Suppose I have two pairs of sequences of random variables $X_n, \tilde{X}_n$ and $Y_n, \tilde{Y}_n$, where $X_n$ and $Y_n$ are not necessarily independent for any $n$, but $\tilde{X}_n$ and $\tilde{Y}_n$ are independent for any $n$. I have the following convergences for two sets $A$ and $B$:

1. $|P(X_n \in A) - P(\tilde{X_n} \in A)| \to 0, n \to \infty$;
2. $|P(Y_n \in B) - P(\tilde{Y_n} \in B)| \to 0, n \to \infty$;

Do this convergences imply the following convergence?

$$ |P \left( \{ X_n \in A \} \& \{Y_n \in B \} \right) - P \left( \{ \tilde{X}_n \in A \} \& \{\tilde{Y}_n \in B \} \right)| \to 0 $$

I was trying to find any method on how to prove it... Any support will be very appreciated.

Context: Suppose I have two pairs of sequences of random variables $X_n, \tilde{X}_n$ and $Y_n, \tilde{Y}_n$, where $X_n$ and $Y_n$ are not necessarily independent for any $n$, but $\tilde{X}_n$ and $\tilde{Y}_n$ are independent for any $n$.

Moreover I know that dependence of $X_n$ and $Y_n$ is getting weaker when $n \to \infty$ in the following sense:

$$ \text{sup}_{A, B} |P(\{X_n \in A\} \& \{Y_n \in B\}) - P(X_n \in A) P(Y_n \in B)| \to 0, n \to \infty. $$

Question: I have the following convergences for two sets $A$ and $B$:

1. $|P(X_n \in A) - P(\tilde{X_n} \in A)| \to 0, n \to \infty$;
2. $|P(Y_n \in B) - P(\tilde{Y_n} \in B)| \to 0, n \to \infty$;

Do this convergences imply the following convergence?

$$ |P \left( \{ X_n \in A \} \& \{Y_n \in B \} \right) - P \left( \{ \tilde{X}_n \in A \} \& \{\tilde{Y}_n \in B \} \right)| \to 0 $$

I was trying to find any method on how to prove it... Any support will be very appreciated.

Suppose I have two pairs of sequences of random variables $X_n, \tilde{X}_n$ and $Y_n, \tilde{Y}_n$, where $X_n$ and $\tilde{X}_n$ are not necessarily independent for any $n$, but $Y_n$ and $\tilde{Y}_n$ are independent for any $n$. I have the following convergences for two sets $A$ and $B$:

1. $|P(X_n \in A) - P(\tilde{X_n} \in A)| \to 0, n \to \infty$;
2. $|P(Y_n \in B) - P(\tilde{Y_n} \in B)| \to 0, n \to \infty$;

Do this convergences imply the following convergence?

$$ |P \left( \{ X_n \in A \} \& \{Y_n \in B \} \right) - P \left( \{ \tilde{X}_n \in A \} \& \{\tilde{Y}_n \in B \} \right)| \to 0 $$

I was trying to find any method on how to prove it... Any support will be very appreciated.

Suppose I have two pairs of sequences of random variables $X_n, \tilde{X}_n$ and $Y_n, \tilde{Y}_n$, where $X_n$ and $Y_n$ are not necessarily independent for any $n$, but $\tilde{X}_n$ and $\tilde{Y}_n$ are independent for any $n$. I have the following convergences for two sets $A$ and $B$:

1. $|P(X_n \in A) - P(\tilde{X_n} \in A)| \to 0, n \to \infty$;
2. $|P(Y_n \in B) - P(\tilde{Y_n} \in B)| \to 0, n \to \infty$;

Do this convergences imply the following convergence?

$$ |P \left( \{ X_n \in A \} \& \{Y_n \in B \} \right) - P \left( \{ \tilde{X}_n \in A \} \& \{\tilde{Y}_n \in B \} \right)| \to 0 $$

I was trying to find any method on how to prove it... Any support will be very appreciated.