OPTIMUM BOUNDS FOR THE DISTRIBUTIONS OF MARTINGALES IN BANACH SPACES

By Iosif Pinelis. 1994

A general device is proposed, which provides for extension of exponential inequalities for sums of independent real-valued random variables to those for martingales in the 2-smooth Banach spaces. This is used to obtain optimum bounds of the Rosenthal-Burkholder and Chung types on moments of the martingales in 2-smooth Banach spaces. In turn, it leads to best-order bounds on moments of sums of independent random vectors in any separable Banach spaces. Although the emphasis is put on infinite-dimensional martingales, most of the results seem to be new even for one-dimensional martingales. Moreover, the bounds on moments of the Rosenthal-Burkholder type seem to be to a certain extent new even for sums of independent real-valued random variables. Analogous inequalities for (one-dimensional) supermartingales are given.

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Citations in the corpus

Byzantine Stochastic Gradient Descent (2018, background): “The following classical concentration result will be useful for us: Lemma 2.4 (Pinelis’ 1994 inequality [28]) .”
Exponential inequalities for martingales with applications (2013, background): “[29] Pinelis, I., 1994.”
DISCRETIZATION ERROR ANALYSIS FOR TIKHONOV REGULARIZATION (2006, unknown): “5 of [36]) gives”
Hoeffding’s inequality for supermartingales (2011, unknown): “Further, inequalities for the probabilities P (Xn > x) have been obtained by Prohorov (1959, [27]), Hoeffding (1963, [18]), Azuma (1967, [1]), Steiger (1967, [29]; 1969, [30]), Freedman (1975, [14]), Nagaev (1979, [22]), Haeusler (1984, [17]), McDiarmid (1989, [21]), Pinelis (1994, [24]), Talagrand ”
Sequential complexities and uniform martingale laws of large numbers (2015, background): “For instance, when the norm is equivalent to a 2-smooth norm (that is, p = 2), Pinelis [21] shows concentration of martingale difference sequences in the Banach space without the extra log(3)(n) term and with better constants.”
Some almost sure results for unbounded functions of intermittent maps and their associated Markov chains (2009, background): “To conclude, it suffices to apply Theorem 3.4 in Pinelis (1994) to the martingale M̄j .”
Moment inequalities for functions of independent random variables (2005, unknown): “) The Burkholder–Rosenthal–Pinelis inequality ([34], Theorem 4.”
Adaptivity of averaged stochastic gradient descent to local strong convexity for logistic regression (2013, methodology): “…the same result: (a) our original somewhat tedious proof based on taking powers of the inequality in Equation (1) and using martingale moment inequalities, (b) a shorter proof later derived by Bach and Moulines (2013), that uses Burkholder-Rosenthal-Pinelis inequality (Pinelis, 1994, Theorem 4.1).”
Error bounds for kernel-based approximations of the Koopman operator (2023, unknown): “Hence, (3.10) follows from Hoeffding’s inequality for Hilbert space-valued random variables [50, Theorem 3.5] (see also [37, Theorem A.5.2]), and”
SPIDER: Near-Optimal Non-Convex Optimization via Stochastic Path Integrated Differential Estimator (2018, methodology): “Concentration Inequality for Vector-valued Martingales We apply a result by Pinelis (1994) and conclude Proposition 2 which is an Azuma-Hoeffding-type concentration inequality. See also Kallenberg & Sztencel (1991), Lemma 4.4 in Zhang (2005) or Theorem 2.1 in Zhang (2005) and the references therein.”