Discrete-time stochastic process: marginals and two-time joint laws from a complete path distribution, with Markov and stationarity checks

Goal and scope

Horizon {0,1,2}, states {0,1}. The complete path support with exact PMF is (0,0,0):1/4, (0,1,1):1/4, (1,0,1):1/8, (1,1,0):1/8, (1,1,1):1/4. (1) Check the support and the PMF, build the probability space and explain why X_0, X_1, X_2 are random variables on the same space; (2) find the marginal law at each time and the joint law of (X_1,X_2); (3) prove that this process is neither stationary nor Markov; (4) construct a path law with the same marginals but a different joint law, showing that the marginal tables are not the whole process.

Treats only the finite-horizon path law specified by G1 and G2 (MKO assumptions a1, a2). Nothing is extended to infinite time, a transition law, stationarity or ergodicity; the Markov property is tested by its definition at time 1 only. The comparison product law Q serves only to show that marginals do not determine the joint law. The canonical MKO and its English edition are not changed.

Givens and symbols

• prob_axioms
• finite_sum_rules
• image_preimage_def
• cartesian_product_def
• function_def

• G1 — Horizon {0,1,2}, states {0,1}. The support lists 5 paths of length 3: ω_1=(0,0,0), ω_2=(0,1,1), ω_3=(1,0,1), ω_4=(1,1,0), ω_5=(1,1,1) (MKO assumption a1: complete support of distinct equal-length finite paths).
• G2 — Exact PMF: p(ω_1)=1/4, p(ω_2)=1/4, p(ω_3)=1/8, p(ω_4)=1/8, p(ω_5)=1/4; paths outside the support have probability 0.
• G3 — Comparison law Q: on {0,1}^3 define Q(x_0,x_1,x_2)=p_0(x_0)p_1(x_1)p_2(x_2), where p_n is the marginal of X_n under the law of G2 (found in S07 to S09).

• \Omega=\{\omega_1,\omega_2,\omega_3,\omega_4,\omega_5\} — the complete path support
• p(\omega) — the exact path PMF
• X_n(\omega)=\omega_n — the coordinate at time n (n=0,1,2)
• P — the probability law induced by p, P(A)=Σ_{ω∈A}p(ω)
• p_n(s)=P(X_n=s) — the marginal at time n
• Q — the product law of the marginals (for comparison)

