Question

The random variable \(X_{n}\) takes the values \(k / n, k=1,2, \ldots, n\), each with probability \(1 / n\). Find its characteristic function and the limit as \(n \rightarrow \infty\). Identify the random variable of the limit characteristic function.

Short answer

The characteristic function for the random variable \(X_n\) is given by \(\phi_{X_n}(t) = \frac{1}{n} \frac{\mathrm{e}^{it/n}(1-\mathrm{e}^{it})}{1-\mathrm{e}^{it/n}}\). Taking the limit as \(n \to \infty\), we find that the limit of the characteristic function is \(1-\mathrm{e}^{it}\), which corresponds to a Bernoulli random variable with a success probability of \(1/2\).

Step-by-step solution

Definition of the characteristic function

The characteristic function of a random variable is a complex-valued function that provides an alternative description of the probability distribution of the random variable. Let \(X\) be a random variable taking values \(x_i\) with probabilities \(p_i\). Then the characteristic function \(\phi_X(t)\) for \(X\) is defined as follows: \[ \phi_X(t) = \mathbb{E}[\mathrm{e}^{itX}] = \sum_{i=1}^{\infty} p_i \mathrm{e}^{itx_i}, \] where \(t\) is a real number, and \(\mathbb{E}\) denotes the expected value operator.

Compute the characteristic function for \(X_n\)

We have the random variable \(X_n\), which takes the values \(\frac{k}{n}, k = 1, 2, \ldots, n\), each with a probability of \(\frac{1}{n}\). Thus, we can write the characteristic function \(\phi_{X_n}(t)\) as: \[ \phi_{X_n}(t) = \sum_{k=1}^{n} \frac{1}{n} \mathrm{e}^{it\frac{k}{n}}. \]

Simplify the characteristic function

We can now simplify the expression for the characteristic function \(\phi_{X_n}(t)\) by factoring out the common term \(\frac{1}{n}\): \[ \phi_{X_n}(t) = \frac{1}{n} \sum_{k=1}^{n} \mathrm{e}^{it\frac{k}{n}}. \] Notice that the sum is a finite geometric series with \(n\) terms, a common ratio of \(\mathrm{e}^{it/n}\), and a first term of \(\mathrm{e}^{it/n}\). We can now find the sum using the formula for the sum of a finite geometric series.

Compute the sum of the geometric series

The sum of a finite geometric series with \(n\) terms, common ratio \(r\), and first term \(a\) is given by: \[ S_n = \frac{a(1-r^n)}{1-r}. \] In our case, \(a = \mathrm{e}^{it/n}\), \(r = \mathrm{e}^{it/n}\), and \(n = n\). So, we can compute the sum: \[ S_n = \frac{\mathrm{e}^{it/n}(1-\mathrm{e}^{it})}{1-\mathrm{e}^{it/n}}. \] Now, substituting back into the expression for \(\phi_{X_n}(t)\), we get: \[ \phi_{X_n}(t) = \frac{1}{n} \frac{\mathrm{e}^{it/n}(1-\mathrm{e}^{it})}{1-\mathrm{e}^{it/n}}. \]

Compute the limit of the characteristic function

Now that we have the characteristic function for \(X_n\), we must compute its limit as \(n \to \infty\): \[ \lim_{n\to\infty} \phi_{X_n}(t) = \lim_{n\to\infty} \frac{1}{n} \frac{\mathrm{e}^{it/n}(1-\mathrm{e}^{it})}{1-\mathrm{e}^{it/n}}. \] As the limit involves an indeterminate form \(\frac{0}{0}\), we can apply L'Hôpital's rule by taking the derivative of the numerator and denominator separately with respect to \(n\). First, let's calculate the derivative of the numerator: \[ \frac{d}{dn} \left( \frac{\mathrm{e}^{it/n}(1-\mathrm{e}^{it})}{n} \right) = \frac{-it\mathrm{e}^{it/n}(1-\mathrm{e}^{it})}{n^2}. \] Next, let's calculate the derivative of the denominator: \[ \frac{d}{dn} \left( 1 - \mathrm{e}^{it/n} \right) = \frac{it\mathrm{e}^{it/n}}{n}. \] Applying L'Hôpital's rule to calculate the limit, we get: \[ \lim_{n\to\infty}\phi_{X_n}(t) = \lim_{n\to\infty}\frac{-it\mathrm{e}^{it/n}(1-\mathrm{e}^{it})}{n^2} \cdot \frac{n}{it\mathrm{e}^{it/n}} = 1-\mathrm{e}^{it}. \]

