, book page 337) we are told:

Definition 7.5.1

A $$k$$-dimensional random vector $$X=(X_1, ..., X_k)$$ is Multivariate Normal if every linear combination of the $$X_j$$ has a Normal distribution. That is, we require

$$t_1X_1 + \cdots + t_kX_k$$

to have a Normal distribution for any constants $$t_1, ..., t_k$$.

Question set-up

Logically speaking this is of the form: $$Q$$ if $$P$$, where:

$$P$$: We have a $$k$$-dimensional random vector $$X=(X_1, ..., X_k)$$ where every linear combination of the $$X_j$$ has a Normal distribution

$$Q$$: $$X=(X_1, ..., X_k)$$ is Multivariate Normal.

So, if $$P$$ then $$Q$$ i.e. $$P \implies Q$$.

Question

If we are told that we have a MVN (i.e. a Bivariate Normal, $$k=2$$) with random vector $$(X, Y)$$ can we conclude that every linear combination of the $$X, Y$$ are Normal? It seems to me we cannot as we are given $$Q$$ is true, not $$P$$.

If Definition 7.5.1 above said "if and only if" then I think we could, but not just with "if"?

Note: I know that it does happen to be the case that every linear combination of the $$X, Y$$ is Normal if $$(X, Y)$$ are MVN...I'm asking if we can logically deduce it from the information given.

Background

This is from homework question 3 b (page 20 of PDF). Provided below for context:

Let $$(X, Y)$$ be Bivariate Normal, with $$X$$ and $$Y$$ marginally $$\mathcal{N}(0, 1)$$ and with correlation $$\rho$$ between $$X$$ and $$Y$$.

Show that $$(X + Y,X − Y )$$ is also Bivariate Normal.

Solution given: the linear combination $$s(X + Y ) + t(X − Y ) = (s + t)X + (s − t)Y$$ is also a linear combination of $$X$$ and $$Y$$, so it is Normal, which shows that $$(X +Y,X −Y )$$ is MVN.

• Definition 7.5.1 is if and only if. Definitions always imply only if, otherwise they wouldn't be definitions. Jan 4, 2022 at 11:02