Question

Let \(\left\\{X_{n}\right\\}\) be a submartingale. Strengthen the maximal inequality, Lemma S.I., to $$ \begin{aligned} \lambda \operatorname{Pr}\left\\{\max _{0 \leq k \leq n} X_{k}>\lambda\right\\} & \leq E\left[X_{n} I\left\\{\max _{0 \leq k \leq n} X_{k}>\lambda\right\\}\right] \\ & \leq E\left[X_{n}^{+}\right] \leq E\left[\left|X_{n}\right|\right], \quad \lambda>0 \end{aligned} $$

Short answer

For a submartingale \(X_n\), we have the stronger maximal inequality, which is given by: \[ \lambda \mathbb{P}\left(\max_{0 \leq k \leq n} X_k > \lambda\right) \leq \mathbb{E}[X_n I\{ \max_{0 \leq k \leq n} X_k > \lambda \}] \leq \mathbb{E}[X_n^+] \leq \mathbb{E}[|X_n|], \quad \lambda > 0. \]

Step-by-step solution

Nonnegative submartingale case

Since \(X_n\) is a submartingale, we know that \(\mathbb{E}[X_{n+1} | \mathcal{F}_n] \geq X_n\) for all nonnegative \(X_n\). Let's consider only nonnegative submartingales in this step. Now we will proceed to show the inequality. Define an event \(A = \{ \max_{0 \leq k \leq n} X_k > \lambda \}\). Then using the fact that \(X_n = \mathbb{E}[X_{n} | \mathcal{F}_A]\) on the event A, \[ \lambda \mathbb{P}(A) \leq \mathbb{E}[X_n I(A)] \leq \mathbb{E}[X_n] . \]

Positive and negative parts

Now, we will generalize the result from Step 1 to arbitrary submartingales. To do this, let's split \(X_n\) into its positive and negative parts, \(X_n = X_n^+ - X_n^-\), where \(X_n^+ = \max(X_n,0)\) and \(X_n^- = \max(-X_n,0)\). Notice that \(X_n^+\) and \(X_n^-\) are nonnegative submartingales. Now, applying the result from Step 1 to the positive part \(X_n^+\), we get \[ \lambda \mathbb{P}\left(\max_{0 \leq k \leq n} X_k^+ > \lambda\right) \leq \mathbb{E}[X_n^+]. \] Similarly, applying the result from Step 1 to the negative part \(X_n^-\), we get \[ \lambda \mathbb{P}\left(\max_{0 \leq k \leq n} X_k^- > \lambda\right) \leq \mathbb{E}[X_n^-]. \] Now, since \(\mathbb{P}\left(\max_{0 \leq k \leq n} X_k > \lambda\right)\) is the same as the union of the previous two events considering the positive and negative parts, adding the two inequalities gives \[ \lambda \mathbb{P}\left(\max_{0 \leq k \leq n} X_k > \lambda\right) \leq \mathbb{E}[X_n^+] + \mathbb{E}[X_n^-] = \mathbb{E}[|X_n|]. \] This completes the proof of the stronger maximal inequality for a submartingale \(X_n\).

Key concepts

Submartingale

Understanding the concept of a submartingale is essential when dealing with stochastic processes. A submartingale is a sequence of random variables, say \( \left\{X_{n}\right\} \), that satisfies a particular 'increasing' condition in terms of expected value. Specifically, for any time \( n \), the expected future value conditioned on the past (\(\mathbb{E}[X_{n+1} | \mathcal{F}_n]\)) is never less than the present value (\(X_n\)).

To put it intuitively, if you consider \( \left\{X_{n}\right\} \) as your wealth over time while gambling, being a submartingale means that, on average, you can expect to be at least as wealthy in the future as you are today, given your current wealth. This concept is fundamental when proving inequalities that give bounds on the probability of a submartingale exceeding certain thresholds.

Expected Value

Delving into the realm of probability theory, the expected value is akin to the long-term average of a random variable. It is represented as \( \mathbb{E}[X] \) for a random variable \( X \). Imagine rolling a fair die: the expected value of the outcome is \( \frac{1+2+3+4+5+6}{6} = 3.5 \), even though you can never actually roll a 3.5.

In the context of a submartingale, the expected value quantifies the anticipated future value, considering what is known so far. This is particularly instrumental in the maximal inequality, which deals with the probability of the process exceeding a certain level multiplicated by this level itself, essentially providing an upper bound that is tied to the expected value.

Nonnegative Random Variables

Random variables are the backbone of probability theory. However, when we specify that a random variable is nonnegative, we're focusing on those that can only take values greater than or equal to zero. Nonnegative random variables are particularly useful because they simplify many probabilistic arguments.

In our problem dealing with submartingales, the maximal inequality initially considers nonnegative submartingales. This is a strategic simplification as it allows for the application of the law of total expectation and paves the way for understanding more complex cases where the random variables can take on both positive and negative values.

Positive and Negative Parts of Random Variables

Every random variable can be broken down into its positive and negative parts, which are themselves nonnegative random variables. For a given random variable \( X \), we define \( X^+ \) as the positive part, where \( X^+ = \max(X,0) \), and \( X^- \) as the negative part, where \( X^- = \max(-X,0) \). This decomposition is pivotal in expanding the scope of certain theorems and inequalities.

For example, these parts play a critical role in the step-by-step solution provided for extending the maximal inequality. By dividing \( X_n \) into \( X_n^+ \) and \( X_n^- \) and applying the established results for nonnegative submartingales to each part separately, we achieve a more comprehensive inequality that holds true for submartingales in general, regardless of whether they take on negative values.