Question

Let \(\xi_{n}\) be nonnegative random variables satisfying $$ E\left[\xi_{n+1} \mid \xi_{1}, \ldots, \xi_{n}\right] \leq \delta_{n}+\xi_{n} $$ where \(\delta_{n} \geq 0\) are constants and \(\Delta=\sum_{n=1}^{\infty} \delta_{n}<\infty .\) Show that with probability one, \(\xi_{n}\) converges to a finite random variable \(\xi\) as \(n \rightarrow \infty\).

Short answer

To show that the sequence \(\xi_n\) converges almost surely to a finite random variable \(\xi\), we first prove that \((\xi_n)\) is a supermartingale using the given conditional expectation inequality. It follows that \(E\left[\xi_{n+1} \mid \xi_{1}, \ldots, \xi_{n}\right] \leq E[\delta_n + \xi_n \mid \xi_1, ..., \xi_n]\), establishing that \((\xi_n)\) is indeed a supermartingale. Next, we apply the Martingale Convergence Theorem and show that \((\xi_n)\) is bounded in \(L^1\). We have \(E[\xi_{k+1}] \leq \sum_{n=1}^{k} E[\delta_n] + E[\xi_1] < \infty\), so the sequence \(\xi_n\) converges almost surely to a finite random variable \(\xi\) as \(n \rightarrow \infty\).

Step-by-step solution

Show that \((\xi_n)\) is a supermartingale

: Recall the definition of a supermartingale - a sequence of random variables \((X_n)_{n \geq 0}\) is a supermartingale if for all \(n\geq 0\), we have: $$ E\left[X_{n+1} \mid X_{1}, \ldots, X_{n}\right] \leq X_n $$ In our case, we know $$ E\left[\xi_{n+1} \mid \xi_{1}, \ldots, \xi_{n}\right] \leq \delta_{n}+\xi_{n} $$ Additionally, since \(\delta_n \geq 0\): $$ \xi_n \leq \delta_n + \xi_n $$ Taking expectations over \(\delta_n + \xi_n\) considering the conditions on the right side of the inequality gives us $$ E[\delta_n + \xi_n \mid \xi_1, ..., \xi_n] = \delta_n + E[\xi_n \mid \xi_1, ..., \xi_n]. $$ Hence, we get $$ E\left[\xi_{n+1} \mid \xi_{1}, \ldots, \xi_{n}\right] \leq E[\delta_n + \xi_n \mid \xi_1, ..., \xi_n]. $$ Thus, \((\xi_n)\) is indeed a supermartingale.

Apply Martingale Convergence Theorem

: Now that we have established that \((\xi_n)\) is a supermartingale, we can apply the Martingale Convergence Theorem, which states that a supermartingale bounded in \(L^1\) (the space of integrable functions) converges almost surely. First, note that by the Monotone Convergence Theorem: $$ E\left[\sum_{n=1}^{\infty} \delta_n\right] = \sum_{n=1}^{\infty} E\left[\delta_n\right] < \infty $$ We also know that $$ E\left[\xi_{n+1} \mid \xi_{1}, \ldots, \xi_{n}\right] \leq \delta_{n}+\xi_{n} \implies \xi_{n+1} \leq \delta_{n}+\xi_{n} $$ Notice that by summing the above inequality for \(n=1,...,k\), we get: $$ \xi_{k+1} \leq \sum_{n=1}^{k} \delta_{n} + \xi_1 $$ Taking expectations, we have: $$ E[\xi_{k+1}] \leq E\left[\sum_{n=1}^{k} \delta_{n} + \xi_1\right] = \sum_{n=1}^{k} E[\delta_n] + E[\xi_1] \leq \sum_{n=1}^{\infty} E[\delta_n] + E[\xi_1] < \infty $$ Therefore, \((\xi_n)\) is indeed bounded in \(L^1\), and by the Martingale Convergence Theorem, our supermartingale converges almost surely (with probability one) to a finite random variable \(\xi\) as \(n \rightarrow \infty\).

Key concepts

Martingale Convergence Theorem

The Martingale Convergence Theorem is a powerful result in probability theory that asserts the conditions under which a martingale or a supermartingale will converge to a limit. Let's say we have a sequence of random variables that depicts a supermartingale. The big question then is: does this sequence converge, and if yes, to what? The theorem provides a reassuring answer: under certain conditions, yes, it does converge!

For a sequence of random variables to qualify as a martingale, the expectation of the next random variable given the past history must equal the present value. For supermartingales, this conditional expectation is less than or equal to the present value. When these random variables are bounded in the space of integrable functions, known as \(L^1\), the Martingale Convergence Theorem tells us they converge almost surely, which means with probability one.

This forms the basis for a vast array of applications in areas such as economics, finance, and various fields of statistics where long-run behavior predictions are crucial.

Nonnegative Random Variables

A nonnegative random variable, as the name suggests, is one that takes on values that are greater than or equal to zero. In the realm of probability and statistics, such variables are important because they often represent quantities like time, distance, or other measurements where negative values would be nonsensical or impossible.

Nonnegative random variables are inherently bounded from below, which introduces interesting properties when considering their long-term behavior. For example, sequences of these types of variables are often easier to handle when it comes to questions of convergence, as their non-negativity implies certain integrability and summability characteristics. These properties come in handy when applying the Martingale Convergence Theorem.

Conditional Expectation

The concept of conditional expectation is akin to peering into a magic crystal ball—but for random variables. It's an expectation value that's calculated with the knowledge of additional information or given a certain condition.

More formally, if you have a random variable and you're given some information about other variables or events, the conditional expectation is your 'best guess' at the random value, considering that information. It's a fundamental concept in probability theory that helps in understanding the future behavior of random sequences, given their past. This concept isn't just a theoretical construct; it has practical applications ranging from predicting stock prices to assessing risks in insurance.

Almost Sure Convergence

Imagine you're betting on a sequence of random events and you'd like to know if, given enough time, you'll eventually reach a point where the outcome doesn't jump around unpredictably anymore. The concept of almost sure convergence deals with this scenario. It means that a sequence of random variables converges to a fixed variable, except perhaps on a set with probability zero.

In more day-to-day terms, think of it like this: if you're flipping a fair coin an infinite number of times, almost sure convergence means that the proportion of heads to total flips settles down to a half as you flip more and more - even though in practice you might see streaks of heads or tails. In our textbook problem, almost sure convergence means that our supermartingale doesn't just dance around aimlessly; it settles down nicely to a finite value as time goes on, with complete certainty except for a negligible chance.