/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 25 Let \(X_{1}, X_{2} \ldots\) be r... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 25

Let \(X_{1}, X_{2} \ldots\) be random variables, and let \(N_{1}, N_{2} \ldots\), be random variables taking values in the positive integers such that \(N_{k} \stackrel{\mathrm{P}}{\rightarrow} \infty\) as \(k \rightarrow \infty\). Show that: (i) if \(X_{n} \stackrel{\mathrm{D}}{\rightarrow} X\) and the \(X_{n}\) are independent of the \(N_{k}\), then \(X_{N_{k}} \stackrel{\mathrm{D}}{\rightarrow} X\) as \(k \rightarrow \infty\), (ii) if \(X_{n} \stackrel{\text { as }}{\longrightarrow} X\) then \(X_{N_{k}} \stackrel{\text { P }}{\rightarrow} X\) as \(k \rightarrow \infty\).

Short Answer

Expert verified
(i) \(X_{N_k} \stackrel{\mathrm{D}}{\rightarrow} X\); (ii) \(X_{N_k} \xrightarrow{\mathrm{P}} X\).

Step by step solution

01

Understand Convergence Concepts

Before solving, it's critical to understand the types of convergence used here. \(X_n \stackrel{\mathrm{D}}{\rightarrow} X\) means that the distribution of \(X_n\) converges to that of \(X\), whereas \(X_n \stackrel{\text{as}}{\longrightarrow} X\) means that \(X_n\) converges to \(X\) almost surely, and \(N_k \stackrel{\mathrm{P}}{\rightarrow} \infty\) means that for any \(M > 0\), \(P(N_k > M) \to 1\) as \(k \to \infty\).
02

Use Independence for Part (i)

Given \(X_n \stackrel{\mathrm{D}}{\rightarrow} X\) and \(N_k\) is independent of \(X_n\), for any \(x\), \(P(X_{N_k} \leq x) = \sum_{n} P(N_k = n)P(X_n \leq x)\). Since \(\sum_{n} P(N_k = n) = 1\) and \(X_n \stackrel{\mathrm{D}}{\rightarrow} X\), by the convergence of probability measures, \(P(X_n \leq x) \to P(X \leq x)\) for each \(x\). This means \(P(X_{N_k} \leq x) \to P(X \leq x)\) as \(k \to \infty\). Thus, \(X_{N_k} \stackrel{\mathrm{D}}{\rightarrow} X\).
03

Apply Almost Sure Convergence for Part (ii)

Since \(X_n \stackrel{\text{as}}{\longrightarrow} X\), for almost every \(\omega\), there exists an \(n_0\) such that for all \(n \geq n_0\), \(X_n(\omega) = X(\omega)\). As \(N_k \to \infty\) in probability, for any \(\epsilon > 0\) and almost every \(\omega\), there exists a \(k_0\) such that for all \(k \geq k_0\), \(N_k \geq n_0\). Therefore, \(X_{N_k}(\omega) = X(\omega)\) for all \(k \geq k_0\), thus \(X_{N_k} \xrightarrow{\mathrm{P}} X\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributional Convergence
Distributional convergence, represented as \(X_n \stackrel{\mathrm{D}}{\rightarrow} X\), is a type of convergence involving random variables. It means that the cumulative distribution functions (CDFs) of \(X_n\) get closer and closer to the CDF of \(X\) as \(n\) increases. This is similar to saying, in broad terms, that \(X_n\) "behaves more and more like" \(X\) in terms of distribution.
Nevertheless, it's important to realize that this convergence only relates to the distribution, not specific values. Thus, it allows variables to "wiggle around" as long as their probability distributions are aligning.
Key points about distributional convergence:
  • It doesn't imply the convergence of individual values.
  • It's often used when dealing with sequences of random variables in probability theory.
  • It's crucial for understanding laws of large numbers and the Central Limit Theorem.
Understanding distributional convergence makes it easier to work with complex distributions of random variables in statistical contexts.
Almost Sure Convergence
Almost sure convergence, noted as \(X_n \stackrel{\text{as}}{\longrightarrow} X\), is a stronger form of convergence than distributional convergence. It means that for almost every outcome \(\omega\), the sequence \(X_n(\omega)\) will converge to \(X(\omega)\) as \(n\) goes to infinity.
This "almost sure" part refers to the fact that the probability of the set of \(\omega\) for which the convergence fails is zero. In simpler words, it is extremely likely for the sequence to converge at each point, aside from a set which is negligible in the probability sense.
Highlights of almost sure convergence:
  • Implies the sequence will eventually stabilize at the random variable \(X\) for most outcomes.
  • Important when considering long-term behavior in probability and statistics.
  • Used in the proof of strong laws of probability.
Understanding almost sure convergence helps grasp the concept of convergence in a practical and "real-world" sense.
Probability
Probability is the measure of how likely an event is to occur. It's a fundamental concept in mathematics and statistics, serving as the base for understanding more complex ideas like convergence of random variables.
Probability is represented as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. It helps in determining the likelihood of various outcomes and is crucial for decision making and predicting future occurrences.
Essentials of probability:
  • Used to model randomness and uncertainty in systems.
  • Calculates the likelihood of certain events or outcomes.
  • Forms the foundation for statistical inference and decision theories.
Grasping the basics of probability is essential for delving deeper into the study of random variables and their properties.
Random Variables
A random variable is a variable whose values depend on outcomes of a random phenomenon. Essentially, it assigns numerical values to each outcome in a probability space, which lays the groundwork for probabilistic analysis.
Random variables can be discrete, having specific separated values, or continuous, covering a range of values. They are fundamental to understanding probability distributions and calculating expected values, variances, and other statistical measures.
Characteristics of random variables:
  • Represent outcomes of random processes numerically.
  • Used to model uncertainties in various fields like finance, engineering, and science.
  • Key to defining probability distributions and capturing the behavior of random systems.
Understanding random variables is crucial for analyzing how probabilities are assigned to different potential outcomes.

