Demonstrate how to prove convergence in distribution when moment generating functions converge in a specified interval | Step-by-Step Solution

Hi there! This is a beautiful and sophisticated problem in probability theory. Let's break it down together so you can really understand how these powerful theorems work together.

What We're Solving:

We want to understand how to prove that a sequence of random variables converges in distribution by using two key tools: Prokhorov's theorem (which helps us find convergent subsequences) and the uniqueness property of moment generating functions (which helps us identify what we're converging to).

The Approach:

• Stage 1 (Prokhorov): We use Prokhorov's theorem to guarantee that some subsequence converges in distribution
• Stage 2 (MGF Uniqueness): We use the uniqueness of MGFs to prove that all possible limit points are the same, which means the entire sequence converges

This is a classic "compactness + uniqueness = convergence" argument that appears throughout mathematics!

Step-by-Step Solution:

Step 1: Set up the problem framework Let {Xₙ} be a sequence of random variables with moment generating functions Mₙ(t). Assume:

• Mₙ(t) → M(t) for all t ∈ (-δ, δ) for some δ > 0
• M(t) is the MGF of some random variable X

Step 2: Apply Prokhorov's theorem (the compactness part)

• First, we need to show the sequence {Xₙ} is tight (the probability mass doesn't "escape to infinity")
• Since MGFs converge in a neighborhood of 0, we can show moments are bounded
• Bounded moments imply tightness (this takes a bit of work with Chebyshev's inequality)
• By Prokhorov's theorem: every subsequence has a further subsequence that converges in distribution

Step 3: Identify what subsequences converge to (the uniqueness part)

• Take any convergent subsequence Xₙₖ ⇒ Y (convergence in distribution)
• By the continuity theorem for MGFs: Mₙₖ(t) → Mᵧ(t) in a neighborhood of 0
• But we also know Mₙₖ(t) → M(t) since it's a subsequence of our original sequence
• Therefore: Mᵧ(t) = M(t) in a neighborhood of 0

Step 4: Apply MGF uniqueness

• Since MGFs uniquely determine distributions (when they exist in a neighborhood of 0)
• We conclude Y has the same distribution as X
• This means every convergent subsequence converges to the same limit!

Step 5: Conclude full sequence convergence

• Every subsequence has a further subsequence converging to X
• All convergent subsequences have the same limit
• This is equivalent to the entire sequence converging: Xₙ ⇒ X

The Answer:

Theorem: If {Xₙ} is a sequence of random variables with MGFs Mₙ(t) such that Mₙ(t) → M(t) for all t in some interval (-δ, δ), and M(t) is the MGF of a random variable X, then Xₙ ⇒ X.

Proof Structure:

• 1. Use MGF convergence to establish tightness
• 2. Apply Prokhorov to get subsequential compactness
• 3. Use MGF uniqueness to show all limit points are identical
• 4. Conclude full sequence convergence

Memory Tip:

Remember this as the "Squeeze Play" theorem! 📝

• Prokhorov squeezes the sequence into a compact space (preventing escape to infinity)
• MGF uniqueness squeezes all possible limits into a single point
• Together, they squeeze the entire sequence to converge to one limit!

The key insight is that we're combining a topological tool (compactness) with an analytical tool (uniqueness) to get a powerful convergence result. This pattern appears in many areas of analysis - it's worth remembering!