Suppose I have two real-valued independent random variables $X$ and $Y$.
What are conditions on $X$ under which the $\gamma$-quantile of $X + Y$ (weakly) exceeds the $\gamma$-quantile of $Y$, for $\gamma \in (0.5, 1)$?
Is there a name for this form of stochastic dominance/order?
Initially, I thought this was equivalent to first-order stochastic dominance. However, that seems to require the above to hold for $\gamma \in (0, 1)$, instead of $\gamma \in (0.5, 1)$ (see Shaked and Shanthikumar, 1.A.12). Whether I merely require it to hold for the quantiles above the median. In fact, I would also be satisfied if it only holds for `large' quantiles $\gamma \in (c, 1)$, for some (not too large) $1/2 < c \ll 1$.
I also further studied the book `Stochastic Orders' by Shaked and Shanthikumar. There I found a closely related concept called the 'dispersion order'. However, this concept seems to be slightly different and stricter than what I require. In particular, it seems to require that
$F^{−1}(\alpha)−F^{−1}(\beta) \leq G^{−1}(\alpha)−G^{−1}(\beta)$,
for any $0 < \alpha \leq \beta < 1$ (3.B.1 in S&S), where $F^{-1}$ and $G^{-1}$ are the quantile functions of $Y$ and $X + Y$.
If we were to fix $\alpha = 1/2$ and assume that $F^{-1}(1/2) = G^{-1}(1/2)$ then this seems to correspond exactly to the type of property I am interested in. However, this would require the median of $Y$ and $X + Y$ to be the same. According to this post, this only seems to be true under quite stringent symmetry conditions. Moreover, it imposes conditions on $Y$, which I would prefer not to make.
I also looked at second order stochastic dominance and the dilation stochastic order, but those seem to be something different from what I am looking for as well...
Another related work is this paper by Townsend and Colonius. They consider the distance between the $\gamma$ and $1-\gamma$ quantiles. For symmetric distributions, this distance is simply twice the $1-\gamma$ quantile, and so seems to coincide with the type of stochastic order. Unfortunately, they do not seem to discuss the properties I am interested in.