0 0 0 0 0 0 0 615.3 833.3 762.8 694.4 742.4 831.3 779.9 583.3 666.7 612.2 0 0 772.4 30 0 obj The latter one will be used in the proof of the MCT. Proof: Let a ∈ R be given, and set "> 0. ?^h-����>�΂���� ,�x �+&�l�Q��-w���֧. 833 611 556 833 833 389 389 778 611 1000 833 833 611 833 722 611 667 778 778 1000 $f_n \to f$ in $L^1(\mu)$ is equivalent to $\int |f_n - f| d\mu \to 0$ and $f \in L^1(\mu)$. 298.4 878 600.2 484.7 503.1 446.4 451.2 468.7 361.1 572.5 484.7 715.9 571.5 490.3 Suﬃcient conditions for the maximum of the limit to be the limit of the maximum are that the convergence is uniform and the parameter space is compact. /Widths[277.8 500 833.3 500 833.3 777.8 277.8 388.9 388.9 500 777.8 277.8 333.3 277.8 However the additive property of integrals is yet to be proved. We do not assume the Xn's are independent of each other. j); this property is known as subadditivity. 778 611 556 722 778 333 333 667 556 944 778 778 611 778 667 556 611 778 722 944 722 Suppose that we have a sequence of random variables that converges in probability to a certain number a, and another sequence that converges in probability to some other number b. /FirstChar 1 666.7 666.7 666.7 666.7 611.1 611.1 444.4 444.4 444.4 444.4 500 500 388.9 388.9 277.8 $s: \Omega \to [0,\infty]$: a simple function. In this case we call it the bounded convergence theorem. << Special cases of the theorem is Markov’s inequality and Chebyshev’s inequality. Properties of probability measures: PDF unavailable: 11: Continuity of probability measure: PDF unavailable: 12: Discrete probability space-finite and countably infinite sample space: ... Monotone Convergence Theorem - 2 : PDF unavailable: 61: Expectation of a … /BaseFont/UGMOXE+MSAM10 /Name/F7 278 444 556 444 444 444 444 444 606 444 556 556 556 556 500 500 500] /Differences[0/Gamma/Delta/Theta/Lambda/Xi/Pi/Sigma/Upsilon/Phi/Psi/Omega/ff/fi/fl/ffi/ffl/dotlessi/dotlessj/grave/acute/caron/breve/macron/ring/cedilla/germandbls/ae/oe/oslash/AE/OE/Oslash/suppress/exclam/quotedblright/numbersign/dollar/percent/ampersand/quoteright/parenleft/parenright/asterisk/plus/comma/hyphen/period/slash/zero/one/two/three/four/five/six/seven/eight/nine/colon/semicolon/exclamdown/equal/questiondown/question/at/A/B/C/D/E/F/G/H/I/J/K/L/M/N/O/P/Q/R/S/T/U/V/W/X/Y/Z/bracketleft/quotedblleft/bracketright/circumflex/dotaccent/quoteleft/a/b/c/d/e/f/g/h/i/j/k/l/m/n/o/p/q/r/s/t/u/v/w/x/y/z/endash/emdash/hungarumlaut/tilde/dieresis/suppress 160/space/Gamma/Delta/Theta/Lambda/Xi/Pi/Sigma/Upsilon/Phi/Psi 173/Omega/ff/fi/fl/ffi/ffl/dotlessi/dotlessj/grave/acute/caron/breve/macron/ring/cedilla/germandbls/ae/oe/oslash/AE/OE/Oslash/suppress/dieresis] $\{f_n\}: \Omega \to \mathbb{R}$: a sequence of measurable functions. /Name/F4 If the real number is a realization of the random variable for every , then we say that the sequence of real numbers is a realization of the sequence of random variables and we write 1 Convergence of random variables We discuss here two notions of convergence for random variables: convergence in probability and convergence in distribution. Convergence Properties ofsome Spike-Triggered Analysis Techniques Liam Paninski Center for Neural Science New York University New York, NY 10003 ... the probability that ourcell emits a spike, given that some observable signal X in the world takes value x. n=1 is said to converge to X in probability, if for any > 0, lim n→∞ P(|X n −X| < ) = 1. Along with convergence theorems, these integral inequalities will be used intensely throughout the probability theory. << 13 0 obj There are several diﬀerent modes of convergence (i.e., ways in which a sequence may converge). Lebesgue’s dominated convergence theorem (DCT) provides a tool for not only monotonically convergent, but general convergent functions that are uniformly dominated by some integrable function. endobj Convergence properties: • Equation (10.1.20) implies that the convergence rate of the algorithm is proportional to α n/2. 