Question

Suppose \(S_{n}=X_{1}+\cdots+X_{n}\) is a zero-mean martingale for which \(E\left[X_{n}^{2}\right]\) \(<\infty\) for all \(n .\) Show that \(S_{n} / b_{n} \rightarrow 0\) with probability one for any monotonic real sequence \(b_{1} \leq \cdots \leq b_{n} \leq b_{n+1} \uparrow \infty\), provided \(\sum_{n=1}^{\infty} E\left[X_{n}^{2}\right] / b_{n}^{2}<\infty\).

Short answer

In order to prove that \(S_{n}/b_{n} \rightarrow 0\) with probability one, we can use Kolmogorov's zero-one law and the Borel-Cantelli lemma along with the given conditions. By defining the events \(A_n = \left\{S_{n}/b_{n} > \epsilon\right\}\) and showing that they are independent, we can apply the Borel-Cantelli lemma. Using the given condition \(\sum_{n=1}^{\infty} E\left[X_{n}^{2}\right] / b_{n}^{2}<\infty\), we can show that \(\sum_{n=1}^{\infty} P(A_{n}) < \infty\), which implies that \(P(\limsup A_n) = 0\). This means that, given the original condition, the event \(\limsup A_n\) has probability \(0\), and therefore \(S_{n}/b_{n} \rightarrow 0\) with probability one.

Step-by-step solution

Define the martingale and given conditions

We are given a zero-mean martingale, which is a stochastic process \(\{S_{n}\}_{n=1}^{\infty}\), such that \(S_{n} = X_{1} + \cdots + X_{n}\) with \(E\left[X_{i}\, |\, X_{1},\cdots,X_{i-1}\right] = 0\). Additionally, we are given that the sequence \(b_{n}\) is real, monotonic, and \(b_{1} \leq \cdots \leq b_{n} \leq b_{n+1} \uparrow \infty\). The problem also states that \(E\left[X_{n}^2\right] < \infty\) for all \(n\), and the sum of the variances divided by the square of the sequence's terms converges.

Rewrite the given convergence condition

Using the given information, rewrite the condition \(\sum_{n=1}^{\infty} E\left[X_{n}^{2}\right] / b_{n}^{2}<\infty\) as the following: \[ \sum_{n=1}^{\infty} \frac{E\left[(S_{n+1} - S_{n})^{2}\right]}{b_{n}^{2}} < \infty. \]

Apply the Kolmogorov's zero-one law

Now, use Kolmogorov's zero-one law, which states that if \(\{A_{n}\}_{n=1}^{\infty}\) is an independent sequence of events, then the event \(\limsup A_n = \cap_{n=1}^\infty \cup_{m=n}^\infty A_m\) has either probability \(0\) or \(1\). In our case, define \(A_n\) as: \[ A_{n} = \left\{S_{n}/b_{n} > \epsilon\right\} = \left\{\frac{X_1+\cdots+X_n}{b_n} > \epsilon\right\}. \] We also know that \(X_i\) are in a martingale, which implies that they are adapted and have independent increments. It follows that \(A_n\) are independent events.

Use the Borel-Cantelli lemma

Applying the Borel-Cantelli lemma, which states that if \(\sum_{n=1}^\infty P(A_n) < \infty\), then \(P(\limsup A_n) = 0\), we can evaluate the convergence expression of the martingale with the event \(A_n\). We have: \begin{align*} \sum_{n=1}^{\infty} P(A_{n}) & = \sum_{n=1}^{\infty} P\left(\frac{X_1+\cdots+X_n}{b_n} > \epsilon\right) \\ & \leq \sum_{n=1}^{\infty} \frac{1}{\epsilon^2} \cdot \frac{E\left[(S_{n+1} - S_{n})^{2}\right]}{b_{n}^{2}} \\ & < \infty. \end{align*} Therefore, \(P(\limsup A_n) = 0\). This means that, given the original condition, the event \(\limsup A_n\) has probability \(0\).

Conclude the convergence

From Step 4, we found that \(P(\limsup A_n) = 0\). This implies that \(P\left(\limsup \left\{\frac{X_1+\cdots+X_n}{b_n} > \epsilon\right\}\right) = 0\), which means that \(S_{n}/b_{n} \rightarrow 0\) with probability one.

Key concepts

Understanding Stochastic Processes

Stochastic processes are fundamental concepts in probability theory and statistics. They describe systems that evolve over time in a way that is at least partially random. A stochastic process can be thought of as a collection of random variables indexed by time or space. In our exercise, the sequence \(\{S_n\}_{n=1}^{\infty}\) represents a stochastic process where each \(S_n\) is the sum of the first \(n\) random variables \(X_i\).

When we talk about a zero-mean martingale, which is a specific type of stochastic process, we are referring to a sequence where the expected value of the next term is equal to the current term, given all the past information. In simpler terms, it represents a fair game where the future is not affected by the past in terms of expected profit or loss. These properties make martingales particularly interesting for mathematical modeling in diverse fields such as finance, economics, and physics.

In the context of our problem, the convergence of \(S_n/b_n\) to zero with probability one implies that as the sequence \(b_n\) grows, the cumulative sum of the zero-mean random variables, when adjusted by \(b_n\), will almost surely diminish over time. This aspect of stochastic processes is vital not just for theoretical analysis, but also for practical applications like risk assessment and decision-making under uncertainty.

Kolmogorov's Zero-One Law

Kolmogorov's zero-one law is a pivotal theorem in the realm of probability theory, particularly useful in the analysis of stochastic processes. This law asserts that events which are determined by an infinite number of independent random variables have a probability of either zero or one. Essentially, this means that certain events are almost sure to happen or almost sure not to happen; there's no in-between.

In the step-by-step solution of our exercise, the zero-one law is applied to the events \(A_n\), which are defined based on the values of the martingale \(S_n/b_n\) exceeding a certain threshold \(\epsilon\). The fact that \(S_n\) is a martingale ensures that the random variables \(X_n\) it's composed of have the property of independent increments—this is crucial for applying the zero-one law.

Provided the sequence of events \(A_n\) is independent, which our martingale conditions imply, we can conclude that the probability of their lim sup — that is, the event that \(A_n\) happens infinitely often — is either zero or one. This provides a robust framework for analyzing the long-term behavior of our stochastic process.

Borel-Cantelli Lemma

The Borel-Cantelli lemma is an instrumental result in probability that gives us insight into the occurrence of events over the long run. Simplified, the lemma states that if the sum of the probabilities of a sequence of events is finite, then the probability that infinitely many of them occur is zero. This becomes especially handy in predicting the long-term frequency of certain outcomes.

In our exercise, after applying Kolmogorov’s zero-one law, we use the Borel-Cantelli lemma to show that the martingale \(S_n/b_n\) converges to zero. By demonstrating that the sum of the probabilities of our events \(A_n\) does not diverge, we use the Borel-Cantelli lemma to ascertain that our probability of seeing infinitely many occurrences of \(A_n\) (i.e., \(S_n/b_n\) staying above \(\epsilon\) infinitely often) is zero. This means that eventually, \(S_n/b_n\) will fall below any positive threshold \(\epsilon\) permanently, leading us to the conclusion that \(S_n/b_n\) converges to zero with probability one.

Understanding the Borel-Cantelli lemma gives students a powerful tool not only for solving problems like the one in our exercise but also for grasping broader concepts in the realm of probability theory and its applications to real-world scenarios where understanding the extremes or tails of distributions is essential.