SELF-NORMALIZED PROCESSES: EXPONENTIAL INEQUALITIES, MOMENT BOUNDS AND ITERATED LOGARITHM LAWS

By Victor H. de la Peña, M. Klass, Tze Leung Lai. 2004

135 citations recorded by the source corpus.

Self-normalized processes arise naturally in statistical applications. Being unit free, they are not affected by scale changes. Moreover, self-normalization often eliminates or weakens moment assumptions. In this paper we present several exponential and moment inequalities, particularly those related to laws of the iterated logarithm, for self-normalized random variables including martingales. Tail probability bounds are also derived. For random variables Bt>0 and At, let Yt(λ)=exp{λAt−λ2Bt2/2}. We develop inequalities for the moments of At/Bt or supt≥0At/{Bt(log logBt)1/2} and variants thereof, when EYt(λ)≤1 or when Yt(λ) is a supermartingale, for all λ belonging to some interval. Our results are valid for a wide class of random processes including continuous martingales with At=Mt and $B_{t}=\sqrt {\langle M\rangle _{t}}$ , and sums of conditionally symmetric variables di with At=∑i=1tdi and $B_{t}=\sqrt{\sum_{i=1}^{t}d_{i}^{2}}$ . A sharp maximal inequality for conditionally symmetric random variables and for continuous local martingales with values in Rm, m≥1, is also established. Another development in this paper is a bounded law of the iterated logarithm for general adapted sequences that are centered at certain truncated conditional expectations and self-normalized by the square root of the sum of squares. The key ingredient in this development is a new exponential supermartingale involving ∑i=1tdi and ∑i=1tdi2. A compact law of the iterated logarithm for self-normalized martingales is also derived in this connection.

Subfields: Finance, Management Science and Operations Research, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/27b5553d558632e7d610ba4fa682d74bb1eac7c7

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Citations in the corpus

Exponential inequalities for martingales with applications (2013, unknown): “[9, 10].”
On the Properties of the Synthetic Control Estimator with Many Periods and Many Controls (2019, background): “4 of de la Peña et al. (2004), there is a constant C1 such that, for any 0 ≤ x ≤ C0k 1 6 ,”
Improved Algorithms for Linear Stochastic Bandits (2011, methodology): “The new inequality is derived using techniques from the theory of self-normalized processes (de la Pe˜na et al., 2004, 2009).”
Uniform, nonparametric, non-asymptotic confidence sequences (2018, methodology): “The literature on self-normalized bounds also makes extensive use of the method of mixtures, sometimes called pseudo-maximization (de la Peña et al., 2004, 2007, 2009).”
Self-normalized limit theorems: A survey (2013, background): “For instance, Csörgö and Hu (2013) [16] established a strong approximation result, Csörgö, Hu and Mei (2013) [17] obtained a strassen-type law of the iterated logarithm, Dembo and Shao (1998, 2006) [23, 24] considered selfnormalized laws of the iterated logarithm under space R, de la Pena, Klass and”
Contextual Combinatorial Bandit and its Application on Diversified Online Recommendation (2014, methodology): “For an introduction to this theory, we refer interested readers to [21, 9].”
Improved Optimistic Algorithms for Logistic Bandits (2020, methodology): “Our analysis follows the steps of the pseudo-maximization principle introduced in de la Pena et al. (2004), used by Abbasi-Yadkori et al. (2011) for the linear bandit and thoroughly detailed in Chapter 20 of Lattimore and Szepesv´ari (2018).”
Time-uniform Chernoff bounds via nonnegative supermartingales (2018, methodology): “• Also in contrast to de la Peña et al. (2004), we allow the exponential pro- cess to be merely upper bounded by a supermartingale, rather than being a supermartingale itself; this permits us to handle bounds on the maximum eigenvalue process of a matrix-valued martingale, using techniques from…”
Estimating means of bounded random variables by betting (2020, background): “[26, 27, 28], and has led to several other followup works [1, 4, 38, 42].”
Time-uniform, nonparametric, nonasymptotic confidence sequences (2018, background): “Lai [41] treats confidence sequences for the parameter of an exponential family using mixture techniques similar to those of Section 3.2.”
Martingales in Sequential Analysis and Time Series, 1945-1985 ∗ (2009, unknown): “Our key observation of de la Peña et al. (2004) was that A2t 2B2t = max θ θAt − θ2B2t 2 .”
Exponential inequalities for self-normalized martingales with applications (2007, background): “Some recent extensions of the above inequalities in a more general framework including discrete-time martingales can also be found in [ 9 ], [10] where the conditionally symmetric assumption is still required for (1.3).”