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In my research on convergence properties of certain Bayesian methods I have encountered $\mathop{\mathrm{arg\,min}}$ limits of the forms \begin{equation} \lim_{n\to\infty} \mathop{\mathrm{arg\,min}}_{\mathbf{x}\in\mathbb{R}^d}\frac{f_n(\mathbf{x})}{g_n(\mathbf{x})} \end{equation} and \begin{equation} \lim_{n\to\infty} \mathop{\mathrm{arg\,min}}_{\mathbf{x}\in\mathbb{R}^d}\big(f_n(\mathbf{x}) + g_n(\mathbf{x})\big) \end{equation} where it is assumed that $f_n,g_n\to 0$ plus $f_n/g_n \to 0$ in the former case and $g_n/f_n \to 0$ in the latter, all in pointwise manner. The functions are real-valued and the minimizers can be assumed unique for each $n$.

I find it fairly intuitive that, at least with some additional assumptions involving maybe monotonicity of the function sequences or convexity, one should be able to discard the effect of $g_n$ at the limit. However, I have been unable to find any references dealing with limits of this type but am quite certain that someone must have addressed the topic. Is anyone able provide references? The only vaguely related question that has caught my eye is thisthis.

In my research on convergence properties of certain Bayesian methods I have encountered $\mathop{\mathrm{arg\,min}}$ limits of the forms \begin{equation} \lim_{n\to\infty} \mathop{\mathrm{arg\,min}}_{\mathbf{x}\in\mathbb{R}^d}\frac{f_n(\mathbf{x})}{g_n(\mathbf{x})} \end{equation} and \begin{equation} \lim_{n\to\infty} \mathop{\mathrm{arg\,min}}_{\mathbf{x}\in\mathbb{R}^d}\big(f_n(\mathbf{x}) + g_n(\mathbf{x})\big) \end{equation} where it is assumed that $f_n,g_n\to 0$ plus $f_n/g_n \to 0$ in the former case and $g_n/f_n \to 0$ in the latter, all in pointwise manner. The functions are real-valued and the minimizers can be assumed unique for each $n$.

I find it fairly intuitive that, at least with some additional assumptions involving maybe monotonicity of the function sequences or convexity, one should be able to discard the effect of $g_n$ at the limit. However, I have been unable to find any references dealing with limits of this type but am quite certain that someone must have addressed the topic. Is anyone able provide references? The only vaguely related question that has caught my eye is this.

In my research on convergence properties of certain Bayesian methods I have encountered $\mathop{\mathrm{arg\,min}}$ limits of the forms \begin{equation} \lim_{n\to\infty} \mathop{\mathrm{arg\,min}}_{\mathbf{x}\in\mathbb{R}^d}\frac{f_n(\mathbf{x})}{g_n(\mathbf{x})} \end{equation} and \begin{equation} \lim_{n\to\infty} \mathop{\mathrm{arg\,min}}_{\mathbf{x}\in\mathbb{R}^d}\big(f_n(\mathbf{x}) + g_n(\mathbf{x})\big) \end{equation} where it is assumed that $f_n,g_n\to 0$ plus $f_n/g_n \to 0$ in the former case and $g_n/f_n \to 0$ in the latter, all in pointwise manner. The functions are real-valued and the minimizers can be assumed unique for each $n$.

I find it fairly intuitive that, at least with some additional assumptions involving maybe monotonicity of the function sequences or convexity, one should be able to discard the effect of $g_n$ at the limit. However, I have been unable to find any references dealing with limits of this type but am quite certain that someone must have addressed the topic. Is anyone able provide references? The only vaguely related question that has caught my eye is this.

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Limits of argmin ratios and sums

In my research on convergence properties of certain Bayesian methods I have encountered $\mathop{\mathrm{arg\,min}}$ limits of the forms \begin{equation} \lim_{n\to\infty} \mathop{\mathrm{arg\,min}}_{\mathbf{x}\in\mathbb{R}^d}\frac{f_n(\mathbf{x})}{g_n(\mathbf{x})} \end{equation} and \begin{equation} \lim_{n\to\infty} \mathop{\mathrm{arg\,min}}_{\mathbf{x}\in\mathbb{R}^d}\big(f_n(\mathbf{x}) + g_n(\mathbf{x})\big) \end{equation} where it is assumed that $f_n,g_n\to 0$ plus $f_n/g_n \to 0$ in the former case and $g_n/f_n \to 0$ in the latter, all in pointwise manner. The functions are real-valued and the minimizers can be assumed unique for each $n$.

I find it fairly intuitive that, at least with some additional assumptions involving maybe monotonicity of the function sequences or convexity, one should be able to discard the effect of $g_n$ at the limit. However, I have been unable to find any references dealing with limits of this type but am quite certain that someone must have addressed the topic. Is anyone able provide references? The only vaguely related question that has caught my eye is this.