Asymmetric-exponent limiting distribution for the two-sided debt model

Prove that, when the transfer-probability function g satisfies distinct asymptotic exponents \(\rho^+\) and \(\rho^-\) as \(k\to\infty\) and \(k\to-\infty\), respectively, with \(g^+\leq g^-\), the scaled stationary wealth distribution converges to a two-sided Gamma distribution whose positive-side and negative-side shape parameters are determined by \(\rho^+\) and \(\rho^-\), respectively.

Background

The paper proves convergence of the scaled stationary wealth distribution when the transfer-probability function has the same second-order exponent $\rho$ on both the positive and negative tails. In the case $g^+>g^-$, the authors state that their argument extends directly to unequal exponents, with the limiting Gamma shape determined by $\rho^+$.

For $g^+\leq g^-$, however, the proof uses separate exponential tiltings of the positive and negative parts. If $\rho^+\neq\rho^-$, the corresponding generating functions have different asymptotic orders, so one side asymptotically dominates the other and the proof no longer applies. Numerical simulations indicate that a two-sided Gamma limit should nevertheless hold, with the two sides governed separately by $\rho^+$ and $\rho^-$.