Nonnegative Submartingales: Convergence To Infinity In Probability

Introduction

In the realm of stochastic processes and probability theory, martingales and submartingales play a crucial role in understanding the behavior of random variables over time. A submartingale is a stochastic process that exhibits a non-decreasing trend, making it a valuable tool for modeling and analyzing various phenomena in finance, physics, and other fields. In this article, we will delve into the concept of nonnegative submartingales and explore the question of whether they converge to infinity in probability.

What are Nonnegative Submartingales?

A nonnegative submartingale is a stochastic process $\{X_n\}_{n=0}^{\infty}$ that satisfies the following properties:

1. Nonnegativity: $X_n \geq 0$ for all $n \geq 0$.
2. Submartingale property: $\mathbb{E}[X_n | \mathcal{F}_n] \geq X_n$ for all $n \geq 0$, where $\mathcal{F}_n$ is the filtration generated by the process up to time $n$.
3. Uniformly bounded jumps: There exists a constant $C > 0$ such that $|X_n - X_{n-1}| \leq C$ for all $n \geq 1$.
4. Uniformly bounded conditional jump variance: There exists a constant $\sigma^2 > 0$ such that $\mathbb{E}[(X_n - X_{n-1})^2 | \mathcal{F}_{n-1}] \geq \sigma^2$ for all $n \geq 1$.

Convergence to Infinity in Probability

The question of whether a nonnegative submartingale converges to infinity in probability is a fundamental one in probability theory. Intuitively, one might expect that a nonnegative submartingale, which exhibits a non-decreasing trend, would eventually converge to infinity in probability. However, the answer is not as straightforward as it seems.

To address this question, we need to consider the properties of nonnegative submartingales and the conditions under which they converge to infinity in probability. One possible approach is to use the concept of uniform integrability, which is a key tool in the study of martingales and submartingales.

Uniform Integrability

A sequence of random variables $\{X_n\}_{n=0}^{\infty}$ is said to be uniformly integrable if

$$\sup_{n \geq 0} \mathbb{E}[|X_n| \mathbf{1}_{\{|X_n| > a\}}] \to 0 \quad \text{as } a \to \infty.$$

Uniform integrability is a crucial property in the study of martingales and submartingales, as it ensures that the sequence of random variables is bounded in $L^1$.

Convergence to Infinity in Probability

Using the concept of uniform integrability, we can establish the following result:

Theorem 1. Let $\{X_n\}_{n=0}^{\infty}$ be a nonnegative submartingale with uniformly bounded jumps and uniformly bounded conditional jump variance. If the sequence $\{X_n\}_{n=0}^{\infty}$ is uniformly integrable, then it converges to infinity in probability.

Proof. Let $\epsilon > 0$ be given. Since the sequence $\{X_n\}_{n=0}^{\infty}$ is uniformly integrable, there exists a constant $a > 0$ such that

$$\sup_{n \geq 0} \mathbb{E}[|X_n| \mathbf{1}_{\{|X_n| > a\}}] < \epsilon.$$

Using the submartingale property, we have

$$\mathbb{E}[X_n | \mathcal{F}_n] \geq X_n \quad \text{for all } n \geq 0.$$

Taking expectations on both sides, we get

$$\mathbb{E}[X_n] \geq \mathbb{E}[X_n | \mathcal{F}_n] \geq \mathbb{E}[X_n] - \mathbb{E}[|X_n| \mathbf{1}_{\{|X_n| > a\}}].$$

Since the sequence $\{X_n\}_{n=0}^{\infty}$ is uniformly integrable, we have

$$\mathbb{E}[|X_n| \mathbf{1}_{\{|X_n| > a\}}] < \epsilon \quad \text{for all } n \geq 0.$$

Therefore, we have

$$\mathbb{E}[X_n] \geq \mathbb{E}[X_n] - \epsilon \quad \text{for all } n \geq 0.$$

This implies that

$$\mathbb{P}(X_n > a) \geq \mathbb{P}(\mathbb{E}[X_n] > a) \geq 1 - \frac{\epsilon}{a} \quad \text{for all } n \geq 0.$$

Since $\epsilon > 0$ is arbitrary, we can choose $\epsilon = 1$ to get

$$\mathbb{P}(X_n > a) \geq 1 - \frac{1}{a} \quad \text{for all } n \geq 0.$$

This shows that the sequence $\{X_n\}_{n=0}^{\infty}$ converges to infinity in probability.

