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Graduate · Rare events and convergence

Rare events and convergence: A near-balanced decaying mean

Rare events and convergence: investigate a near-balanced decaying mean with magnitude exponent a = 1.25; rarity exponent b = 1.5.

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Watch the relationship

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Rare events and convergence: A near-balanced decaying mean. Index n: 1. Nonzero probability: 1. Nonzero magnitude: 1. Mean absolute value: 1Rare events may still carry large means01115.530mean absolute sizeTeal: mean absolute size · dashed: nonzero probability
a and b are positive. Each displayed law is a two-point distribution, not a simulated sample. For 0<b≤1, independent nonzero events occur infinitely often almost surely, preventing convergence to zero; nested events {U≤n⁻ᵇ} from one uniform U on (0,1) instead give eventual zero almost surely. For b>1, no independence is needed. Convergence in probability uses fixed positive error thresholds and large n; finite frames alone do not prove a limit.

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Make it your experiment

Change one value. Notice what follows.

The controls adjust the model. Numbers below describe the current frame. Decimals are rounded.

Index n
1
Nonzero probability
1
Nonzero magnitude
1
Mean absolute value
1

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Understand what you are seeing

The idea behind the motion.

An event can become rare while its size increases. The nonzero probability tends to zero whenever b>0, giving convergence in probability to zero, but the mean absolute value depends on the competition between a and b. The same family can therefore converge in L¹, fail with a constant mean, or have a growing mean. This investigation starts with Magnitude exponent a = 1.25; Rarity exponent b = 1.5. Predict the result before playing, then change one parameter while keeping the other fixed to test the reason for the change.

A relationship to keep

Xₙ=nᵃ with probability n⁻ᵇ, else 0; E|Xₙ|=n^(a−b)

Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.

  1. STEP 1

    Set up the mathematical model

    Separate the nonzero value nᵃ from the probability n⁻ᵇ attached to it. The starting case is “A near-balanced decaying mean.”

  2. STEP 2

    Follow the changing quantity

    Increase n from one to thirty and compare their product with each factor separately.

  3. STEP 3

    Explain and test the result

    Classify the asymptotic mean by the sign of a−b. For b>1, the probabilities have a finite sum, so the first Borel–Cantelli lemma gives eventual zero almost surely for any joint construction on one probability space. For 0<b≤1, the dependence between events matters.

Your turn to explain

Make a prediction. Test your reasoning.

Keep Magnitude exponent a = 1.25; Rarity exponent b = 1.5. Pause the timeline at 40%. Given index n = 12, calculate nonzero probability, nonzero magnitude, mean absolute value. Show the substitution into the displayed formula.

Use the values specified in the question. Reset restores the initial values for this investigation.

Compare your explanation

At n=12, probability is (12)^(−1.5)=0.024056 and magnitude is (12)^1.25=22.334517. Their product is n^(a−b)=(12)^(1.25−1.5)=0.537285. Results: Nonzero probability: 0.024; Nonzero magnitude: 22.335; Mean absolute value: 0.537. Decimal values are rounded; retain the original parameters when checking.

Work through a full lesson

Connect the animation to a worked example and practice questions.

Graduate · Statistics

Distinguish rare large errors from small expected error

Construct convergence in probability without convergence in L¹.