Let $p\in [0,1/2]$, and define $\xi$ as the symmetric random variable such that $$ \xi = \begin{cases} 1 & \text{ w.p. } p\\ 0 & \text{ w.p. } 1-2p\\ -1 & \text{ w.p. } p \end{cases} $$ so that $\mathbb{E}[\xi]=0$, $\mathbb{E}[\xi^2]=2p$, and for $p=1/2$ $\xi$ is just a Rademacher r.v.

I am interested in tight (but manageable/usable) bounds on the subgaussian parameter of $\xi$, as a function of $p$ (and, if possible, references to cite directly). In particular, $\xi$ is always $1$-subgaussian, but what can we say for $p\to 0$?

he MGF of $\xi$ is easy to calculate as $$ \mathbb{E} e^{t\xi} = 1+2p(\cosh t - 1), \qquad t\geq 0 $$ so it boils down to studying the quantity $$ \kappa(p) = \sup_{t> 0}\frac{1+2p(\cosh t - 1)}{t^2} $$

The case of a binary random variable (Bernoulli) was given in [Theorem 2.1, 1]. Is the analogue for the above symmetrized version known?

[1] Buldigīn, V. V.; Moskvichova, K. K. Sub-Gaussian norm of a binary random variable. (Ukrainian) ; translated from Teor. Ĭmovīr. Mat. Stat. No. 86 (2011), 28--42 Theory Probab. Math. Statist. No. 86, (2013), 33--49 [pdf]