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MA3H2 Markov Processes and Percolation Theory (Support Class 1) Yulong Lu (yulong.lu@warwick.ac.uk) 1 Simple Random Variables Poisson distribution. Let λ ∈ [0, ∞)) and let X : Ω → N0 be such that P(X = n) = e−λ λn , n! n ∈ N0 Then X is called the Poisson random variable with parameter λ, or X ∼ Poi(λ). E[X] = Var[X] = λ. Poisson distribution is usually used to model the probability of a given number of events occurring in a fixed interval of time, for example, the number of buses passing certain bus stop per hour. Exponential distribution. Let θ > 0 and let X be a nonnegative random variable such that Z x θe−θt dt for x > 0 P(X ≤ x) = 0 Then X is an exponential random variable with parameter θ, or X ∼ Exp(θ). E[X] = 1/θ, Var[X] = 1/θ2 . Exponential distribution is the distribution that describes the time between events in a Poisson process. In the bus example, the waiting time between two buses is exponential distributed. θr r−1 −θx Gamma distribution. Let θ, r > 0 and let Γθ,r be the distribution on [0, ∞) with density Γ(r) x e . Then Γθ,r is called the Gamma distribution with scaled parameter θ and shape parameter P r. E Γθ,r = r/θ, Var Γθ,r = r/θ2 . In particular, if {Xi }ni=1 are i.i.d random variables of Exp(θ), then ni=1 Xi ∼ Γθ,n . 2 Borel-Cantelli Lemma Let (Ω, F, P) be a probability space. Consider a sequence of subsets {An } of Ω. We define ∞ lim sup An = ∩∞ n=1 ∪m=n Am = {ω ∈ Ω : #{n ∈ N : ω ∈ An } = ∞} and ∞ lim inf An = ∪∞ / An } < ∞} n=1 ∩m=n Am = {ω ∈ Ω : #{n ∈ N : ω ∈ In other words, the limit superior is the event where infinitely many of the An occur. On the other hand, the limit inferior is the event where eventually all of the An occur. It is clear that lim sup 1An = 1lim sup An , lim inf 1An = 1lim inf An where 1A (x) is the indicator function of set A. It is more convenient to write lim sup An = {ω : ω ∈ An i.o.}, where i.o. means infinitely often. P Theorem 2.1. Borel-Cantelli lemma 1 (BC1) If ∞ n=1 P(An ) < ∞, then P(An i.o.) = 0. Proof. By definition and the summability of P(An ), P(An i.o.) = P(∩∞ n=1 ∪∞ m=n Am ) = lim P(∪∞ m=n Am ) n→∞ 1 ≤ lim n→∞ ∞ X m=n P(Am ) = 0 (BC1) provides a sufficient condition to verify P(An i.o.) = 0, which is very useful to prove some almost sure results in probability. Recall that a family of random variables {Xn } converge to X in probability means, for any ε > 0, P(|Xn − X| > ε) → 0 as n → ∞. Xn converges to X almost surely (a.s.) is to say P(ω : |Xn (ω) − X(ω)| → 0) = 1. Proposition 2.1. Xn → X a.s. if and only if for any ε > 0, P(|Xn − X| > ε i.o.) = 0. Proof. By definition, Xn → Xa.s. ⇐⇒ P(ω : |Xn (ω) − X(ω)| → 0) = 1 ⇐⇒ P(∀ε > 0, ∃N, s.t.|Xn − X| ≤ ε for n > N ) = 1 ∞ ⇐⇒ P(∩εk ∪∞ n=1 ∩m=n {|Xm − X| ≤ εk }) = 1 ∞ ⇐⇒ ∀ε > 0, P(∩∞ n=1 ∪m=n {|Xm − X| > ε}) = 0 ⇐⇒ ∀ε > 0, P(|Xn − X| > ε i.o.) = 0 The following result gives an application of (BC1). Proposition 2.2. Xn → X in probability if and only if for every subsequence Xn(m) such that there is a further subsequence Xn(mk ) that converges almost surely to X. Proof. Let εk be a sequence of positive numbers converging to zero. For each k, there is an n(mk ) > n(mk−1 ) so that P(|Xn(mk ) − X| > εk ) ≤ 2−k . Noticing that ∞ X P(|Xn(mk ) − X| > εk ) < ∞, k=1 (BC1) implies P(|Xn(mk ) − X| > εk i.o. ) = 0, i.e., Xn(mk ) → X a.s. On the other hand, if for every subsequence Xn(m) there is a further subsequence Xn(mk ) → X a.s. , then for any δ > 0, any subsequence of the sequence {yn } with yn := P(|Xn − X| > δ) has a convergent further subsequence (with limit zero), thus yn → 0. P Theorem 2.2. Borel-Cantelli lemma 2 (BC2). Assume that An are independent, then ∞ n=1 P(An ) = ∞ implies P(An , i.o.) = 1. Proof. We only need to show P({An , i.o.}c ) = 0. Indeed, by the independence of An c P({An , i.o.