Return to Question

Consider a sequence of conditional pdf's $p_n(y | x)$ on a Polish space $X \times Y$, endowed with its Borel sigma algebra. Suppose, as $n\rightarrow \infty$, in $\mathbb{L}_1$ (the following densities can be seen as mixtures of normals with weights functions $\omega_{k,n}(X)$, $\omega^*_k(X)$ that sum up to 1 almost surely for fixed $X$ and $n$):

$$ \sum_{k=1}^{\infty} \omega_{k,n} (X) \frac{1}{\sqrt{2 \pi \sigma^2_k}} \exp\left( -\frac{(y - \beta_{0,k,n} - \beta_{1,k,n} X)^2}{2 \sigma^2_k} \right) = p_n(y | X) \stackrel{\mathbb{L}_1}{\rightarrow} p_0(y | X) = \sum_{k=1}^{\infty} \omega^*_k (X) \frac{1}{\sqrt{2 \pi \sigma^2_{k,*}}} \exp\left( -\frac{(y - \beta^*_{0,k} - \beta^*_{1,k} X)^2}{2 \sigma^2_{k,*}} \right) $$

is then true that for a sequence of measurable function $f_n \stackrel{\mathbb{L}_1}{\rightarrow} f_0$ also the "perturbed" conditional expectations converge as follows:

$$ \sum_{k=1}^{\infty} \omega_{k,n} (X) (\beta_{0,k,n} + \beta_{1,k,n} (X+f_n(X))) \stackrel{\mathbb{L}_1}{\rightarrow} \sum_{k=1}^{\infty} \omega^*_k (X) (\beta^*_{0,k} + \beta^*_{1,k} (X+f_0(X))) $$

PS: this question is related to this, but it is not the same, as I believe it is harder.

Consider a sequence of conditional pdf's $p_n(y | x)$ on a Polish space $X \times Y$, endowed with its Borel sigma algebra. Suppose, as $n\rightarrow \infty$, in $\mathbb{L}_1$ (the following densities can be seen as mixtures of normals with weights functions $\omega_{k,n}(X)$, $\omega^*_k(X)$ that sum up to 1 almost surely for fixed $X$ and $n$):

$$ \sum_{k=1}^{\infty} \omega_{k,n} (X) \frac{1}{\sqrt{2 \pi \sigma^2_k}} \exp\left( -\frac{(y - \beta_{0,k,n} - \beta_{1,k,n} X)^2}{2 \sigma^2_k} \right) = p_n(y | X) \stackrel{\mathbb{L}_1}{\rightarrow} p_0(y | X) = \sum_{k=1}^{\infty} \omega^*_k (X) \frac{1}{\sqrt{2 \pi \sigma^2_{k,*}}} \exp\left( -\frac{(y - \beta^*_{0,k} - \beta^*_{1,k} X)^2}{2 \sigma^2_{k,*}} \right) $$

is then true that for a sequence of measurable function $f_n \stackrel{\mathbb{L}_1}{\rightarrow} f_0$ also the "perturbed" conditional expectations converge as follows:

$$ \sum_{k=1}^{\infty} \omega_{k,n} (X) (\beta_{0,k,n} + \beta_{1,k,n} (X+f_n(X))) \stackrel{\mathbb{L}_1}{\rightarrow} \sum_{k=1}^{\infty} \omega^*_k (X) (\beta^*_{0,k} + \beta^*_{1,k} (X+f_0(X))) $$

Consider a sequence of conditional pdf's $p_n(y | x)$ on a Polish space $X \times Y$, endowed with its Borel sigma algebra. Suppose, as $n\rightarrow \infty$, in $\mathbb{L}_1$ ($N(a,b)$ denotes the normal distribution means $a$ and variance $b$ and weights functions $\omega_{k,n}(X)$ that sum up to 1):

$$ \sum_{k=1}^{\infty} \omega_{k,n} (X) N(\beta_{0,k,n} + \beta_{1,k,n} X, \sigma^2_k) = p_n(y | X) \stackrel{\mathbb{L}_1}{\rightarrow} p_0(y | X) = \sum_{k=1}^{\infty} \omega^*_k (X) N(\beta^*_{0,k} + \beta^*_{1,k} X, \sigma^2_{k,*}) $$

is then true that for a sequence of measurable function $f_n \stackrel{\mathbb{L}_1}{\rightarrow} f_0$ also the "perturbed" conditional expectations converge as follows:

$$ \sum_{k=1}^{\infty} \omega_{k,n} (X) (\beta_{0,k,n} + \beta_{1,k,n} (X+f_n(X))) \stackrel{\mathbb{L}_1}{\rightarrow} \sum_{k=1}^{\infty} \omega^*_k (X) (\beta^*_{0,k} + \beta^*_{1,k} (X+f_0(X))) $$

PS: this question is related to this, but it is not the same, as I believe it is harder.

