Strong Limit Theorems for Increments of Random Fields
After reconsidering the oscillating behaviour of sums of i.i.d. random variables we study the oscillating behavior for sums over i.i.d. random fields under exact moment conditions. This summarizes several papers published jointly with A. Gut (Uppsala).
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Strong Limit Theorems for Increments of Random Fields Ulrich Stadtmuller ¨
Abstract After reconsidering the oscillating behaviour of sums of i.i.d. random variables we study the oscillating behavior for sums over i.i.d. random fields under exact moment conditions. This summarizes several papers published jointly with A. Gut (Uppsala).
11.1 Introduction We shall consider a classical scenario, namely i.i.d. random variables X; Xi , i 2 N and we shall impose appropriate moment conditions later on. As usually we denote by n X Sn D Xj ; n2N j D1
the partial sums of these random variables and begin with an overview of almost sure limit theorems on Sn . In all kind of statistics and questions averages play an important role and averages are just of the form Sn =n and almost sure limit theorems deal with such averages. Typically in this situation there is an equivalence between such limit results and appropriate moment conditions. In order to demonstrate this we begin with the strong law of large numbers (SLLN). Fore more details see, e.g., the book [210]. Theorem 11.1 (SLLN (Kolmogorov)). Sn a:s: ! 0 ” EjX j < 1 and EX D 0: n U. Stadtm¨uller () Ulm University, Ulm, Germany e-mail: [email protected] E. Spodarev (ed.), Stochastic Geometry, Spatial Statistics and Random Fields, Lecture Notes in Mathematics 2068, DOI 10.1007/978-3-642-33305-7 11, © Springer-Verlag Berlin Heidelberg 2013
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Remark 11.1. The ) direction is formulated somewhat sloppily, here and throughout it should be read as follows: if lim supn!1 jSnn j < 1 then EjX j < 1, hence by the converse conclusion the limit exists and is then EX: This result was extended as follows. Theorem 11.2 (Marcinkiewicz–Zygmund ’37). For 0 < r < 2 Sn a:s: ! 0 ” E.jX jr / < 1 and .EX D 0 provided r 1/: n1=r By the CLT it follows that the result fails to hold for r D 2: Next, we go on with the speed of convergence in the SLLN, the famous law of iterated logarithm (LIL). Theorem 11.3 (LIL (Hartmann–Wintner ’44, Strassen ’66)). It holds that Sn a:s: lim sup p D 1 ” EjX j2 D 1; EX D 0: n!1 2n log log n Remark 11.2. Obviously under these moment conditions Sn a:s: lim inf p D 1 n!1 2n log log n p and any point in Œ1; 1 is an a.s. limit point of the sequence .Sn = 2n log log n/: A corresponding remark applies to related theorems below. A somewhat different topic are limit laws for increments of sums of i.i.d. random variables. That is we shall study the almost sure oscillation behaviour of partial sums. This oscillation behaviour is interesting itself. We shall start with the following result where we denote by logC x D maxf1; log xg; x > 0: Theorem 11.4 (Chow ’73, Lai ’74 [123, 124, 319]). For 0 < ˛ < 1 we have a SLLN SnCn˛ Sn a:s: ! 0 ” E.jX j1=˛ / < 1; EX D 0; n˛ and a law of single logarithm (LSL) SnCn˛ Sn a:s: p lim sup p D 1 ˛ ” E.jX j2=˛ .logC jX j/1=˛ / < 1 n!1 2n˛ log n and EX D 0; EX 2 D 1: Remark 11.3. 1. Increments of sums can be considered as special weighted sums of random variables. SnCn˛ Sn D
1 X kD1
wk n Xk
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