Existence of random variable given first $k$ moments A sequence of real numbers $\{m_k\}$ is the list of moments of some real random variable if and only if the infinite Hankel matrix $$\left(\begin{matrix}
m_0 & m_1 & m_2 & \cdots     \\
m_1 & m_2 & m_3 & \cdots  \\
m_2 & m_3 & m_4 & \cdots  \\
\vdots & \vdots & \vdots & \ddots  \\
\end{matrix}\right)$$ is positive definite.  (Source: https://en.wikipedia.org/wiki/Hamburger_moment_problem)
My question is, given only the first $k$ moments, is it sufficient that the top left $k \times k$ minor of the Hankel matrix be positive definite for there to exist a real random variable with those first $k$ moments? 
In other words, can a $k \times k$ positive definite Hankel matrix always be extended to an infinite positive definite Hankel matrix?

Yes, this works. There should really be an easy direct argument, but all I can think of right now is the following: The finite moment problem $\int x^n\, d\mu(x)=m_n$, $n=0,1,\ldots , k$, can be solved in the same way as the full problem $n\ge 0$. Namely, run Gram-Schmidt on $1,x,\ldots , x^N$ (with $2N=k$); the orthogonal polynomials will satisfy a three term recurrence
$$
a_n p_{n+1} + a_{n-1}p_{n-1} + b_n p_n = xp_n ,
$$
and the spectral measures of the associated Jacobi matrix will solve the moment problem.
In particular, you can extend to the half line $n\ge 1$ by just making up coefficients $a_n,b_n$ for $n\ge N$ at will, and any such measure will have the given moments $m_0,\ldots , m_k$ (because these only depend on the first coefficients). Its subsequent moments will give you the desired extension.
In fact, there is a description of all solutions to a finite moment problem (sometimes called the Nevanlinna parametrization), which more or less works like this.
