Modes of Convergence Theorem Almost sure convergence in implies convergence in probability The statement X na:s: X is equivalent to the the fact that for any >0, PfjX n Xj> in nitely ofteng= 0 0 = fjX n Xj> in nitely ofteng= \1 n=1 [1 m= fjX m Xj> g Thus, [1 m=n fjX m Xj> g is a decreasing sequence of events to converging 0 with P(0) = 0
answer is that both almost-sure and mean-square convergence imply convergence in probability, which in turn implies convergence in distribution On the other hand, almost-sure and mean-square convergence do not imply each other Proposition7 1 Almost-sure convergence implies convergence in probability Proof: If X n →a s X, then Plimsup
convergence We want to know which modes of convergence imply which The answer is that both almost-sure and mean-square convergence imply convergence in probability, which in turn implies convergence in distribution On the other hand, almost-sure and mean-square convergence do not imply each other
Lecture-14: Almost sure convergence of random variables 1 Almost sure convergence Consider a probability space (W,F,P) Recall that a random variable X is an F-measurable function on the sample space W such that X 1( ¥,x] 2F for all x 2R A sequence of random variables (Xn(w) : n 2N) is hence a sequence of F-measurable functions
While convergence in distribution only says something about distributions, convergence in probability, almost sure convergence and convergence in L p are statements implying that X n and X are asymptotically “close” Almost sure convergence implies convergence in probability, as can be easily seen from my definitions above Convergence in
Convergence almost surely requires that the probability that there exists at least a k ≥ n such that Xk deviates from X by at least tends to 0 as ntends to infinity (for every > 0) This demonstrates that an ≥pn and, consequently, that almost sure convergence implies convergence in probability To better explain this notion of almost sure
For almost sure convergence, convergence in probability and convergence in distribution, if X n converges to Xand if gis a continuous then g(X n) converges to g(X) Convergence in distribution di ers from the other modes of convergence in that it is based not on a direct comparison of the random variables X
Hence, almost sure convergence is a statement about the “infinite dimensional” event A Note that A is the collection of sample outcomes ω on which the numerical sequence Xn(ω) converges to X∞(ω) as n → ∞ Almost sure convergence has alternative names: Convergence with probability one, Convergence almost ev-
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