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The map $(x_n)\mapsto (\lim x_n,x_1-\lim x_n,x_2-\lim x_n,\dots )$ proves $c\cong c_0$, equivalently $c_0\oplus \mathbb{F}\cong c_0$.

An increasing real sequence converges exactly when it is bounded above. For a bounded sequence, its limit is the supremum of its set of terms.

For a positive sequence converging to $L\ge 0$, its cumulative geometric means converge to the same limit.

The Cesàro mean of the first $n\ge 1$ terms of a sequence is their arithmetic mean. It is a cumulative average: increasing $n$ includes the whole initial segment, rather than a fixed-length sliding window. For $a_k=(-1{)}^k$, the even-length averages are zero and the odd-length averages are $-1/n$, so they tend to zero even though the sequence does not converge.

For a periodic sequence with period $p\ge 1$ and one-cycle arithmetic mean $M$, split an initial segment into full cycles and a remainder of fewer than $p$ terms. The full cycles contribute exactly $M$ per term; the remainder contributes a fixed bounded error divided by $n$. Thus the Cesàro means tend to $M$. The cycle $0,2,1$ in the bounded partial sums need not be Cesàro summable illustration has arithmetic mean one. Treating it as a sequence of partial sums gives a series with first-order Cesàro sum one; averaging a sequence and summing its terms remain different operations.

independent indicators with success probabilities $1/n$ converge to zero in Probability, but successes occur infinitely often almost surely by the second Borel--Cantelli lemma.

From $X_n\to X$ in Probability, choose $n_k$ with
$\mathbb{P}(|X_{n_k}-X|>2^{-k})<2^{-k}$. The first Borel--Cantelli lemma then gives $X_{n_k}\to X$ almost surely.

Convergence in probability is equivalent toThe forward implication splits at a fixed error threshold; the reverse implication is Markov's inequality.

If $X_n$ converges in Probability to both $X$ and $Y$, then $X=Y$ almost surely. Indeed, the triangle inequality and a union bound show that $\mathbb{P}(|X-Y|>\epsilon )=0$ for every $\epsilon >0$.

Let $X_1,X_2,\dots$ be real independent and identically distributed random variables with $\mathbb{E}|X_1|<\infty$ and expected value $\mu =\mathbb{E}[X_1]$. The weak law of large numbers states that their sample mean has convergence in probability to $\mu$: for every fixed $\epsilon >0$,Thus a large sample is unlikely to give an average with a specified fixed-size error. The strong law of large numbers makes the stronger statement of almost sure convergence along an entire sequence. Under the displayed assumptions both laws apply; their names distinguish the conclusions, not two competing predictions.

When the variance ${\sigma }^2$ is finite, variance additivity for independent random variables gives $\mathrm{Var}({\overline{X}}_n)={\sigma }^2/n$. The Chebyshev inequality immediately supplies a useful quantitative form and a brief proof:For independent trials with a Bernoulli distribution of parameter $p=0.2$, the expected value is $0.2$ and the variance is $0.16$. At $n=1000$, the chance of an error of at least $0.1$ is therefore at most $0.016$. The exact binomial distribution gives a much smaller probability, as the figure under the strong law of large numbers shows. The bound is conservative, but it is valid for every sample size under these assumptions and does not require a normal approximation.

The weak law of large numbers addresses a fixed tolerance $\epsilon$. The central limit theorem instead resolves errors on the shrinking scale $\sigma /\sqrt{n}$ and gives their approximate normal distribution. In particular, the weak law of large numbers alone supplies neither a bell-shaped error law nor a universal rate of approach. For example, a Pareto distribution with minimum one and shape $3/2$ has expected value three but infinite variance: both the weak law of large numbers and strong law of large numbers apply, while the finite-variance statement of the central limit theorem does not.

If $X_n\stackrel{d}{\to }X$ and a measurable function $g$ is continuous at $X$ almost surely, then $g(X_n)\stackrel{d}{\to }g(X)$. The same principle holds for random elements of metric spaces.

The Einstein tensor isThe contracted Bianchi identity and metric compatibility give
${\nabla }^{\nu }{G}_{\mu \nu }=0$, ensuring compatibility of the Einstein equations with covariant stress-energy conservation.

An adapted process $M$ is a local martingale when there are stopping times $T_n\uparrow \infty$ almost surely such that each stopped process $M_{t\wedge T_n}$ is a martingale. Every martingale is local, while a positive local martingale is a supermartingale.

A birth-death process moves only between neighbouring nonnegative integer states, with state-dependent birth and death rates.

For integer batch size $k\ge 1$ and rates $\lambda ,\beta \ge 0$, a Batch-birth linear-death process has transitionsWriting $P_n(t)=\mathbb{P}(N_t=n)$ and taking $P_j=0$ for $j<0$, its Master equation is

For a finite-state chain with transition semigroup $P(t)=e^{tQ}$, a probability row vector $\pi$ is invariant when $\pi P(t)=\pi$ for every $t\ge 0$. Equivalently,

Explosion occurs when infinitely many jumps take place in finite time.

Explosion is determined by whether the successive holding times of the continuous-time Markov chain have a finite sum.

The jump chain records the successive states visited by a Continuous-time Markov chain while discarding its holding times.

For $h>0$, the fixed-time skeleton has matrix $P(h)$. If $q_i$ is the holding rate, then for $nh\le t\le (n+1)h$,Thus ${\int }_0^{\infty }{p}_{ii}(t)dt$ diverges exactly when ${\sum }_n{p}_{ii}(nh)$ does, proving recurrence equivalence for an irreducible chain.

Fix a finite embedded jump path $i_0,\dots ,i_n$ whose holding rates $q_{i_k}$ are positive and finite. Conditional on those successive states, the complete holding times are independent exponential random variables with the corresponding rates. To observe exactly these $n$ jumps by a fixed time $t$, the final holding time is censored rather than completed.

Write $H_{i_0\cdots i_n}(t)$ for this conditional holding-time probability. With $u_n=t-{\sum }_{k<n}{u}_k$,The completed holding times supply their density factors; the last portion supplies only an exponential survival factor. The fixed observation time imposes the simplex constraint.

Multiplying by the terminal rate completes the symmetric product of all $n+1$ rates. Reversing the time portions and the state sequence preserves the simplex integral, with unit absolute Jacobian. HenceThe right-hand holding kernel can be defined by this formula even if the reverse embedded path is not feasible; its conditional path interpretation requires that feasibility. For actual unconditioned path probabilities, the holding kernel must also be multiplied by the product of jump-chain transition probabilities. Those products need not agree under reversal, so the holding identity alone is not a reversibility theorem for the chain.

For transition rates $q_{nm}$, the generator acts on a function $f$ byFor a test function in the generator domain, under the integrability conditions for Dynkin's formula and differentiation,Choosing polynomial functions $f$ gives differential equations for the moments.