Rules and sources

• prob_axioms · prerequisite — Probability axioms: P(A)≥0; P(Ω)=1; the probability of a (countable) union of pairwise disjoint events is the sum of their probabilities. Hence P(∅)=0 and P(Aᶜ)=1−P(A).
• finite_sum_rules · prerequisite — Finite sums: Σ_{i=1}^{0} a_i=0 and Σ_{i=1}^{n+1} a_i=Σ_{i=1}^{n} a_i + a_{n+1}; Σ(a_i+b_i)=Σa_i+Σb_i; Σ c·a_i=c·Σa_i.
• image_preimage_def · prerequisite — Image and preimage: f(S)={f(a) ∣ a∈S}; f⁻¹(T)={a∈A ∣ f(a)∈T}. The preimage notation does not presuppose that f has an inverse.
• cartesian_product_def · prerequisite — Cartesian product: A×B is the set of ordered pairs (a,b) with a∈A and b∈B; (a,b)=(c,d) ⇔ a=c and b=d.
• function_def · prerequisite — Function: f:A→B assigns to each a∈A exactly one f(a)∈B. Two functions are equal ⇔ they have the same domain and codomain and agree at every a.
• def_unfold · prerequisite — Unfolding a definition: replacing a defined term by its definition (or folding it back) does not change whether a statement is true.
• case_analysis · prerequisite — Proof by cases: if P∨Q holds and both P⇒R and Q⇒R are proved, then R holds. The cases must cover every possibility.
• numeric_evaluation · prerequisite — Exact evaluation of an explicit integer or rational expression; independently recomputable.
• set_operations_def · prerequisite — Set operations: x∈A∪B ⇔ x∈A or x∈B; x∈A∩B ⇔ x∈A and x∈B; x∈A∖B ⇔ x∈A and x∉B; in a universe U, Aᶜ=U∖A.
• set_builder · prerequisite — Set-builder notation: x∈{y∈A ∣ P(y)} if and only if x∈A and P(x).
• contrapositive · prerequisite — Contrapositive: P⇒Q is equivalent to ¬Q⇒¬P. The converse Q⇒P is in general not equivalent to P⇒Q.
• frac_rules · prerequisite — Fraction rules (all denominators nonzero): a/c+b/c=(a+b)/c; (a/b)(c/d)=(ac)/(bd); (a/b)/(c/d)=(ad)/(bc) (also needing c≠0); a/b=(ak)/(bk) for k≠0.
• quantifier_negation · prerequisite — Negating quantifiers: ¬(∀x, P(x)) is equivalent to ∃x, ¬P(x), and ¬(∃x, P(x)) is equivalent to ∀x, ¬P(x).
• distributive · prerequisite — Distributivity: u(v+w)=uv+uw; with commutativity also (v+w)u=vu+wu.
• mul_one · prerequisite — Multiplicative identity: 1·u=u.
• and_intro_elim · prerequisite — Conjunction: from P and Q infer P∧Q; from P∧Q infer P and infer Q.
• mko_process_def · cited — A discrete-time stochastic process (X_n) is a family of random variables on the same probability space. A complete path distribution over a finite horizon specifies the joint law, from which time marginals and two-time joint distributions are derived. Neither the Markov property nor stationarity follows automatically. · mko-discrete-stochastic-process
• p_cond_prob · prerequisite — Conditional probability: for P(B)>0, P(A∣B)=P(A∩B)/P(B); it is undefined when P(B)=0.
• p_markov_def · prerequisite — Markov property (at time 1): for all h, i, j with P(X_0=h, X_1=i)>0, P(X_2=j ∣ X_1=i, X_0=h)=P(X_2=j ∣ X_1=i). A Markov process must satisfy such a condition at every time.
• p_stationary_nec · prerequisite — (Strict) stationarity: a time shift changes no joint law; one necessary condition is that all one-time marginal laws coincide.
• L_marginal · lemma — For n∈{0,1,2} and s∈{0,1}: P(X_n=s)=Σ_{ω∈Ω, ω_n=s} p(ω); likewise the two-time joint P(X_m=s, X_n=t)=Σ_{ω_m=s, ω_n=t} p(ω).
• L_product_marg · lemma — If q_0, q_1, q_2 are probability vectors on {0,1} and Q(x_0,x_1,x_2)=q_0(x_0)q_1(x_1)q_2(x_2), then the time-n marginal of Q is q_n.

Main derivation

(X_n)_{n\in\{0,1,2\}}\ \text{process}

X_0,X_1,X_2:\Omega\to\{0,1\}\ \text{on one}\ (\Omega,\mathcal{F},P);\ \text{law}=\text{path PMF}

\omega_1,\omega_2,\omega_3,\omega_4,\omega_5\in\{0,1\}^3

\omega_k\ \text{pairwise distinct},\ \text{each of length}\ 3

p=(\tfrac14,\tfrac14,\tfrac18,\tfrac18,\tfrac14)

p(\omega_k)\ge0,\ \ \sum_{k=1}^{5}p(\omega_k)=1

\Omega,\ p

(\Omega,\mathcal{P}(\Omega),P),\ \ P(A)=\sum_{\omega\in A}p(\omega)\ \text{satisfies the axioms}

X_n(\omega)=\omega_n

X_0,X_1,X_2\ \text{random variables on the same}\ (\Omega,\mathcal{P}(\Omega),P)

P(X_n=s)

\sum_{\omega\in\Omega,\ \omega_n=s}p(\omega)

P(X_0=0),\ P(X_0=1)

p_0=(\tfrac12,\tfrac12)

P(X_1=0),\ P(X_1=1)

p_1=(\tfrac38,\tfrac58)

P(X_2=0),\ P(X_2=1)

p_2=(\tfrac38,\tfrac58)

P(X_1=s,\ X_2=t)

(0,0):\tfrac14,\ (0,1):\tfrac18,\ (1,0):\tfrac18,\ (1,1):\tfrac12

p_0=(\tfrac12,\tfrac12),\ p_1=(\tfrac38,\tfrac58)

p_0\neq p_1\ \Rightarrow\ \text{not stationary}

P(X_2=1\mid X_1=1,\ X_0=0)

\frac{P(\{\omega_2\})}{P(\{\omega_2\})}=\frac{1/4}{1/4}=1

P(X_2=1\mid X_1=1,\ X_0=1)

