Assume that $X_1$ is Gaussian, say $X_1 \sim N(\mu_1,\sigma_1^2)$. Then, the assumption is that
- The conditional density of $X_2$ given $X_1=x_1$ is a Gaussian density. Note that we are not specifying what the mean and variance of this conditional density are and and how they depend on $x_1$ or $\mu_1$ or $\sigma_1^2$; just that the conditional density is a Gaussian density.
- The conditional density of $X_3$ given that $X_1+X_2=y$ is a Gaussian density (once again, no specification about how this density depends on $y$ or on any of the parameters).
- More generally, the conditional density of $X_i$ given that
$X_1+X_2+\cdots + X_{i-1} =z$ is a Gaussian density (once again, no specification about how this density depends on $z$ or on any of the parameters).
The question asked is "Is the unconditional density of $X_n$ a Gaussian density?"
Let's start with $X_2$. We have that
$$f_{X_2\mid X_1}(x_2 \mid X_1=x_1) = \frac{1}{h(x_1,\mu_1,\sigma_1)\sqrt{2\pi}}
\exp\left(-\frac{1}{2}\left(\frac{x_2-g(x_1,\mu_1,\sigma_1)}{h(x_1,\mu_1,\sigma_1)}\right)^2\right)$$
where $g(x_1,\mu_1,\sigma_1)$ and $h(x_1,\mu_1,\sigma_1)$ are functions that describe how the mean and the standard deviation of this conditionally Gaussian density depend on $x_1$ and the mean and standard
deviation of $X_1$. Since $g$ and $h$ are essentially arbitrary (we do need to insist that $h$ be nonnegative, of course, but that is a minor restriction), it is going to be the exception rather than the rule
that $X_1$ and $X_2$ are jointly Gaussian. Thus, in general, the answer to the question asked is NO for the case $n=2$ and since the same argument can be extended to larger values of $n$, the answer to the
question "Is the unconditional density of $X_n$ a Gaussian density?"
is No, not in general.
What about the exceptional case? If $g$ and $h$ can be expressed in the form
$$g(x_1,\mu_1,\sigma_1)
= \mu_2 + {\rho}{\sigma_2}\left(\frac{x_1-\mu_1}{\sigma_1}\right),
\quad h(x_1,\mu_1,\sigma_1) = \sigma_2\sqrt{1-\rho^2},\tag{1}$$
where $\sigma_2 > 0$ and $|\rho|\leq 1$, then $X_1$ and $X_2$ are
indeed jointly Gaussian and $X_2 \sim N(\mu_2,\sigma_2^2)$. It
follows that $X_1+X_2$ is also a Gaussian random variable.
So, $X_3$ is conditionally Gaussian given $X_1+X_2 = y$, and if
the amazing relationships described in $(1)$ apply mutatis mutandis in this case, then we get that $X_3$ and $X_1+X_2$ are jointly Gaussian,
that $X_3$ is unconditionally Gaussian, and that $X_1+X_2+X_3$
is Gaussian. Continuing to believe that that such fortuitous coincidences
will continue to occur exactly when we need them, we arrive at
the conclusion that $X_n$ is unconditionally a Gaussian random variable, and indeed that $X_1, X_2, \ldots, X_n$ are jointly Gaussian random variables. But when $n > 8$ early in the morning, you need to be better than the Red Queen in Lewis Carroll's (Through the Looking Glass) who believed as many as eight impossible things before breakfast.