0 0 688 0 778 618 0 0 547 0 778 0 0 0 880 778 0 702 0 667 466 881 724 750 0 0 0 0 778 778 778 667 611 500 444 444 444 444 444 444 638 407 389 389 389 389 278 278 278 << We study convergence properties of the mixed strategies that result from a general class of optimal no regret learning strategies in a repeated game setting where the stage game is any 2 by 2 competitive game (i.e. 667 667 667 333 606 333 606 500 278 500 611 444 611 500 389 556 611 333 333 611 333 The –rst two axioms essentially bound how events are weighed. In addition, since our major interest throughout the textbook is convergence of random variables and its rate, we need our toolbox for it. Lecture notes for STAT 547C: Topics in Probability (draft) Ben Bloem-Reddy November 17, 2020 Contents 1 Sets, classes of subsets, and measurable spaces5 The second one is trivial by continuity of measure $\phi$ defined as in the first one. n!1 0. /LastChar 226 1.1 Convergence in Probability We begin with a very useful inequality. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 500 0 278] /Encoding 31 0 R 0 0 0 0 0 0 0 333 208 250 278 371 500 500 840 778 278 333 333 389 606 250 333 250 Finally, we state Markov-Chebyshev inequality. Fatou’s lemma, another important convergence theorem can be directly derived by the MCT. /FirstChar 33 So in words, convergence in probability means that almost all of the probability mass of the random variable Yn, when n is large, that probability mass get concentrated within a narrow band around the limit of the random variable. /Subtype/Type1 /LastChar 196 >> random variables with mean EXi = μ < ∞, then the average sequence defined by ¯ Xn = X1 + X2 +... + Xn n Theorem 2. /Name/F8 424 331 827 0 0 667 0 278 500 500 500 500 606 500 333 747 333 500 606 333 747 333 Measurable Functions 24 3. As per mathematicians, “close” implies either providing the upper bound on the distance between the two Xn and X, or, taking a limit. The concept of convergence in probability is based on the following intuition: two random variables are "close to each other" if there is a high probability that their difference is very small. {X n}∞ n=1 is said to converge to X in distribution, if at all points x where P(X ≤ x) is continuous, lim n→∞ P(X n ≤ x) = P(X ≤ x). << /Type/Font Earlier, I mentioned that to prove the additive property of the Lebesgue integrals, we need the monotone convergence theorem. 400 606 300 300 333 611 641 250 333 300 488 500 750 750 750 444 778 778 778 778 778 $Y = c \in \mathbb{R}$). 1277.8 811.1 811.1 875 875 666.7 666.7 666.7 666.7 666.7 666.7 888.9 888.9 888.9 R ANDOM V ECTORS The material here is mostly from • J. 147/quotedblleft/quotedblright/bullet/endash/emdash/tilde/trademark/scaron/guilsinglright/oe | Find, read and cite all the research you need on ResearchGate Proposition 1 (Markov’s Inequality). >> 5. /Type/Font 277.8 500 555.6 444.4 555.6 444.4 305.6 500 555.6 277.8 305.6 527.8 277.8 833.3 555.6 /Type/Font 277.8 500] Proposition7.5 Convergence in probability implies convergence in distribution. 9 >> We begin with convergence in probability. 833.3 1444.4 1277.8 555.6 1111.1 1111.1 1111.1 1111.1 1111.1 944.4 1277.8 555.6 1000 >> << /Widths[791.7 583.3 583.3 638.9 638.9 638.9 638.9 805.6 805.6 805.6 805.6 1277.8 Comment: In the above example Xn → X in probability, so that the latter does not imply convergence in the mean square either. $\{f_n\}_{n\in\mathbb{N}}: \Omega \to [0,\infty]$: a sequence of measurable functions. 7 0 obj 40 0 obj 27 0 obj /Subtype/Type1 New content will be added above the current area of focus upon selection Construction and Extension of Measures 12 3. >> Since the expectation $EX$ is defined as a mere integral, all of the above theorems can be applied. It follows that convergence with probability 1, convergence in probability, and convergence in mean all imply convergence in distribution, so the latter mode of convergence is indeed the weakest. /Encoding 7 0 R 42 0 obj 774 611 556 763 832 337 333 726 611 946 831 786 604 786 668 525 613 778 722 1000 /LastChar 196 /FontDescriptor 15 0 R stream If X n!a.s. $\{f_n\}: \Omega \to [0,\infty]$: a sequence of measurable functions. /Subtype/Type1 Mappings and σ-Fields 21 2. Comparison between convergence in probability and weak convergence 175 Chapter 6. probability zero with respect to the measur We V.e have motivated a definition of weak convergence in terms of convergence of probability measures. 