Predicting the Future

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Predicting the Future

stdatomic memory orders. Compare relaxed, consume, acquire, release, sequence consistent mem order

Predicting the Future

Conditional Expectation Mean square error and Orthogonality, 𝐸[𝑋𝑌F]=𝑋𝐸[𝑌F] Martingale Theory

Predicting the Future

Deriving the conditional expectation from joint probability density function Martingale Theory

Predicting the Future

Proof Bayes’ Formula and Conditional Probability formula Martingale Theory

Predicting the Future

Bitcoin SV 51% attack Random Walk Markov Chain process modelling

Predicting the Future

Conditional Expectation and Radon-Nikodym Theorem Martingale Theory

Predicting the Future

C++ 11 to 20 Boost Coroutine2. What Why and How. Learn Boost Coroutine async programming by example

Predicting the Future

C++ 11 to 20 Dead lock prevention using stdlock

Predicting the Future

Sufficient and necessary conditions for Almost sure convergence

Predicting the Future

Prove Integrability implies finite almost surely

Predicting the Future

C++ 11 to 20, Comparing different function types Function pointer, stdfunction, functor, lambda

Predicting the Future

Prove Kolmogorov's 0-1 law If 𝑋1,𝑋2,... are independent and 𝐴∈Tail σ field then 𝑃(𝐴)=0 or 𝑃(𝐴)=1.

Predicting the Future

All pairwise correlations equal corr(X1,X2) = corr(X1,X3) = corr(X2,X3) = ρ. What's the range of ρ

Predicting the Future

Definition of Tail 𝜎-field and Examples, lim S_n, limsup S_n, lim S_nn etc.

Predicting the Future

Monte Carlo to the rescue! Solving the Expected sum of distances in a triangle Question in C++!

Predicting the Future

Brain teasers! - Expected sum of distances in a triangle

Predicting the Future

Prove Central Limit Theorem using characteristic function

Predicting the Future

Prove 𝜙𝑛(𝑡)→𝜙(𝑡) and 𝜙(𝑡) is continuous at 0 indicates 𝐹𝑛 is tight & 𝐹𝑛⇒𝐹 where F is a distribution

Predicting the Future

Theorem Sufficient condition for tightness (Bounded in probability)

Predicting the Future

Every subseq’s limit func 𝐹 in Helly’s selection theorem is a distribution function iff 𝐹𝑛 is tight

Predicting the Future

Prove Helly’s selection theorem

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Definition Tightness or Bounded in Probability

Predicting the Future

Prove If 𝑋𝑛⇒𝑋∞ then 𝜙𝑛(𝑡)→𝜙(𝑡) for all 𝑡.

Predicting the Future

Prove Slutsky’s theorem. If 𝑋𝑛⇒𝑋, 𝑌𝑛→𝑐 in prob, 𝑍𝑛→𝑑 in prob, 𝑍𝑛+𝑌𝑛𝑋𝑛⇒𝑑+𝑐𝑋. If 𝑐≠0, 𝑍𝑛+𝑋𝑛𝑌𝑛⇒𝑑+𝑋𝑐.

Predicting the Future

𝑔 be a measurable function with 0 measure discont pts, 𝑔(𝑋𝑛)⇒𝑔(𝑋∞). If 𝑔 is bounded, 𝐸𝑔(𝑋𝑛)→𝐸𝑔(𝑋∞).

Predicting the Future

If 𝑋𝑛⇒𝑋 and 𝑌𝑛−𝑋𝑛→0 in prob, then 𝑌𝑛⇒𝑋 in dist. If 𝑋𝑛⇒𝑋, 𝑍𝑛→0 in prob, then 𝑋𝑛+𝑍𝑛⇒𝑋 in dist.

Predicting the Future

Proof Portmanteau Lemmas

Predicting the Future

Proof 𝑋𝑛⇒𝑏 shows 𝑋𝑛→𝑏 in prob. 𝑋𝑛→𝑋 in prob shows 𝑋𝑛⇒𝑋. But 𝑋𝑛⇒𝑋 not indicate 𝑋𝑛→𝑋 in general.

Predicting the Future

Proof 𝑋𝑛+𝑌𝑛→𝑋+𝑌 in probability, 𝑋𝑛𝑌𝑛→𝑋𝑌 in probability

Predicting the Future