} ) = P(∪∞ n=1 ≤ lim n→∞ ∩∞ m=n ∞ Y Acm ) = c lim P(∩∞ m=n Am ) n→∞ exp (−P(Am )) = lim exp − n→∞ m=n ∞ Y = lim n→∞ ∞ X (1 − P(Am )) m=n ! P(Am ) =0 m=n where we have used the fact that 1 − x ≤ e−x when x > 0 for the inequality. Theorem 2.3. Strong law of large P numbers (SLLN). Let {Xi } be an i.i.d sequence with E|Xi | < ∞. Assume that E|Xi | = µ and Sn = ni=1 Xi . Then Sn /n → µ a.s. as n → ∞. 2 As an application of (BC2), the following result shows that E|Xi | < ∞ is necessary for SLLN. Proposition P 2.3. If {Xn } is a family of i.i.d random variables with E|X1 | = ∞, then P(|Xn | ≥ n, i.o.) = 1. So if Sn = ni=1 Xi then P(limn→∞ Sn /n exits ∈ (−∞, ∞)) = 0. Proof. Note that Z ∞ P(|X1 | > x)dx ≤ E|X1 | = 0 ∞ X P(|X1 | > n). n=0 If E|X1 | = ∞ and are i.i.d, by (BC2) P(|Xn | ≥ n, i.o.) = 1. Moreover, observe that Sn Sn+1 Sn Xn+1 − = − n n+1 n(n + 1) n + 1 and on C := {ω : limn→∞ Sn /n exists ∈ (−∞, ∞)}, Sn /(n(n + 1)) → 0. Thus on C ∩ {ω : |Xn | ≥ n i.o. }, we have Sn Sn+1 2 n − n + 1 > 3 i.o. contradicting the fact that ω ∈ C. Thus {ω : |Xn | ≥ n i.o } ∩ C = ∅ and since P(|Xn | ≥ n, i.o.) = 1, we have P(C) = 0. 3 Kolmogorov’s Extension Theorem In this section, we discuss the problem of constructing stochastic processes which are infinite collections {Xt : t ∈ I} of random variables. In general, this construction starts from the finite distribution of the process. Then Kolmogorov’s Extension Theorem guarantees the existence of a consistent measure on some infinite dimensional space. Let (E, E) be a measurable space and let T be an index set. We denote by E T the set of all functions f from T to E. For every t ∈ T , define πt : E T → E to be the evaluation (projection) map πt (f ) := f (t) Then one can define a σ-algebra E T on E T which is the smallest σ-algebra such that each πt is (E T /E)measurable. Definition 3.1. (Stochastic process) Let T be an index set, (E, E) a measurable space and (Ω, F, P) a probability space. A stochastic process is defined as a collection {Xt : t ∈ T } of (E, E)-valued random variables carried by the triple (Ω, F, P). Let X be a stochastic process, then for ω ∈ Ω, the map X(ω) : t ∈ T 7−→ Xt (ω) ∈ E is called the sample function of X. Here is another view of X, that is the map X : ω ∈ Ω 7−→ X(ω) ∈ E T or an (E T , E T )-valued random variable. Definition 3.2. (Law of the stochastic process) The law of the stochastic process is defined as the probability measure µ : P ◦ X −1 on (E T , E T ). For any non-empty subset S of T , we define πS X : Ω 7−→ E S via (πS X)(ω) := πS (X(ω)) = X(ω)|S and µS := P ◦ (πS X)−1 on (E S , E S ). 3 Definition 3.3. (Finite dimensional distribution) Let Fin(T ) be the set of non-empty finite set of T . The probability measures {µS : S ∈ Fin (T )} are called the finite-dimensional distribution of X. Definition 3.4. (Compatibility condition) Let U ⊂ V ∈ Fin (T ) and let πUV denote the restriction map from E V to E U , then we say that the finite-dimensional distribution {µS } has the Compatibility condition if µU = µV ◦ (πUV )−1 Theorem 3.1. (Kolmogorov’s Extension Theorem) Let E be a compact metrisable space, and let E = B(E). Let T be a set. Suppose that for each S in Fin (T ), there exits a probability measure µS on (E S , E S ) and that the measures {µS : S ∈ Fin (T )} are compatible, i.e. µU = µV ◦ (πUV )−1 holds whenever U, V ∈ Fin(T ) for U ⊂ V . Then there exists a unique measure µ on (E T , E T ) such that µS = µ ◦ πS−1 on (E S , E S ) Remark 3.1. The compact metrisable condition on (E, E) in Theorem 3.1 can be weaken to be a topological space endowed with the Borel σ-algebra. Example (Gaussian processes). Let C be a symmetric positive quadratic form on RN . Then there exists a unique Gaussian processes with index set N, state space R, such that for all finite J ∈ N, the finite dimensional distributions are |J|-dimensional Gaussian vectors with covariance C J . Here C J is the finite-dimensional sub-matrices of C. References [1] R. Durrett, Probability: Theory and Examples. 3nd Edition, Cambridge University Press, 2010. [2] L.C.G. Rogers and D. Williams, Diffusions, Markov Processes, and Martingales. 2nd Edition, Vol 1-2, Cambridge University Press, 2000. 4