We consider a randomized urn model with objects of finitely many colors. The replacement matrices are random, and are conditionally independent of the color chosen given the past. Further, the conditional expectations of the replacement matrices are close to an almost surely irreducible matrix. We obtain almost sure and $L^1$ convergence of the configuration vector, the proportion vector and the count vector. We show that first moment is sufficient for i.i.d. replacement matrices independent of past color choices. This significantly improves the similar results for urn models obtained in Athreya and Ney (1972) requiring $L{log}_{+}L$ moments. For more general adaptive sequence of replacement matrices, a little more than $L{log}_{+}L$ condition is required. Similar results based on $L^1$ moment assumption alone has been considered independently and in parallel in Zhang (2018). Finally, using the result, we study a delayed elephant random walk on the nonnegative orthant in $d$ dimension with random memory.

我们考虑一个具有有限多种颜色的对象的随机瓮模型。替换矩阵是随机的，并且有条件地独立于过去选择的颜色。此外，替换矩阵的条件期望接近于几乎可以肯定的不可约矩阵。我们几乎可以肯定并且${\mathrm{大号}}^1$配置向量、比例向量和计数向量的收敛。我们表明，对于独立于过去颜色选择的独立同分布替换矩阵，第一时刻就足够了。这显着改进了在 Athreya 和 Ney (1972) 中获得的瓮模型的类似结果，要求$\mathrm{大号}{日志}_{+}\mathrm{大号}$时刻。对于更一般的替换矩阵自适应序列，略多于$\mathrm{大号}{日志}_{+}\mathrm{大号}$条件是必需的。类似的结果基于${\mathrm{大号}}^1$Zhang (2018) 单独和并行地考虑了矩假设。最后，利用该结果，我们研究了在非负正交上的延迟大象随机游走$d$具有随机记忆的维度。