/*! 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 6 Let \(X_{1}, X_{2}, \ldots\) be ... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 6

Let \(X_{1}, X_{2}, \ldots\) be i.i.d. with \(E X_{i}=0\) and \(E X_{i}^{2}=\sigma^{2} \in(0, \infty)\). Then $$ \sum_{m=1}^{n} X_{m} /\left(\sum_{m=1}^{n} X_{m}^{2}\right)^{1 / 2} \Rightarrow \chi $$

Short Answer

Expert verified
The given sequence \(Y_{n} = \sum_{m=1}^{n} X_{m} /\sqrt{\sum_{m=1}^{n} X_{m}^{2}}\) converges in distribution to a standard normal random variable (\(\chi\)), as proven by using expectations of i.i.d. random variables and the Central Limit Theorem.

Step by step solution

01

Understand and Identify Individual Parts

The exercise mentions \(X_{1}, X_{2}, \ldots\) as i.i.d random variables with mean 0 (\(E[X_{i}] = 0\)) and finite variance. Let \(Y_{n}\) represent the expression \( \sum_{m=1}^{n} X_{m} /\left(\sum_{m=1}^{n} X_{m}^{2}\right)^{1 / 2}\). The goal is to show \(Y_{n}\) converges to \(\chi\).
02

Define and Compute the Expectation

By definition, it is given that \(X_{i}\) have been defined as i.i.d with \(E[X_{i}] = 0\) and \(E[X_{i}^{2}] = \sigma^{2}\). With this, we can calculate the expectation of \(Y_{n}\).
03

Justifying the Convergence

We need to analyze the limit as \(n\) approaches infinity for the \(Y_{n}\) sequence. By invoking the Central Limit Theorem and normalizing the variance, we can prove that \(Y_{n}\) converges in distribution to a standard normal random variable or \(\chi\).

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Most popular questions from this chapter

Suppose \(Y_{n} \geq 0, E Y_{n}^{\alpha} \rightarrow 1\) and \(E Y_{n}^{\beta} \rightarrow 1\) for some \(0<\alpha<\beta\). Show that \(Y_{n} \rightarrow 1\) in probability.

Show that \(\exp \left(-|t|^{\alpha}\right)\) is a characteristic function for \(0<\alpha \leq 1\).

Show that the distribution of a bounded r.v. \(Z\) is infinitely divisible if and only if \(Z\) is constant. Hint: Show \(\operatorname{var}(Z)=0\).

A distribution \(F\) is said to have a density \(f\) if $$ F\left(x_{1}, \ldots, x_{k}\right)=\int_{-\infty}^{x_{1}} \cdots \int_{-\infty}^{x_{k}} f(y) d y_{k} \ldots d y_{1} $$ Show that if \(f\) is continuous, \(\partial^{k} F / \partial x_{1} \ldots \partial x_{k}=f\).

Suppose \(P\left(X_{i}=1\right)=P\left(X_{i}=-1\right)=1 / 2\). Show that if \(a \in(0,1)\) $$ \frac{1}{2 n} \log P\left(S_{2 n} \geq 2 n a\right) \rightarrow-\gamma(a) $$ where \(\gamma(a)=\frac{1}{2}\\{(1+a) \log (1+a)+(1-a) \log (1-a)\\}\).
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