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 
$$
\kappa(p) = \sup_{t> 0}\frac{\log(1+2p(\cosh t - 1))}{t^2}
$$

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]](https://www.ams.org/journals/tpms/2013-86-00/S0094-9000-2013-00887-4/S0094-9000-2013-00887-4.pdf)