275 1000 666.7 666.7 888.9 888.9 0 0 555.6 555.6 666.7 500 722.2 722.2 777.8 777.8 416.7 416.7 416.7 416.7 1111.1 1111.1 1000 1000 500 500 1000 777.8] /FirstChar 1 12/03/2020 ∙ by Vidya Muthukumar, et al. /Type/Font In addition, since our major interest throughout the textbook is convergence of random variables and its rate, we need our toolbox for it. /BaseFont/XPWLTX+URWPalladioL-Roma Let X be a non-negative random variable, that is, P(X ≥ 0) = 1. One has to think of all theXt’s andZ = 0. 883 582 546 601 560 395 424 326 603 565 834 516 556 500 333 606 333 606 0 0 0 278 $X \ge 0$ a.s., $a > 0$ $\implies P(X \ge a) \le EX/a.$, $a > 0$ $\implies P(X \ge a) \le EX^2/a^2.$. 277.8 305.6 500 500 500 500 500 750 444.4 500 722.2 777.8 500 902.8 1013.9 777.8 >> Properties Convergence in probability implies convergence in distribution. Lebesgue–Stieltjes Measures 18 Chapter 2. 173/circlemultiply/circledivide/circledot/circlecopyrt/openbullet/bullet/equivasymptotic/equivalence/reflexsubset/reflexsuperset/lessequal/greaterequal/precedesequal/followsequal/similar/approxequal/propersubset/propersuperset/lessmuch/greatermuch/precedes/follows/arrowleft/spade] 0 0 0 0 0 0 0 333 333 250 333 500 500 500 889 778 278 333 333 389 606 250 333 250 556 444 500 463 389 389 333 556 500 722 500 500 444 333 606 333 606 0 0 0 278 500 endobj << /LastChar 255 667 667 333 606 333 606 500 278 444 463 407 500 389 278 500 500 278 278 444 278 778 0 0 0 528 542 602 458 466 589 611 521 263 589 483 605 583 500 0 678 444 500 563 524 In this section we assume a measure space $(\Omega, \mathcal{F}, \mu)$ with finite measure $\mu$. 500 500 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 625 833.3 /LastChar 229 /Type/Font << 1000 1000 1055.6 1055.6 1055.6 777.8 666.7 666.7 450 450 450 450 777.8 777.8 0 0 endobj 160/space/Gamma/Delta/Theta/Lambda/Xi/Pi/Sigma/Upsilon/Phi/Psi 173/Omega/alpha/beta/gamma/delta/epsilon1/zeta/eta/theta/iota/kappa/lambda/mu/nu/xi/pi/rho/sigma/tau/upsilon/phi/chi/psi/tie] 777.8 777.8 0 0 1000 1000 777.8 722.2 888.9 611.1 1000 1000 1000 1000 833.3 833.3 Convergence in Probability. Suppose B is the Borel σ-algebr n a of R and let V and V be probability measures o B).n (ß Le, t dB denote the boundary of any set BeB. A special case of Hölder’s inequality is the Cauchy-Schwarz inequality: $\|fg\|_1 \le \|f\|_2 \|g\|_2.$. /Differences[1/dotaccent/fi/fl/fraction/hungarumlaut/Lslash/lslash/ogonek/ring 11/breve/minus For instance, if $X_n \to X$ a.s. and $|X_n| \le Y$ for some $Y$ such that $E|Y| < \infty$, then by DCT $EX_n \to EX$ as $n\to\infty$. /FirstChar 32 296 500 500 500 500 500 500 500 500 500 500 250 250 606 606 606 500 747 722 611 667 777.8 777.8 1000 1000 777.8 777.8 1000 777.8] 778 778 778 667 604 556 500 500 500 500 500 500 758 444 479 479 479 479 287 287 287 However the additive property of integrals is yet to be proved. 333 333 556 611 556 556 556 556 556 606 556 611 611 611 611 556 611 556] /Name/F2 >> This post series is based on the textbook Probability: Theory and Examples, 5th edition (Durrett, 2019) and the lecture at Seoul National University, Republic of Korea (instructor: Prof. Johan Lim). 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Convergence properties of the DCT is where the sequence$ { f_n } $) ( iv ) is way! A real number prove the yet to be proved second one is trivial by continuity of measure$ \phi defined... 175 chapter 6 by the rate of change of the MCT allows to! Theorems only on the one hand PDF | on Jan 1, 1994, Léon and! Directly derived by the MCT if X1, X2, X3, ⋯ are i.i.d \varphi \mathbb!