Lecure 4: Cheybyshev Inequality, Borel-Cantelli lemmas & related issues

The speaker revisits the concepts of random variables, distribution functions, and density functions, emphasizing their importance in understanding the behavior of random variables. The normal distribution, also known as the Gaussian distribution, is highlighted as a fundamental probability distribution with a specific density function. The speaker clarifies that the course's focus is on applying probabilistic tools in finance rather than teaching basic probability and statistics. The introduction of Chebyshev's lemma and the Borel-Cantelli lemma is announced as essential for analyzing sequences of events in finance.

The speaker explains Chebyshev's lemma, which provides a method to estimate the probability of a random variable deviating from its mean by a certain amount. The lemma is applicable to any random variable and uses the concept of the modulus of the variable. The proof is outlined, demonstrating how the integral of the modulus raised to a power p can be used to derive the inequality. The lemma is a precursor to discussing the behavior of sequences of events, which is crucial for financial analysis.

The speaker delves into the Borel-Cantelli lemma, which addresses the concept of a sample point occurring infinitely often within a sequence of events. The construction of the set of sample points that belong to infinitely many events in the sequence is discussed, along with the properties of the Sigma-algebra. The lemma's conclusion is that if the sum of the probabilities of the events in the sequence is finite, then the probability of a sample point occurring in infinitely many events is zero, indicating that almost all sample points belong to only finitely many events in the sequence.

The speaker introduces the concepts of convergence in probability and almost sure convergence of random variables. Convergence in probability is defined as the probability that the sequence of random variables deviates from a certain value approaching zero as the sequence progresses. Almost sure convergence means that, except for a set of measure zero, the sequence of random variables converges to a specific value for almost all sample points. The speaker also hints at the importance of these concepts in understanding central limit theorems and their applications in finance.

The speaker explores the relationship between convergence in probability and almost sure convergence. It is established that if a sequence of random variables converges in probability, then there exists a subsequence that converges almost surely to the same limit. The proof involves selecting subsequences that satisfy the definition of convergence in probability for any given epsilon, and then applying the Borel-Cantelli lemma to show that the subsequence converges almost surely.

The speaker applies the Borel-Cantelli lemma to demonstrate the convergence of a subsequence of random variables almost surely. By constructing a series of events where the probability of the event occurring infinitely often is zero, the lemma is used to show that for almost all sample points, the subsequence converges to the desired limit. This reinforces the idea that while a sequence may converge in probability, a subsequence can be identified that converges almost surely.

The speaker concludes the discussion by connecting the concepts of probability and convergence to financial mathematics. The importance of understanding Brownian motion, which is central to stock market dynamics and the Ito calculus used in derivative pricing, is emphasized. The speaker anticipates further exploration of these topics, including central limit theorems, in subsequent lectures.