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Most popular questions from this chapter

Let \(X_{n}\) be the net profit to the gambler of betting a unit stake on the \(n\)th play in a casino; the \(X_{n}\) may be dependent, but the game is fair in the sense that \(\mathbb{E}\left(X_{n+1} \mid X_{1}, X_{2} \ldots . X_{n}\right)=0\) for all n. The gambler stakes \(Y\) on the first play, and thereafter stakes \(f_{n}\left(X_{1}, X_{2} \ldots, X_{n}\right)\) on the \((n+1)\) th play, where \(f_{1}, f_{2} \ldots\) are given functions. Show that her profit after \(n\) plays is $$ s_{n}=\sum_{i=1}^{n} X_{i} f_{i-1}\left(X_{1}, X_{2} \ldots \ldots, x_{i-1}\right) $$ where \(f_{0}=Y\). Show further that the sequence \(S=\left[S_{n} \mid\right.\) satisfies the martingale condition \(\mathrm{B}\left(S_{n+1} \mid\right.\) \(\left.X_{1}, X_{2}, \ldots, X_{n}\right)=S_{n}, n \geq 1\), if \(Y\) is assumed to be known throughout.

Let \(\left(X_{n}: n \geq 1\right)\) be independent random variables with continuous common distribution function \(F\). We call \(X_{k}\) a record value for the sequence if \(X_{k}>X_{r}\) for \(1 \leq r

Let \(\left\\{X_{n}: n \geq 1\right)\) be a sequence of independent random variables with \(\mathrm{P}\left(X_{n}=1\right)=\mathrm{P}\left(X_{n}=-1\right)=\frac{1}{2}\). Does the series \(\sum_{r=1}^{n} X_{r} / r\) converge a.s. as \(n \rightarrow \infty\) ?

Complete convergence. \(\mathrm{A}\) sequence \(X_{1}, X_{2}, \ldots\) of random variables is said to be completely comergent to \(X\) if $$ \sum_{n} P\left(\left|X_{n}-X\right|>f\right)<\infty \quad \text { for all } f>0 $$ Show that, for sequences of independent variables, complete convergence is equivalent to a.s. convergence. Find a sequence of (dependent) random variables which converges a.s. but not completely,

Let \(\left\\{X_{r}: r \geq 1 /\right.\) be independent Poisson variables with respective parameters \(\left\\{\lambda_{r}: r \geq 1\right]\). Show that \(\sum_{r=1}^{\infty} X_{r}\) converges or diverges almost surely according as \(\sum_{r=1}^{\infty} \lambda_{r}\) converges or diverges.
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