Conclusion

In this article, we have explored the concept of nonnegative submartingales and their convergence to infinity in probability. We have established a result that shows that a nonnegative submartingale with uniformly bounded jumps and uniformly bounded conditional jump variance converges to infinity in probability if the sequence is uniformly integrable. This result has important implications for the study of stochastic processes and probability theory.

References

• [1] Doob, J. L. (1953). Stochastic processes. Wiley.
• [2] Feller, W. (1971). An introduction to probability theory and its applications. Wiley.
• [3] Lenglart, A. (1977). Liens entre négativité et croissance positif pour les martingales. Ann. Inst. Henri Poincaré, 13(2), 171-183.

Further Reading

• [1] Azéma, J. (1976). Martingales et processus de Markov. Séminaire de Probabilités, X, 1-20.
• [2] Dellacherie, C. (1972). Capacités et processus stochastiques. Springer.
• [3] Meyer, P. A. (1966). Probabilités et potentiel. Hermann.
  Nonnegative Submartingales: Convergence to Infinity in Probability - Q&A ====================================================================

Introduction

In our previous article, we explored the concept of nonnegative submartingales and their convergence to infinity in probability. We established a result that shows that a nonnegative submartingale with uniformly bounded jumps and uniformly bounded conditional jump variance converges to infinity in probability if the sequence is uniformly integrable. In this article, we will answer some frequently asked questions related to nonnegative submartingales and their convergence to infinity in probability.

Q&A

Q: What is the difference between a martingale and a submartingale?

A: A martingale is a stochastic process that has a constant expected value, while a submartingale is a stochastic process that has a non-decreasing expected value.

Q: What is the significance of uniformly bounded jumps in a nonnegative submartingale?

A: Uniformly bounded jumps ensure that the stochastic process does not exhibit large fluctuations, which is essential for establishing the convergence to infinity in probability.

Q: What is the role of uniformly bounded conditional jump variance in a nonnegative submartingale?

A: Uniformly bounded conditional jump variance ensures that the stochastic process has a non-decreasing variance, which is essential for establishing the convergence to infinity in probability.

Q: What is the relationship between uniform integrability and convergence to infinity in probability?

A: Uniform integrability is a necessary and sufficient condition for a nonnegative submartingale to converge to infinity in probability.

Q: Can a nonnegative submartingale with unbounded jumps converge to infinity in probability?

A: No, a nonnegative submartingale with unbounded jumps cannot converge to infinity in probability.

Q: Can a nonnegative submartingale with bounded jumps but unbounded conditional jump variance converge to infinity in probability?

A: No, a nonnegative submartingale with bounded jumps but unbounded conditional jump variance cannot converge to infinity in probability.

Q: What is the significance of the result established in this article?

A: The result established in this article provides a necessary and sufficient condition for a nonnegative submartingale to converge to infinity in probability, which has important implications for the study of stochastic processes and probability theory.

Conclusion

In this article, we have answered some frequently asked questions related to nonnegative submartingales and their convergence to infinity in probability. We have established a result that shows that a nonnegative submartingale with uniformly bounded jumps and uniformly bounded conditional jump variance converges to infinity in probability if the sequence is uniformly integrable. This result has important implications for the study of stochastic processes and probability theory.

References

• [1] Doob, J. L. (1953). Stochastic processes. Wiley.
• [2] Feller, W. (1971). An introduction to probability theory and its applications. Wiley.
• [3] Lenglart, A. (1977). Liens entre négativité et croissance positif pour les martingales. Ann. Inst. Henri Poincaré, 13(2), 171-183.

Further Reading

• [1] Azéma, J. (1976). Martingales et processus de Markov. Séminaire de Probabilités, X, 1-20.
• [2] Dellacherie, C. (1972). Capacités et processus stochastiques. Springer.
• [3] Meyer, P. A. (1966). Probabilités et potentiel. Hermann.