Identify the random variable of the limit characteristic function

We have found that the limit of the characteristic function for \(X_n\) as \(n \to \infty\) is given by \(1-\mathrm{e}^{it}\). This characteristic function corresponds to a Bernoulli random variable with a success probability of \(p=\frac{1}{2}\), as its characteristic function is given by \(\phi(t)=1-p+p\mathrm{e}^{it}=1-\frac{1}{2}+\frac{1}{2}\mathrm{e}^{it}=1-\mathrm{e}^{it}\). Therefore, the random variable of the limit characteristic function is a Bernoulli random variable with a success probability of \(1/2\).

Key concepts

Stochastic Processes and Their Characteristics

A stochastic process is a collection of random variables representing the evolution of some system of random values over time. It is akin to a random journey, where each step you take is uncertain and governed by probability laws. This is crucial in many fields, such as finance for stock prices, in physics for particle movement, or in biology for population dynamics. The problem provided involves a sequence \(X_n\), which is a simple illustration of a stochastic process where each random variable \(X_n\) has a different probability distribution as \(n\) changes.

In the analysis of stochastic processes, the characteristic function plays a vital role as it encapsulates all the probabilistic information about a random variable. This makes it a powerful tool for understanding the limiting behavior of sequences of random variables, as seen with \(X_n\) when \(n \rightarrow \infty\), and helps identify the underlying distribution, demonstrating its importance in stochastic process theory.

Probability Distribution and Characteristic Functions

The concept of a probability distribution describes how the probabilities are distributed over the outcomes of a random variable. It's a map detailing the landscape of chance—it tells you how likely you are to find a particular outcome in the territory of randomness.

In our exercise, we encounter the characteristic function, an alternate but equivalent description of a probability distribution. This function transforms probability into a language of complex exponential functions, providing a new perspective from which to analyze features such as the expected values, variances, and convergence of random variables—similar to changing from a street map view to a satellite view for a new angle of observation.

The Beauty of Geometric Series in Probability

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the ratio. It is a recurrent theme in various branches of mathematics, including probability. The sum of a finite geometric series can be compactly expressed, which makes it exceptionally useful for computing the characteristic function in our exercise.

The characteristic function for \(X_n\) involves a sum that forms a geometric series. To find the sum and thus the characteristic function, you cleverly apply the geometric series sum formula, turning a potentially complex problem into a straightforward computation. This illuminates the role geometric series play in streamlining probabilistic calculations.

L'Hôpital's Rule to the Rescue in Probability Limits

In calculus, L'Hôpital's rule is a standard way to evaluate limits of indeterminate forms such as \(0/0\) or \(\infty/\infty\). When you face a deterministic dead-end, this rule can provide a path forward. This scenario emerges in the exercise as we calculate the limit of the characteristic function for \(X_n\) as \(n\) tends toward infinity.

By differentiating the numerator and denominator with respect to \(n\), L'Hôpital's rule simplifies an otherwise complex limit, revealing the behavior of the characteristic function at the horizons of large \(n\). It's a powerful mathematical detour when direct roadways to limits are blocked by indeterminate forms.

Bernoulli Random Variable - A Pillar of Discrete Probability

A Bernoulli random variable is the simplest kind of discrete random variable, having only two possible outcomes—usually termed 'success' and 'failure'. In the grand casino of probability, it's akin to the flip of a fair coin. It relates deeply to the binomial distribution, where it describes a single trial with only two outcomes.

The characteristic function derived in our exercise converges to that of a Bernoulli random variable, as \(n\) grows without bounds. By recognizing this limiting characteristic function, we extend our understanding of how complex stochastic behaviors can often be traced back to simple, foundational elements of probability theory such as the Bernoulli random variable.