Question

Show that if \(\lim _{r \rightarrow \infty} \mathbf{X}_{r}\) exists, then it is unique.

Short answer

Question: Prove that if the limit of the sequence of random variables, \(\mathbf{X}_r\), as \(r\) approaches infinity exists, then it is unique. Answer: To prove the uniqueness of the limit, we assumed two different limits existed, \(\mathbf{A}\) and \(\mathbf{B}\), and their corresponding cumulative distribution functions (CDFs), \(F_A(x)\) and \(F_B(x)\). Using the definition of the limit, we set up an equation with an arbitrarily small \(\epsilon > 0\). Then, we used the triangle inequality to show that the difference between the CDFs, \(F_A(x)\) and \(F_B(x)\), is equal to zero, meaning that they are the same. Therefore, \(\mathbf{A} = \mathbf{B}\) and the limit of the sequence \(\mathbf{X}_r\) is unique if it exists.

Step-by-step solution

Assume two different limits exist

Assume that the limit of the sequence \(\mathbf{X}_r\) as \(r\) approaches infinity exists, and that these limits are two different random variables \(\mathbf{A}\) and \(\mathbf{B}\).

Define cumulative distribution functions

Let \(F_A(x)\) be the cumulative distribution function (CDF) of the random variable \(\mathbf{A}\), and \(F_B(x)\) be the CDF of the random variable \(\mathbf{B}\). Define \(F_r(x)\) as the CDF of \(\mathbf{X}_r\).

Use the definition of the limit to set up an equation

Since \(\lim_{r\to\infty} \mathbf{X}_r = \mathbf{A}\) and \(\lim_{r\to\infty} \mathbf{X}_r = \mathbf{B}\), their CDFs must also converge. This means that for any \(\epsilon > 0\), there exists an \(N\) such that for all \(r \ge N\), we have both \(|F_r(x) - F_A(x)| < \epsilon\) and \(|F_r(x) - F_B(x)| < \epsilon\).

Prove that the CDFs must be equal

Using the triangle inequality, we have: \(|F_A(x) - F_B(x)| = |(F_A(x) - F_r(x)) + (F_r(x) - F_B(x))| \le |F_A(x) - F_r(x)| + |F_r(x) - F_B(x)| < 2\epsilon\) Since \(\epsilon > 0\) can be arbitrarily small, it must be the case that \(|F_A(x) - F_B(x)| = 0\), and therefore, \(F_A(x) = F_B(x)\).

Conclude that the two limits are equal

Since we showed that both \(\mathbf{A}\) and \(\mathbf{B}\) have the same cumulative distribution function (\(F_A(x) = F_B(x)\)), it must be the case that \(\mathbf{A} = \mathbf{B}\). Thus, if the limit of the sequence \(\mathbf{X}_r\) exists, it is unique.

Key concepts

Cumulative distribution function

A cumulative distribution function, often abbreviated as CDF, is a crucial concept in probability theory and statistics. It is used to describe the distribution of random variables. Essentially, a CDF tells us the probability that a random variable is less than or equal to a certain value, denoted as:\[ F(x) = P(X \leq x) \]CDFs are important because they provide a comprehensive way to describe the distribution of random variables in a non-negative, non-decreasing function that approaches 1 as the variable goes to infinity. In the step-by-step solution, when we discuss the CDFs of the random variables \(\mathbf{A}\), \(\mathbf{B}\), and \(\mathbf{X}_r\), we are addressing their probability distributions - the manner in which their values spread over possible outcomes.

• Each random variable has its own CDF. It is a unique representation of the probability distribution.
• The use of CDFs in limits is powerful, allowing us to infer direct relationships about convergence of sequences and functions.

Understanding these CDFs allows us to manipulate and analyze the behavior of random variables in an intuitive fashion.

Uniqueness of limits

The uniqueness of limits is a fundamental concept in analysis, especially when dealing with sequences and functions. It states that if a sequence converges, it can converge to only one limit. This means, if \[ \lim_{r \rightarrow \infty} \mathbf{X}_r = \mathbf{A} \]and also \[ \lim_{r \rightarrow \infty} \mathbf{X}_r = \mathbf{B} \]then \(\mathbf{A}\) must equal \(\mathbf{B}\). This ensures the limit is unique.In the solution, uniqueness is enforced by showing that the CDFs \(F_A(x)\) and \(F_B(x)\) must be identical, which implies \(\mathbf{A}\) and \(\mathbf{B}\) cannot be different. This adheres to the core idea that if two convergent sequences have the same limit, their cumulative distributions align perfectly.

• The concept of uniqueness eliminates ambiguity in mathematical analysis.
• It provides clarity and predictability in modelling and solving problems.

When applying this to limits of sequences, as done in the step-by-step solution, it ensures that if there are two different potential limits, they must be reconciled into one, confirming the unique path of convergence.

Triangle inequality

The triangle inequality is a foundational concept in mathematics, particularly in the context of metric spaces and analysis. It states that for any real numbers \(a, b\), the following should always hold:\[ |a + b| \leq |a| + |b| \]This principle is intuitively understood like the sides of a triangle: the sum of any two sides must be greater than or equal to the third side.When applied to cumulative distribution functions in the solution, the triangle inequality helps show that the difference between \(F_A(x)\) and \(F_B(x)\) can be bounded by the sum of their individual differences with \(F_r(x)\). Mathematically,\[ |F_A(x) - F_B(x)| \leq |F_A(x) - F_r(x)| + |F_r(x) - F_B(x)| \]This relationship is used to demonstrate that if \(F_A(x)\) and \(F_B(x)\) are both within a small \(\epsilon\) range of \(F_r(x)\), they must eventually converge, resulting in \(F_A(x) = F_B(x)\).

• The triangle inequality is crucial in analysis for establishing bounds and convergence.
• It simplifies complex problems by breaking them into more manageable parts.

Thus, triangle inequality is a key tool in proving results like the uniqueness of limits by exploiting the properties of distances.