\frac{P(\{\omega_5\})}{P(\{\omega_4,\omega_5\})}=\frac{1/4}{3/8}=\frac23

P(X_2=1\mid X_1=1)=\frac{1/2}{5/8}=\frac45

1\neq\tfrac23:\ \text{the Markov condition fails at time 1}

Q(x_0,x_1,x_2)=p_0(x_0)p_1(x_1)p_2(x_2)

Q\ \text{has marginals}\ p_0,p_1,p_2,\ \text{but}\ Q(X_1=1,X_2=1)=\tfrac{25}{64}\neq\tfrac12

\text{S05 to S15}

p_0,p_1,p_2,\ \text{joint}(X_1,X_2);\ \text{not stationary};\ \text{not Markov};\ \text{marginals do not determine the law}

Substeps and lemmas

A\cap B=\emptyset

P(A\cup B)=\sum_{\omega\in A\cup B}p(\omega)=\sum_{\omega\in A}p(\omega)+\sum_{\omega\in B}p(\omega)

\{X_n=s\}=X_n^{-1}(\{s\})

\{\omega\in\Omega\mid\omega_n=s\},\ \ P(X_n=s)=\sum_{\omega_n=s}p(\omega)

\{X_0=0\}\cap\{X_1=1\}

\{\omega_2\};\quad \{X_0=1\}\cap\{X_1=1\}=\{\omega_4,\omega_5\};\quad \{X_1=1\}=\{\omega_2,\omega_4,\omega_5\}

\sum_{x_1,x_2}q_0(x_0)q_1(x_1)q_2(x_2)

q_0(x_0)\Big(\sum_{x_1}q_1(x_1)\Big)\Big(\sum_{x_2}q_2(x_2)\Big)=q_0(x_0)\cdot1\cdot1

Q(X_1=1,X_2=1)=\sum_{x_0}p_0(x_0)p_1(1)p_2(1)

1\cdot\tfrac58\cdot\tfrac58=\tfrac{25}{64}

Reverse audits

• S16 · premise_audit · S05, S10, S11, S14, S15 — Only conjoins conclusions already proved; every conclusion concerns only the finite-horizon instance of G1 and G2.
• S15.2 · inverse_equivalence · distributive, numeric_evaluation — An exact rational computation, recomputed by N15 and N16.
• S15.1 · inverse_equivalence · distributive, finite_sum_rules — Expanding and factoring undo each other, so both directions hold; the final step needs each q_n to sum to 1.
• S15 · premise_audit · L_product_marg, S10 — One example with equal marginals and different joints refutes 'marginals determine the process'. Whether Q is Markov, or any other property of Q, is not needed here.
• S14 · premise_audit · p_markov_def, S12, S13 — One failing triple (h,i,j) refutes the Markov property; both conditioning events have positive probability (1/4 and 3/8), so the comparison is legitimate.
• S13 · inverse_equivalence · p_cond_prob, numeric_evaluation — The denominator is positive, so the definition is an equality; the value is recomputed by N12.
• S12.1 · inverse_equivalence · set_operations_def, G1 — All 5 paths are enumerated, and the definition of intersection holds both ways.
• S12 · inverse_equivalence · p_cond_prob, numeric_evaluation — With a positive denominator the definition of conditional probability is an equality, both sides having the same value; with denominator 0 the step could not be taken.
• S11 · premise_audit · p_stationary_nec, contrapositive — Uses only the contrapositive of a necessary condition; the converse (equal marginals imply stationarity) is not claimed and fails in general.
• S10 · inverse_equivalence · L_marginal, numeric_evaluation — Row and column sums of the joint table return the marginals, a consistency check; conversely the marginals do not determine the joint table uniquely (S15).
• S09 · inverse_equivalence · S06, numeric_evaluation — Computed as in S08; equal marginals do not mean X_1=X_2 (see error control 3).
• S08 · inverse_equivalence · S06, numeric_evaluation — 3/8+5/8=1, and the two groups of paths partition Ω.
• S07 · inverse_equivalence · S06, numeric_evaluation — Each number is recomputed by N2, N3; the two groups of paths partition Ω exactly, so their total of 1 is an independent check.
• S06.1 · inverse_equivalence · image_preimage_def, set_operations_def — The preimage and the set-builder description correspond both ways; X_n need not be invertible.
• S06 · inverse_equivalence · L_marginal, image_preimage_def — Both sides are two expressions of one number: the left is the probability of the event {X_n=s}, the right is the sum obtained by unfolding the definition of P.
• S05 · premise_audit · S04, function_def, image_preimage_def — The power-set event family makes measurability automatic; with a general event family the preimages would need to be checked separately.
• S04.1 · premise_audit · finite_sum_rules, set_operations_def — Disjointness is necessary; with overlap the mass of shared paths would be counted twice.
• S04 · premise_audit · S03, prob_axioms, finite_sum_rules — Needs the distinctness of S02, otherwise one path could be counted twice; finiteness reduces countable additivity to finite additivity (at most 5 nonempty disjoint events).
• S03 · premise_audit · G2, numeric_evaluation — Both nonnegativity and total 1 must be checked; a total only approximately 1 would not be a valid PMF.
• S02 · premise_audit · G1, cartesian_product_def — The grouping covers all 10 pairs: pairs in different groups are separated by an earlier coordinate and pairs in the same group by a later one. Distinctness ensures each path's mass is counted once.
• S01 · premise_audit · mko_process_def — Only unfolds the definition; 'one probability space' is the key point: if each X_n lived on a different space, there would be no dependence across time to speak of.

