What are the conditions for convergence for the following problem

Question

Following is a part of a bigger problem that I am trying to solve. Let $\alpha_t , \beta_t, k $ be random variables as a function of time $(t)$, such that $ k(\alpha_t +e_1(t)) = \beta_t +e_2(t)$. Where $\alpha_t, \beta_t$ are measurable with errors $e_1(t)$ and $e_2(t)$, respectively. The exact values of these errors are not known but they are bounded from above and below with known bounds. If we use recursive Bayesian method to estimate $k$, then under what conditions on $e_1(t), e_2(t), \alpha_t, \beta_t$, will $k$ converge to its true value $k_0$?