Discrete Stochastic Processes (Draft of the 2nd Edition by Robert G. Gallager

By Robert G. Gallager

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72), we see that "µ ∂2 # Sn lim E = 0. 73) −X n→1 n As a result, we say that Sn /n converges in mean square to X. This convergence in mean square says that the sample average, Sn /n, differs from the mean, X, by a random variable whose standard deviation approaches 0 with increasing n. , a sequence of functions) approaching a constant is clearly much more involved than a sequence of numbers approaching a constant. The laws of large numbers bring out this central idea in a more fundamental, and usually more useful, way.

We discuss four types of convergence in what follows, convergence in distribution, in probability, in mean square, and with probability 1. For the first three, we first recall the type of large number result with that type of convergence and then give the general definition. For convergence with probability 1 (WP1), we first define this type of convergence and then provide some understanding of what it means. This will then be used to state and prove the SLLN. 80) says æ Z z Ω µ 2∂ Sn − nX 1 −x √ √ exp lim Pr ≤z = dx for every z ∈ R.

Are IID rv’s satisfying E [|X|] < 1. Then for any ≤ > 0, ΩØ æ Ø Ø Sn Ø lim Pr Ø − E [X] Ø > ≤ = 0. 89) n→1 n Proof: We use a truncation argument; such arguments are used frequently in dealing with rv’s that have infinite variance. 33. 14) by   Xi ˘i = E [X] + b X  E [X] − b for E [X] − b ≤ Xi ≤ E [X] + b for Xi > E [X] + b for Xi < E [X] − b. 90) ˘ i are IID and, because of the truncation, must have a finite The truncated variables X second moment. Also the first moment approaches the mean of the original variables Xi 30 Central limit theorems also hold in many of these more general situations, but they do not hold as widely as the WLLN.

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