On Lin's condition for products of random variables

TL;DR

This paper explores Lin's condition for products of independent random variables, providing a new proof and demonstrating the necessity of independence for the condition to hold.

Contribution

It offers a new proof that Lin's condition is preserved under multiplication of independent variables and clarifies that independence is essential for this property.

Findings

Lin's condition is preserved under multiplication of independent variables.

Without independence, Lin's condition may not hold for the product.

The paper provides a new proof of the main result.

Abstract

The paper presents an elaboration of some results on Lin's conditions. A new proof of the fact that if densities of independent random variables $\xi_1$ and $\xi_2$ satisfy Lin's condition, the same is true for their product is presented. Also, it is shown that without the condition of independence, the statement is no longer valid.