Question

assume of (x n, y n ) that x converges in distribution and y to c. show that (x.,, yn) (x, c). this is an example of problem 29.2

Answer 1

The question addresses the topic of convergence in distribution of random variables in probability theory, particularly of a sequence of vectors to a vector with a constant component.

Step-by-step explanation:

The student's question concerns the convergence in distribution of a sequence of random variables. Specifically, if a sequence of random vectors (xn, yn) is such that xn converges in distribution to x and yn converges to a constant c, one needs to show that the sequence of vectors converges in distribution to the vector (x, c). This is a demonstration of a concept in probability theory related to the convergence of random variables.

To answer this, we apply the concept that if yn converges to a constant c, it is basically saying that the distribution of yn becomes concentrated at the point c as n increases. For convergence in distribution, if xn converges in distribution to x, then for any continuous bounded function g, the expected value of g(xn) converges to the expected value of g(x).

Here, we consider a function g on R2 that splits into two separate functions such that g(xn, yn) = h1(xn) + h2(yn), where h1 and h2 are continuous. Then, by the properties of convergence in distribution and constants, limn → ∞ E[g(xn, yn)] = E[g(x, c)], which implies that (xn, yn) converges in distribution to (x, c).