Theory Overview

In probability, a sequence of random variables can converge in different ways. This experiment explores a famous example that highlights the difference between two important types:

• Convergence in Probability: The sequence \(X_n\) converges to \(X\) if, as \(n\) gets large, the probability that \(X_n\) is far from \(X\) becomes vanishingly small.
• Almost Sure Convergence: A much stronger condition. For any specific outcome \(\omega\), the sequence of numbers \(X_n(\omega)\) must converge to \(X(\omega)\). This means individual sample paths must eventually "settle down" at the limit.

Procedure

• Read the experiment setting which defines the "moving, shrinking rectangles" sequence \(X_n\).
• Enter a value for \(\omega\) in the range (0, 1). This represents picking a single outcome from all possibilities.
• Enter a block number \(k\). Each block contains \(2^{k-1}\) random variables.
• Click "Check". The tool will find the specific random variable \(X_n\) within that block where its value is 1 for your chosen \(\omega\).
• Observe how for any \(\omega\), you can always find an \(X_n=1\) in any block, no matter how large \(k\) is. This proves the sequence of outcomes \(X_n(\omega)\) never settles to 0, and thus does not converge almost surely.