Let X1, X2, …be independent and identically distributed...

Let $X_{1}, X_{2}, \ldots$ be independent and identically distributed random variables with expectation 0 and variance $\sigma^{2}<\infty$. Let
$$
Z_{n}=\left(\sum_{i=1}^{n} X_{i}\right)^{2}-n \sigma^{2}
$$
Show that $Z_{1}, Z_{2}, \ldots$ is a martingale.

Key Concepts

Martingale

A martingale is a stochastic process where the conditional expectation of the next value, given the history of past values up to the current time, is equal to the present value. This concept is central in probability theory and stochastic processes because it formalizes a “fair game” scenario where, on average, no net gain or loss is expected over time.

Conditional Expectation

Conditional expectation is the expected value of a random variable given the knowledge of some other event or sigma-algebra. Within the context of martingales, it is used to formalize the idea that the future evolution of the process, when informed by past information, preserves an unbiased nature.

Filtration

A filtration is an increasing sequence of sigma-algebras that represents the accumulation of information over time in a stochastic process. It provides the formal framework inside which conditional expectations are taken, ensuring that 'the past' is properly accounted for when verifying the martingale property.

Independence

Independence of random variables ensures that the future increments of the process are not influenced by past values. This property simplifies the computation of conditional expectations, as independent random variables can be handled separately, which is essential in proving that a process satisfies the martingale condition.

Linearity of Expectation

Linearity of expectation is a fundamental property stating that the expected value of a sum of random variables is equal to the sum of their expected values. This property is used in the manipulation and simplification of the expressions involved when proving the martingale condition.

Variance

Variance measures the dispersion of a random variable around its mean. In the presented problem, the finite variance condition is used when adjusting the squared sum of the independent random variables to center the process, ensuring that the key properties hold to form a martingale.

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