# Convergence of Vector argmax

Suppose we have some vector v in $$\mathbb{R}^d$$. At each timestep t, one of its d elements, chosen randomly, is changed by an external process that moves v towards some unkown v* with high probability (in a single dimension). The only thing we can be sure about that process is that, as t goes to infinity, the changes/deltas go to 0, and v goes to v*. How can we assert with high/arbitrary probability that, at some timestep T, argmax(v) won't change anymore (like some statistical test)?

I'm also very open to being pointed to the appropriate topics so I can solve this myself. I just don't know what to look for exactly.

• Would it be reasonable to store a moving average and a moving variance for each element and check if the average corresponding to the argmax is significantly above all other averages? Sounds a bit heuristic to me, I'd like a more principled approach if possible. – rcpinto May 17 at 21:13