Convergence in Distribution

FOR SOLVED PREVIOUS PAPERS OF ISS KINDLY CONTACT US ON OUR WHATSAPP NUMBER 9009368238

FOR SOLVED PREVIOUS PAPERS OF ISS KINDLY CONTACT US ON OUR WHATSAPP NUMBER 9009368238

---

---

Convergence in Distribution

In probability theory, the concept of convergence in distribution is one of the fundamental modes of convergence for sequences of random variables. It describes how a sequence of random variables approaches a limiting distribution, even if the individual variables do not converge in a pointwise sense. This concept is crucial for understanding the behavior of estimators, the Central Limit Theorem, and many other statistical and probabilistic phenomena. In this blog, we’ll explore convergence in distribution in detail, including its definition, properties, and real-world applications.

---

1. What is Convergence in Distribution?

Convergence in distribution (also called weak convergence) describes the behavior of a sequence of random variables ( {X_n} ) as their distributions approach a limiting distribution. Unlike other modes of convergence (e.g., almost sure convergence or convergence in probability), convergence in distribution focuses on the convergence of the cumulative distribution functions (CDFs) rather than the random variables themselves.

Definition:

A sequence of random variables ( {X_n} ) converges in distribution to a random variable ( X ) if:
[
\lim_{n \to \infty} F_{X_n}(x) = F_X(x)
]
for all points ( x ) where ( F_X(x) ) is continuous. Here:

• ( F_{X_n}(x) ) is the CDF of ( X_n ).
• ( F_X(x) ) is the CDF of ( X ).

We denote this as:
[
X_n \xrightarrow{d} X
]

---

2. Key Properties of Convergence in Distribution

1. Focus on Distributions:

• Convergence in distribution is concerned with the convergence of the CDFs, not the random variables themselves. The random variables ( X_n ) and ( X ) do not need to be defined on the same probability space.

1. Continuity Points:

• The convergence ( F_{X_n}(x) \to F_X(x) ) is required only at points where ( F_X(x) ) is continuous. This allows for flexibility in handling discrete and mixed distributions.

1. Relation to Other Modes of Convergence:

• Convergence in distribution is weaker than almost sure convergence, convergence in probability, and convergence in ( L^p )-norm. However, it is often easier to verify in practice.

1. Portmanteau Lemma:

• Convergence in distribution is equivalent to any of the following:
  • ( E[g(X_n)] \to E[g(X)] ) for all bounded, continuous functions ( g ).
  • ( P(X_n \in A) \to P(X \in A) ) for all sets ( A ) with ( P(X \in \partial A) = 0 ), where ( \partial A ) is the boundary of ( A ).

---

3. Example: Applying Convergence in Distribution

Let’s walk through an example to see how convergence in distribution works in practice.

Problem:

Suppose ( {X_n} ) is a sequence of random variables where each ( X_n ) follows a Bernoulli distribution with parameter ( p_n = \frac{1}{n} ). Show that ( X_n \xrightarrow{d} X ), where ( X ) is a degenerate random variable with ( P(X = 0) = 1 ).

Solution:

1. Define the CDFs:

• The CDF of ( X_n ) is:
  [
  F_{X_n}(x) = \begin{cases}
  0 & \text{if } x < 0 \
  1 – p_n & \text{if } 0 \leq x < 1 \
  1 & \text{if } x \geq 1
  \end{cases}
  ]
• The CDF of ( X ) is:
  [
  F_X(x) = \begin{cases}
  0 & \text{if } x < 0 \
  1 & \text{if } x \geq 0
  \end{cases}
  ]

1. Compute the Limit:

• For ( x < 0 ):
  [
  \lim_{n \to \infty} F_{X_n}(x) = 0 = F_X(x)
  ]
• For ( x \geq 0 ):
  [
  \lim_{n \to \infty} F_{X_n}(x) = \lim_{n \to \infty} (1 – p_n) = 1 = F_X(x)
  ]

1. Conclusion:
   Since ( F_{X_n}(x) \to F_X(x) ) for all ( x ), we have ( X_n \xrightarrow{d} X ).

---

4. Applications of Convergence in Distribution

Convergence in distribution is widely used in various fields to analyze the behavior of sequences of random variables. Here are some examples:

a. Central Limit Theorem (CLT):

• The CLT states that the sum (or average) of a large number of independent, identically distributed random variables converges in distribution to a Normal distribution:
  [
  \frac{S_n – n\mu}{\sigma \sqrt{n}} \xrightarrow{d} N(0, 1)
  ]
  where ( S_n = X_1 + X_2 + \dots + X_n ), ( \mu = E[X_i] ), and ( \sigma^2 = \text{Var}(X_i) ).

b. Statistical Estimation:

• Many estimators, such as the sample mean or sample variance, converge in distribution to a limiting distribution (often Normal) as the sample size increases.

c. Stochastic Processes:

• In the study of stochastic processes, convergence in distribution is used to analyze the limiting behavior of processes like random walks or Brownian motion.

d. Finance:

• In financial modeling, the distribution of asset returns or portfolio values often converges to a limiting distribution as the number of observations or time steps increases.

---

5. Key Takeaways

• Convergence in distribution describes the convergence of the CDFs of a sequence of random variables to a limiting CDF.
• It is weaker than other modes of convergence but is often easier to verify.
• It is widely used in the Central Limit Theorem, statistical estimation, stochastic processes, and financial modeling.

---

6. Why Does Convergence in Distribution Matter?

Convergence in distribution is a powerful tool for understanding the limiting behavior of sequences of random variables. By understanding it, you can:

• Analyze the behavior of estimators and statistical tests.
• Understand the Central Limit Theorem and its applications.
• Model and analyze stochastic processes and financial data.

---

Conclusion

Convergence in distribution is a fundamental concept in probability and statistics, offering a way to understand the limiting behavior of sequences of random variables. Whether you’re analyzing the Central Limit Theorem, building statistical models, or studying stochastic processes, convergence in distribution provides the mathematical framework to analyze and predict outcomes. By mastering this concept, you’ll be well-equipped to tackle a wide range of problems in science, engineering, and beyond.

---