Consider a sequence of conditional pdf's $p_n(y | x)$ on a Polish space $X \times Y$, endowed with its Borel sigma algebra. Suppose, as $n\rightarrow \infty$, in $\mathbb{L}_1$ (the following densities can be seen as mixtures of normals with weights functions $\omega_{k,n}(X)$, $\omega^*_k(X)$ that sum up to 1 almost surely for fixed $X$ and $n$):

$$ \sum_{k=1}^{\infty} \omega_{k,n} (X) \frac{1}{\sqrt{2 \pi \sigma^2_k}} \exp\left( -\frac{(y - \beta_{0,k,n} - \beta_{1,k,n} X)^2}{2 \sigma^2_k} \right) = p_n(y | X) \stackrel{\mathbb{L}_1}{\rightarrow} p_0(y | X) = \sum_{k=1}^{\infty} \omega^*_k (X) \frac{1}{\sqrt{2 \pi \sigma^2_{k,*}}} \exp\left( -\frac{(y - \beta^*_{0,k} - \beta^*_{1,k} X)^2}{2 \sigma^2_{k,*}} \right) $$

is then true that for a sequence of measurable function $f_n \stackrel{\mathbb{L}_1}{\rightarrow} f_0$ also the "perturbed" conditional expectations converge as follows:

$$ \sum_{k=1}^{\infty} \omega_{k,n} (X) (\beta_{0,k,n} + \beta_{1,k,n} (X+f_n(X))) \stackrel{\mathbb{L}_1}{\rightarrow} \sum_{k=1}^{\infty} \omega^*_k (X) (\beta^*_{0,k} + \beta^*_{1,k} (X+f_0(X))) $$

PS: this question is related to this, but it is not the same, as I believe it is harder.

Consider a sequence of conditional pdf's $p_n(y | x)$ on a Polish space $X \times Y$, endowed with its Borel sigma algebra. Suppose, as $n\rightarrow \infty$, in $\mathbb{L}_1$ ($N(a,b)$ denotes the normal distribution means $a$ and variance $b$ and weights functions $\omega_{k,n}(X)$ that sum up to 1):

$$ \sum_k \omega_{k,n} (X) N(\beta_{0,k,n} + \beta_{1,k,n} X, \sigma^2_k) = p_n(y | X) \stackrel{\mathbb{L}_1}{\rightarrow} p_0(y | X) = \sum_k \omega^*_k (X) N(\beta^*_{0,k} + \beta^*_{1,k} X, \sigma^2_{k,*}) $$

is then true that for a sequence of measurable function $f_n \stackrel{\mathbb{L}_1}{\rightarrow} f_0$ also the "perturbed" conditional expectations converge as follows:

$$ \sum_k \omega_{k,n} (X) (\beta_{0,k,n} + \beta_{1,k,n} (X+f_n(X))) \stackrel{\mathbb{L}_1}{\rightarrow} \sum_k \omega^*_k (X) (\beta^*_{0,k} + \beta^*_{1,k} (X+f_0(X))) $$

PS: this question is related to this, but it is not the same, as I believe it is harder.

Consider a sequence of conditional pdf's $p_n(y | x)$ on a Polish space $X \times Y$, endowed with its Borel sigma algebra. Suppose, as $n\rightarrow \infty$, in $\mathbb{L}_1$ ($N(a,b)$ denotes the normal distribution means $a$ and variance $b$ and weights functions $\omega_{k,n}(X)$ that sum up to 1):

$$ \sum_{k=1}^{\infty} \omega_{k,n} (X) N(\beta_{0,k,n} + \beta_{1,k,n} X, \sigma^2_k) = p_n(y | X) \stackrel{\mathbb{L}_1}{\rightarrow} p_0(y | X) = \sum_{k=1}^{\infty} \omega^*_k (X) N(\beta^*_{0,k} + \beta^*_{1,k} X, \sigma^2_{k,*}) $$

is then true that for a sequence of measurable function $f_n \stackrel{\mathbb{L}_1}{\rightarrow} f_0$ also the "perturbed" conditional expectations converge as follows:

$$ \sum_{k=1}^{\infty} \omega_{k,n} (X) (\beta_{0,k,n} + \beta_{1,k,n} (X+f_n(X))) \stackrel{\mathbb{L}_1}{\rightarrow} \sum_{k=1}^{\infty} \omega^*_k (X) (\beta^*_{0,k} + \beta^*_{1,k} (X+f_0(X))) $$

PS: this question is related to this, but it is not the same, as I believe it is harder.