General-jump extension of exponential traveling-wave and speed-bound results

Let $J$ denote the independent jump-size distribution, and let $v_{**}$ be the minimum selected speed. A mean-field limit (MFL) is the deterministic limiting particle-system dynamics, and a traveling-wave solution (TWS) is a traveling-wave MFL. For an MFL initial state $f(0)$, write $f_x(0)$ for its distribution function and define its right-tail exponent through $-\log(1-f_x(0))/x$. Let $v(\zeta)$ denote the speed bound associated with tail exponent $\zeta$, and let $\zeta_{**}$ be the critical exponent. Also write $\mathring f^n(\infty)$ for the stationary distribution of the centered process and $\phi^{**}$ for the TWS with speed $v_{**}$. General-jump conjecture. For generally distributed jump sizes, analogs of the exponential-jump traveling-wave existence theorem and the corresponding MFL speed bounds should hold; in particular, if the jump-size distribution satisfies $\alpha>0$, then: (i) the unique TWS exists for every $v\ge v_{**}$ and does not exist for $v<v_{**}$; (ii) if the right-tail exponent is lower bounded by $\zeta\leq\zeta_{**}$, namely

$$\liminf_{x\to\infty}\frac{-\log(1-f_x(0))}{x}\geq\zeta,$$

then the MFL average speed is at most $v(\zeta)$, and if the liminf is at least $\zeta_{**}$, the average speed equals $v_{**}$; (iii) if the right-tail exponent is upper bounded by $\zeta\leq\zeta_{**}$, namely

$$\limsup_{x\to\infty}\frac{-\log(1-f_x(0))}{x}\leq\zeta,$$

then the MFL average speed is at least $v(\zeta)$; and (iv)

$$\mathring f^n(\infty)\Rightarrow\phi^{**}.$$

Here $\phi^{**}$ is the TWS with speed $v_{**}$. The conjecture extends the paper's exponential-jump traveling-wave and speed estimates to general jump-size distributions, and predicts convergence of centered stationary processes to the critical traveling wave.