Implicaciones de $\lim\limits_{n\to∞}\left[nP_j-\smash[b]{\frac{\exp(x_j)}{1+\frac1n\sum\limits_{k=1}^n\exp(x_k)}}\right]=0\ (\forall j\in\mathbb N)$

Question

Consideremos que el siguiente límite se mantiene $\forall j\in \mathbb{N}$ : $$ \lim_{n\rightarrow \infty}\left[ nP_j-\frac{\exp(x_j)}{1+\frac{1}{n}\smash[b]{\sum\limits_{k=1}^n} \exp(x_k)}\right]=0, $$ donde

• $P_j\in [0,1]$ $\forall j\in \mathbb{N}$ ,
• $x_j\in [-\overline{M}, \overline{M}]$ $\forall j\in \mathbb{N}$ con $\overline{M}<\infty$ .

Quiero demostrar que

$$ \lim_{n\rightarrow \infty}\left[\frac{1}{n}\sum_{j=1}^n n P_j-\frac{1}{n}\sum_{j=1}^ n\frac{\exp(x_j)}{1+\frac{1}{n}\smash[b]{\sum\limits_{k=1}^n} \exp(x_k)}\right]=0. $$

¿Alguna pista?

Answer 1

$\def\e{\mathrm{e}}$ Se demostrará que $$ \lim_{n \to ∞} \left( \frac{1}{n} \sum_{j = 1}^n nP_j - \frac{1}{n} \sum_{j = 1}^n \frac{\e^{x_j}}{\smash[b]{\displaystyle 1 + \frac{1}{n} \sum_{k = 1}^n \e^{x_k}}} \right) = \color{red}{-1}. $$

Denote $y_j = \e^{x_j}$ , $\displaystyle T_n = \frac{1}{n} \sum_{j = 1}^n y_j$ . Desde $|x_j| \leqslant \overline{M}$ para todos $j$ entonces existe $M > 0$ tal que $0 \leqslant y_j \leqslant M$ para todos $j$ . Por lo tanto, para cualquier $j$ , $$ 0 \leqslant \frac{\e^{x_j}}{\displaystyle 1 + \frac{1}{n} \sum_{k = 1}^n \e^{x_k}} = \frac{y_j}{1 + T_n} \leqslant y_j \leqslant M, \quad \forall n \in \mathbb{N}_+ $$ entonces $$ 0 = \lim_{n \to ∞} \frac{1}{n} · \lim_{n \to ∞} \left( nP_j - \frac{y_j}{1 + T_n} \right) = \lim_{n \to ∞} \left( P_j - \frac{1}{n} · \frac{y_j}{1 + T_n} \right) = P_j, $$ lo que implica $$ 0 = \lim_{n \to ∞} \left( nP_j - \frac{y_j}{1 + T_n} \right) = -y_j \lim_{n \to ∞} \frac{1}{1 + T_n}, $$ entonces $T_n > 0 \Rightarrow \lim\limits_{n \to ∞} T_n = +\infty$ . Por lo tanto, \begin{align*} &\mathrel{\phantom{=}}{} \lim_{n \to ∞} \left( \frac{1}{n} \sum_{j = 1}^n nP_j - \frac{1}{n} \sum_{j = 1}^k \frac{\e^{x_j}}{\smash[b]{\displaystyle 1 + \frac{1}{n} \sum_{k = 1}^n \e^{x_k}}} \right) = -\lim_{n \to ∞} \frac{1}{n} \sum_{j = 1}^n \frac{\e^{x_j}}{\displaystyle 1 + \frac{1}{n} \sum_{k = 1}^n \e^{x_k}}\\ &= -\lim_{n \to ∞} \frac{\displaystyle \frac{1}{n} \sum_{j = 1}^n \e^{x_j}}{\displaystyle 1 + \frac{1}{n} \sum_{k = 1}^n \e^{x_k}} = -\lim_{n \to ∞} \frac{\displaystyle \frac{1}{n} \sum_{j = 1}^n y_j}{\displaystyle 1 + \frac{1}{n} \sum_{k = 1}^n y_k} = -\lim_{n \to ∞} \frac{T_n}{1 + T_n} = -1. \end{align*}