Let X1, X2, …, Xn be independent random variables, each with...

Let $X_{1}, X_{2}, \ldots, X_{n}$ be independent random variables, each with characteristic function $\phi(t)$. Obtain the characteristic function of $$ Y_{n}=a_{n}+b_{n}\left(X_{1}+X_{2}+\cdots+X_{n}\right) $$ where $a_{n}$ and $b_{n}$ are arbitrary real numbers. Suppose that $\phi(t)=e^{-|t|^{\alpha}}$, where $0<\alpha \leq 2$. Determine $a_{n}$ and $b_{n}$ such that $Y_{n}$ has the same distribution as $X_{1}$ for $n=1,2, \ldots .$ Find the probability density functions of $X_{1}$ when $\alpha=1$ and when $\alpha=2 .$ (Oxford 1980F)

Let $X_{1}, X_{2}, \ldots, X_{n}$ be independent random variables, each with characteristic function $\phi(t)$. Obtain the characteristic function of
$$
Y_{n}=a_{n}+b_{n}\left(X_{1}+X_{2}+\cdots+X_{n}\right)
$$
where $a_{n}$ and $b_{n}$ are arbitrary real numbers.
Suppose that $\phi(t)=e^{-|t|^{\alpha}}$, where $0<\alpha \leq 2$. Determine $a_{n}$ and $b_{n}$ such that $Y_{n}$ has the same distribution as $X_{1}$ for $n=1,2, \ldots .$ Find the probability density functions of $X_{1}$ when $\alpha=1$ and when $\alpha=2 .$ (Oxford 1980F)

Key Concepts

Characteristic Function

A characteristic function uniquely defines the probability distribution of a random variable by encoding the distribution's moments (when they exist) via a Fourier transform. It is an essential tool in probability theory because it simplifies the study of sums of independent random variables through the property that the characteristic function of a sum is the product of the individual characteristic functions.

Linear Transformation of Random Variables

When a random variable is subjected to a linear transformation, by scaling and/or translating, its characteristic function is modified in a predictable way. Specifically, if Y = a + bX, then the characteristic function of Y becomes e^(ia t)?(b t), where ?(t) is the characteristic function of X. This property is fundamental in finding the distributions of transformed sums of random variables.

Stable Distributions

Stable distributions are a class of probability distributions that remain qualitatively unchanged under convolution, aside from scaling and location shifts. This means that a properly scaled and shifted sum of independent identically distributed random variables from a stable distribution will follow the same type of distribution. They are crucial for understanding the behavior of sums of heavy-tailed random variables and for the generalized central limit theorem.

Scaling and Self-Similarity

The scaling property of stable distributions implies that when summing independent copies of the distribution, one can choose scaling parameters to maintain the same distributional form. This self-similarity is evident in the condition that a scale factor proportional to n^(-1/?) and no translation (or an appropriate a_n) will yield a stable distribution identical to the original, a key property in problems involving limit distributions and renormalization.

Cauchy Distribution

The Cauchy distribution is a special case of stable distributions with ? = 1. It is characterized by heavy tails and a lack of finite moments (such as the mean and variance). Its probability density function is given by 1/(?(1+x^2)), and it serves as a classic example of a distribution that remains invariant (up to scale and shift) under summation of independent copies.

Normal Distribution

The normal distribution, associated with ? = 2 in the context of stable laws, is perhaps the most well-known example of a stable distribution. Its characteristic function is an exponential of a quadratic form, and it features light tails and finite moments of all orders. The normal distribution’s invariance under the summing of independent identical distributions and its scaling properties are central to the classical Central Limit Theorem.

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