Final answer:
When a sequence of functions fn converges weakly to a function f, and are dominated by an integrable function, their densities converge to the density of f in the limit as the sequence progresses.
Step-by-step explanation:
The student's question pertains to the convergence of sequence of functions, specifically in the context of probability density functions (pdfs). When a function fn converges weakly to a function f, and if each fn is dominated by an integrable function, we can use the Dominated Convergence Theorem (DCT) to show that the integration of fn converges to the integration of f. The density of f can therefore be understood as the limit of the densities of fn as n approaches infinity. Continuous probability density functions have the property where the probability is equivalent to the area under the pdf curve, which is bounded by the constraint that the total area under the pdf equals one.
A continuous probability density function is a function f(x) for which the area between the curve and the x-axis represents the probability of a continuous random variable falling within a certain range. Therefore, if a sequence of such functions fn converges to f, and their integrals are dominated by an integrable function, it follows that the densities of fn will also converge to that of f. This is due to the fact that the integrals of these functions represent the cumulative probabilities, and hence their limits reflect the convergence in density.
Therefore, when we want to find the density for the limiting function f, we consider the densities of fn and demonstrate that as n approaches infinity, the densities fn approach the density of f, providing us with a density function that is consistent with the weak convergence of the fn to f.