Error controls and negative controls

Marginal tables are not the whole process

The misconception named by the MKO is to take the collection of time marginals as the complete stochastic process. The product law Q of S15 has exactly the same three marginals as the original law, yet gives P(X_1=1,X_2=1)=25/64 instead of 1/2. Dependence across time is recorded only in the complete path law (MKO paragraph 0), and the marginals discard it.

Do not assert Markov automatically

The Markov property is an obligation about conditional distributions and must be checked from the definition (MKO paragraph 2). Here P(X_2=1∣X_1=1,X_0=0)=1 but P(X_2=1∣X_1=1,X_0=1)=2/3, so the past still matters given the present. Also the conditioning events must have positive probability to be compared; the two used here have probabilities 1/4 and 3/8.

Equal marginals do not mean equal variables

X_1 and X_2 have the same marginal (3/8,5/8), but they are not the same random variable: on ω_3, X_1=0 and X_2=1, and on ω_4, X_1=1 and X_2=0, so P(X_1≠X_2)=1/8+1/8=1/4 (N17). Equal distributions mean equal laws, not pointwise equality.

• Knowing the three marginal tables of X_0, X_1, X_2 determines the whole process → reject — The product law Q has the same marginals but Q(X_1=1,X_2=1)=25/64≠1/2 (S15).
• Every finite-horizon discrete process is Markov → reject — Here P(X_2=1∣X_1=1,X_0=0)=1≠2/3=P(X_2=1∣X_1=1,X_0=1) (S12 to S14).
• X_1 and X_2 have the same distribution, so X_1=X_2 almost surely → reject — P(X_1≠X_2)=P({ω_3,ω_4})=1/4>0.
• From the path law on {0,1,2} one can read off a transition law valid for all times → reject — MKO assumption a2 states that a finite-horizon joint law does not extend to infinite time or a transition law; moreover here even the transition at time 1 depends on X_0.

Result and limitations

Marginals p_0=(1/2,1/2), p_1=(3/8,5/8), p_2=(3/8,5/8); the joint law of (X_1,X_2) is (0,0):1/4, (0,1):1/8, (1,0):1/8, (1,1):1/2. The process is not stationary (p_0≠p_1) and not Markov (P(X_2=1∣X_1=1,X_0=0)=1, P(X_2=1∣X_1=1,X_0=1)=2/3, P(X_2=1∣X_1=1)=4/5). The product law with the same marginals gives P(X_1=1,X_2=1)=25/64≠1/2, so the marginal tables are not the whole process.

• The Markov property is tested at time 1 only; nothing beyond the three-point horizon {0,1,2} is discussed. Time 0 has no earlier observed past within this horizon; no infinite-horizon Markov claim is made.
• Whether the product law Q is Markov is irrelevant to the conclusions and is not proved.
• Finite Markov chains, transition matrices and path probabilities (the MKO's formal next) belong to a later entry.

Verification and source binding

17 declared checks for this guide; only the listed expressions are covered.

• source commit: b28a5edbd46279b1db2a3f3f48bb3f447e18899f
• adopted source SHA-256: 494d4109b4e53c43ae761b91728e2a7845b2d0d29b33ffa8731e9bd3c6b60e80
• original source SHA-256: 0c576baf78d0d575087097afb891ccdf82353c877594afdc55475e74c390656e
• Lean: NOT_RUN · independent AI: NOT_MEASURED

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