Carleman's condition for Catalan numbers

Question

I've been going through some elementary random matrix theory and in these lecture notes we say a probability measure is completely determined by its moments so long as it obey's Carleman's condition i.e. the sequence of moments $m_k$ are such that $$ \sum_{k \geq 1} m_k^{-1/(2k)}= \infty. $$ Later on when the semicircle distribution is brought up, whose moments are 0 for odd moments and the Catalan numbers for even numbers, they use a complex analytic method to prove that the moments are completely determined by the probability measure which is natural following Hadamard's formula. However I was wondering if a pure real analysis proof could still be constructed as I haven't found anything online, i.e. is the criterion $$ \sum_{k \geq 1} C_k^{-1/(2k)}= \infty $$ obeyed without analytic continuation? I've tried using Sterling's Formula which I hoped would cancel the exponent as well but no success as this series tends to zero.

Answer 1

Asymptotically, $C_k \sim \frac{4^k}{k^{3/2}\sqrt{\pi}}$. Therefore, $$ \lim_{k \to \infty} C_k^{-\frac{1}{2k}} = \frac{1}{2} \lim_{k \to \infty} \pi^{\frac{1}{4 k}} k^{\frac{3}{4k}} = \frac{1}{2} \lim_{k \to \infty} \pi^{\frac{1}{4 k}} \exp\left(\frac{3 \log k}{4 k}\right) = \frac{1}{2} \left( 1 \cdot \exp(0) \right) = \frac{1}{2} $$ Since the terms in the series do not in fact tend toward zero, the series diverges.