Question

If \(\left\\{X_{j}\right\\}\) is a sequence of independent identically distributed random variables with mean 0 and variance 1 , the distributions of $$ \sum_{1}^{n} X_{j} /\left(\sum_{1}^{n} X_{j}^{2}\right)^{1 / 2} \quad \text { and } \quad \sqrt{n} \sum_{1}^{n} X_{j} / \sum_{1}^{n} X_{j}^{2} $$ both converge vaguely to the standard normal distribution.

Short answer

Both expressions converge vaguely to the standard normal distribution as \( n \to \infty \).

Step-by-step solution

Understanding the Problem

We are given a sequence of independent and identically distributed (i.i.d.) random variables \( \left\{X_{j}\right\} \) with mean 0 and variance 1. We need to analyze the distributions of two specific sums to see if they converge to the standard normal distribution as the number of terms \( n \) goes to infinity.

Expression Setup

Examine each expression separately. For the first expression, \( \sum_{j=1}^{n} X_{j} / \left(\sum_{j=1}^{n} X_{j}^{2}\right)^{1/2} \), this is known as a standardized sum of random variables. The second expression is \( \sqrt{n} \sum_{j=1}^{n} X_{j} / \sum_{j=1}^{n} X_{j}^{2} \). Both need to be evaluated for distribution convergence.

Analyzing First Expression

The first expression is similar to the normalization appearing in the Central Limit Theorem (CLT). It essentially represents the variance stability property because \( \text{Var}(\sum_{j=1}^{n} X_{j}) = n \) and we are normalizing this by \(\left(\sum_{j=1}^{n} X_{j}^{2}\right)^{1/2}\). By CLT, this sum converges to a normal distribution as \( n \) approaches infinity.

Analyzing Second Expression

The second expression can be viewed as a likelihood ratio after scaling by \( \sqrt{n} \). Here, as \( n \) increases, \( \sum_{j=1}^{n} X_{j}^{2} \) converges to \( n \) (since the expected value is 1 and the sum contains \( n \) terms). Due to self-normalization and ratio scaling effects after dividing by \( n \) (scaled), this also approaches the standard normal distribution limit.

Conclusion on Convergence

Thus, both expressions demonstrate that under the law of large numbers and the Central Limit Theorem, the standardizing and scaling effects lead both ratios to converge vaguely to the standard normal distribution.

Key concepts

Independent Identically Distributed (i.i.d) Random Variables

Understanding independent identically distributed (i.i.d) random variables is crucial in probability and statistics. These variables are both independent from one another and identically distributed. This means that:

• Each random variable in the sequence does not affect the other. For example, rolling two dice simultaneously results in independent outcomes.
• The probability distribution of each random variable is the same. If you have multiple dice, each die exhibits the same behavior — six outcomes, each equally likely.

The concept of i.i.d variables is pivotal for simplifying the analysis of complex stochastic processes. It helps in utilizing tools such as the Central Limit Theorem, which is often used when analyzing the convergence of sequences of random variables.

Standard Normal Distribution

The standard normal distribution is a special case of the normal distribution. It has a mean of 0 and a standard deviation of 1. This is often represented by the notation \( N(0,1) \). The standard normal distribution is symmetric around the mean, with the following key characteristics:

• The curve is bell-shaped and continuous.
• Approximately 68% of the data falls within one standard deviation of the mean.
• Approximately 95% of the data falls within two standard deviations, while about 99.7% falls within three.

In statistical applications, raw data is often transformed to fit this standard form, making it easier to apply statistical inference techniques. The central limit theorem guarantees that, for a large number of i.i.d. variables, their sum tends to follow a normal distribution, even if the original variables are not normally distributed.

Self-normalization

Self-normalization is an approach used in probability that allows for more robust assessments of data when scaling factors can be variable. It refers to techniques where sums are standardized using properties intrinsic to the data. Such as in the exercise given, standardizing a sum by dividing by its own variability-based measure (like the square root of the sum of squared variables) is a prominent example.

Self-normalized statistics are particularly useful when handling datasets that may not fit the traditional prerequisites for the Central Limit Theorem, offering efficiency in estimating distributions' convergence. This method utilizes data-driven measures for normalization, thus being adaptable to the characteristics of the specific data set.

Ultimately, this adapts as a balance against fluctuating variances and assists in achieving convergence to the standard normal distribution, as outlined in various statistical analyses.