By S. M. Ulam, A. R. Bednarek, Françoise Ulam

Many of the rules awarded continue their significance this present day, and . . . are totally fundmental, either from a historic and from a systematic viewpoint.--Gian-Carlo Rota, Massachusetts Institute of expertise

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**Sample text**

Hence, using that r(u) = r(−u) we get that nV[mn ] = r(0) + 2 n−1 ∑ (n − u)r(u). 11) n−1 ∞ ∞ Now, if ∑t=0 r(t) is convergent, Sn = ∑t=0 r(t) → ∑t=0 r(t) = S, say, which 1 n 3 implies that n ∑k=1 Sk → S , as n → ∞. , nV [mn )] → r(0) + ∞ 2 ∑∞ 1 r(u) = ∑−∞ r(u). (c) The first statement follows from (a) and (b), since E[(mn − m)2 ] = V[mn ] + E[mn − m]2 → 0. The second statement is a direct consequence of Chebyshev’s inequality, P(|mn − m| > ε ) ≤ E[(mn − m)2 ] . ε2 The consequences of the theorem are extremely important for data analysis with dependent observations.

Since E[X (t)] = 0, the variance is E[X (t)2 ] − 02 = E[A2 ] · E[(cos(2π f0 t + φ ))2 ] = E[A2 ] · E[(sin(2π f0 t + φ ))2 ], because a randomly shifted cosine function can not be distinguished from a randomly shifted sine function. But cos2 x + sin2 x = 1, so the two expectations are equal to one half each. 2 Superposition of random harmonic oscillations A cosine function with random amplitude and phase is not very useful as a model for a random function of time. However, adding many independent such cosine functions will make a big difference, and lead to very general and useful stochastic processes.

1 Daily average temperature in Målilla during January, for 1988–1997. The fat curves mark the years 1992 and 1997. Data: Carl-Erik Fröberg. measures of the dependence over time. The statistical definitions are simple, but the practical interpretation can be complicated. We illustrate this by the simple concepts of “average temperature” and “day-to-day” correlation. 1 (“Daily temperature”). 1 shows the daily average temperature in the Swedish village of Målilla during the month of January for the ten years 1988–1997.