Conjectures on tail asymptotics of the marginal stationary distribution for a multidimensional SRBM

Abstract

We are concerned with the stationary distribution of a d-dimensional semimartingale reflecting Brownian motion on a nonnegative orthant, provided it is stable, and conjecture about the tail decay rate of its marginal distribution in an arbitrary direction. Due to recent studies, the conjecture is true for d = 2. We show its validity for the skew symmetric case for a general d.

Light tail asymptotics in multidimensional reflecting processes for queueing networks

“…Miyazawa (2003) conjectured the decay rates for the generalized Jackson network. Miyazawa and Kobayashi (2010) make a similar conjecture for an SRBM, which is in the same line as that conjectured in Sect. 6.2.…”

“…This suggests a similar characterization for d ≥ 3. These ideas are also considered for the multidimensional SRBM in Miyazawa and Kobayashi (2010). However, they remain as conjectures for d ≥ 3.…”

Reflecting Brownian Motion in Two Dimensions: Exact Asymptotics for the Stationary Distribution

“…Except for these special cases and the one studied in [25], the exact asymptotics for two-dimensional SRBMs are not known. A part of the present results have recently been conjectured by Miyazawa and Kobayashi [24], which also includes conjectures for SRBMs in d ≥ 3 dimensions.…”

Reflecting Brownian motion in two dimensions: Exact asymptotics for the stationary distribution

“…Except for these special cases and the one studied in [26], the exact asymptotics for two-dimensional SRBMs are not known. A part of the present results have recently been conjectured by Miyazawa and Kobayashi [25], which also includes conjectures for SRBMs in d ≥ 3 dimensions.…”

“…The results in this paper may be extended to cover SRBMs in d dimensions, where d ≥ 3. Indeed, Miyazawa and Kobayashi [25] have conjectured such extensions. We hope the technical results in the present paper will be useful to prove these challenging conjectures.…”