We consider the invariant measure of homogeneous random walks in the quarter-plane. In particular, we consider measures that can be expressed as a countably infinite sum of geometric terms which individually satisfy the interior balance equations. We demonstrate that the compensation approach is the only method that may lead to such a type of invariant measure. In particular, we show that if a countably infinite sum of geometric terms is an invariant measure, then the geometric terms in an invariant measure must be the union of at most six pairwise-coupled sets of countably infinite cardinality each. We further show that for such invariant measure to be a countably infinite sum of geometric terms, the random walk cannot have transitions to the north, northeast or east. Finally, we show that for a countably infinite weighted sum of geometric terms to be an invariant measure at least one of the weights must be negative.

中文翻译：

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四分之一平面随机游走补偿方法的必要条件

我们考虑四分之一平面中均匀随机游走的不变测度。特别是，我们考虑可以表示为几何项的可数无限和的度量，这些几何项分别满足内部平衡方程。我们证明了补偿方法是唯一可能导致这种类型不变度量的方法。特别地，我们表明，如果几何项的可数无限和是不变测度，那么不变测度中的几何项必须是至多六个成对耦合的可数无限基数集的并集。我们进一步表明，对于这样的不变测度是几何项的可数无限和，随机游走不能向北、东北或东过渡。最后，