Learn with Amar
Teaching video
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Graduate chapters and video availability01 · Read and understand
What you will learn
- Construct convergence in probability without convergence in L¹.
- Justify the conclusion "E|Xₙ|=n·(1/n)=1, so L¹ convergence fails" using the stated assumptions.
Before you start
Random variables and expectation.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Let Xₙ=n with probability 1/n and zero otherwise. Does Xₙ converge to zero in probability and in L¹?
Why this math matters
Construct convergence in probability without convergence in L¹. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- The stated two-point law holds for every n.
- No cross-n dependence is needed for these two conclusions.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Distinguish rare large errors from small expected error
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Let Xₙ=n with probability 1/n and zero otherwise. Does Xₙ converge to zero in probability and in L¹?
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
For fixed ε>0 and n>ε, P(|Xₙ|>ε)=1/n
Large values become increasingly rare.
Work through the mathematics
The tail probability tends to zero
This verifies convergence in probability.
Check the conclusion
E|Xₙ|=n·(1/n)=1, so L¹ convergence fails
The increasing size exactly offsets the decreasing probability.
The result
E|Xₙ|=n·(1/n)=1, so L¹ convergence fails
The increasing size exactly offsets the decreasing probability.
Common mistakes to catch
- A small probability of error need not mean a small mean error.
- Almost-sure claims require a joint construction across n.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
What is E[Xₙ²]?
Show a hint
Square the nonzero value before weighting.
Reveal answer and explanation
n
n²·(1/n)=n.
Practice 2
Does convergence in L¹ imply convergence in probability?
Show a hint
Apply Markov's inequality to the absolute error.
Reveal answer and explanation
Yes
P(|Xₙ−X|>ε)≤E|Xₙ−X|/ε tends to zero.
Take the idea with you
Use this example to question an error metric that ignores the size of rare failures.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Change probability measures with likelihood weights
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