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Chapter 6: Problem 1

Let \(X\) be a random variable on \((\Omega, \mathcal{A}, \mathbf{P})\) and let $$ \Lambda(t):=\log \left(\mathbf{E}\left[e^{t X}\right]\right) \quad \text { for all } t \in \mathbb{R} $$ Show that \(D:=\\{t \in \mathbb{R}: \Lambda(t)<\infty\\}\) is a nonempty interval and that \(\Lambda\) is infinitely often differentiable in the interior of \(D\).

Short Answer

Expert verified
The set \( D \) is a nonempty interval, and \( \Lambda \) is infinitely differentiable in the interior of \( D \).

Step by step solution

01

Define the Set D

The set \( D = \{ t \in \mathbb{R} : \Lambda(t) < \infty \} \) consists of values of \( t \) for which the moment-generating function \( \mathbf{E}[e^{tX}] \) is finite. To show \( D \) is nonempty, note that \( \Lambda(0) = \log(\mathbf{E}[e^{0 \cdot X}]) = \log(1) = 0 \), which is finite. Thus, \( 0 \in D \), proving that \( D \) is nonempty.
02

Show D is an Interval

For \( D \) to be an interval, if \( t_1, t_2 \in D \) and \( s \) is between \( t_1 \) and \( t_2 \), we must have \( s \in D \). Consider \( \Lambda(s) = \log(\mathbf{E}[e^{sX}]) \). By H枚lder鈥檚 inequality, \( \mathbf{E}[e^{sX}] \leq \mathbf{E}[e^{t_1X}]^{\frac{t_2-s}{t_2-t_1}} \mathbf{E}[e^{t_2X}]^{\frac{s-t_1}{t_2-t_1}} \). If \( \mathbf{E}[e^{t_1 X}] < \infty \) and \( \mathbf{E}[e^{t_2 X}] < \infty \), then \( \mathbf{E}[e^{sX}] < \infty \), ensuring \( \Lambda(s) < \infty \). Thus, \( D \) is an interval.
03

Prove Infinite Differentiability of \( \Lambda \) in the Interior

The function \( \Lambda(t) = \log(\mathbf{E}[e^{tX}]) \) is the logarithm of a moment-generating function which, under the finite conditions (i.e., \( t \in \text{int}(D) \)), is differentiable. The derivatives exist since the moment-generating function is smooth where it's finite, assuring differentiability for all orders. The derivatives can be found using Leibniz's rule since derivatives of \( \mathbf{E}[e^{tx}] \) lead to terms involving moments which are ensured to be finite.

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Key Concepts

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

Random Variable
A random variable is a fundamental concept in probability and statistics, representing outcomes of a random phenomenon. In simple terms, it's a variable whose value is determined by the outcome of a random event.

Some characteristics of random variables include:
  • **Types**: They can be discrete, dealing with distinct outcomes like rolling a die, or continuous, involving a range of outcomes such as the temperature on a given day.
  • **Probability Distribution**: It describes how probabilities are assigned to each outcome or interval of outcomes. For example, a fair die has a uniform distribution over its six faces.
  • **Expectation**: This is the theoretical mean of a random variable, providing a measure of the central tendency.
Understanding random variables is crucial, as they form the building blocks for more advanced concepts such as moment-generating functions, which help in analyzing the distribution of the variable's outcomes.
Differentiability
Differentiability refers to the property of a function to have a derivative at each point in its domain. If a function is differentiable, it implies the function is smooth and has a well-defined tangent at each point.

Here are some key concepts about differentiability:
  • **Derivative**: The derivative of a function at a point measures the rate at which the function value changes as its input changes.
  • **Continuity**: A function must be continuous to be differentiable at a point. However, continuity alone does not ensure differentiability.
  • **Infinitely Differentiable**: A function is termed infinitely differentiable if it can be differentiated an infinite number of times and all these derivatives exist.
In the context of the moment-generating function, differentiability in the interval is ensured when the function is smooth, meaning all its derivatives exist and are continuous within that interval. This smoothness allows us to use tools like the Taylor series expansion to analyze and approximate the function.
Logarithm Function
The logarithm function plays a significant role in expanding our understanding of mathematical operations, specifically when dealing with product-related problems. It inverts the process of exponentiation and is particularly useful to simplify complex problems involving multiplication.

Here are a few important aspects of the logarithm function:
  • **Base of Logarithm**: The base can be any positive number, but common bases include base 10 (common logarithm) and base \( e \) (natural logarithm).
  • **Properties**: Logarithms transform multiplication into addition, division into subtraction, powers into multipliers, and roots into divisors.
  • **Inverse Function**: The logarithmic function is the inverse of the exponential function. If \( y = \log_b(x) \), then \( b^y = x \).
In the given exercise, the function \(\Lambda(t) = \log(\mathbf{E}[e^{tX}])\) is the logarithm of the moment-generating function, which facilitates easier differentiation due to the properties of the logarithm. It allows us to assess the growth behavior of expected values in a manageable way.
H枚lder's Inequality
H枚lder's inequality is a fundamental result in analysis, providing a generalized form of the Cauchy-Schwarz inequality. It plays a crucial role in various fields, including functional analysis and probability.

The main features of H枚lder's inequality include:
  • **Description**: It describes a relationship between integrals of products of functions and the integrals of powers of those functions.
  • **Formulation**: If \( f \) and \( g \) are measurable functions, then \( \int |fg| \leq \left(\int |f|^p\right)^{1/p} \left(\int |g|^q\right)^{1/q} \) for \( 1/p + 1/q = 1 \). This assumes the integrals on the right exist and are finite.
  • **Applications**: It's extensively used in proving results involving Lebesgue integration and spaces like \( L^p \) spaces.
In the context of the exercise, H枚lder's inequality helps assure that the moment-generating function remains finite over an interval, thereby ensuring that the function \( \Lambda(t) \) exists and is differentiable over this domain.

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Most popular questions from this chapter

Let \(X_{1}, X_{2}, \ldots\) be independent, square integrable, centered random variables with \(\sum_{i=1}^{\infty} \operatorname{Var}\left[X_{i}\right]<\infty .\) Show that there exists a square integrable \(X\) with \(X=\lim _{n \rightarrow \infty} \sum_{i=1}^{n} X_{i}\) almost surely.

(Egorov's theorem (1911), [41]) Let \((\Omega, \mathcal{A}, \mu)\) be a finite measure space and let \(f_{1}, f_{2}, \ldots\) be measurable functions that converge to some \(f\) almost everywhere. Show that, for every \(\varepsilon>0\), there is a set \(A \in \mathcal{A}\) with \(\mu(\Omega \backslash A)<\varepsilon\) and \(\sup _{\omega \in A}\left|f_{n}(\omega)-f(\omega)\right| \stackrel{n \rightarrow \infty}{\longrightarrow} 0\)

Give an example of a sequence that (i) converges in \(L^{1}\) but not almost everywhere, (ii) converges almost everywhere but not in \(L^{1}\).
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