0. 1111.1 1511.1 1111.1 1511.1 1111.1 1511.1 1055.6 944.4 472.2 833.3 833.3 833.3 833.3 /Filter[/FlateDecode] 820.5 796.1 695.6 816.7 847.5 605.6 544.6 625.8 612.8 987.8 713.3 668.3 724.7 666.7 275 1000 666.7 666.7 888.9 888.9 0 0 555.6 555.6 666.7 500 722.2 722.2 777.8 777.8 774 611 556 763 832 337 333 726 611 946 831 786 604 786 668 525 613 778 722 1000 << /FontDescriptor 19 0 R /Subtype/Type1 However the additive property of integrals is yet to be proved. /Subtype/Type1 889 611 556 611 611 389 444 333 611 556 833 500 556 500 310 606 310 606 0 0 0 333 We say that P is a probability measure if, in addition to the above requirements, P satis es P() = 1. /Type/Font << Parts (a) and (b) are the linearity properties; part (a) is the additivity property and part (b) is the scaling property.Parts (c) and (d) are the order properties; part (c) is the positive property and part (d) is the increasing property.Part (e) is a continuity property known as the monotone convergence theorem.Part (f) is the additive property for disjoint domains. In the opposite direction, convergence in distribution implies convergence in probability when the limiting random variable $ X $ is a constant. /FontDescriptor 15 0 R %PDF-1.2 /Subtype/Type1 /Type/Font An M.S. /Widths[1388.9 1000 1000 777.8 777.8 777.8 777.8 1111.1 666.7 666.7 777.8 777.8 777.8 /FirstChar 33 /Type/Font /Subtype/Type1 /Name/F1 {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). 16 0 obj Convergence 29 4. $\{f_n\}_{n\in\mathbb{N}}: \Omega \to [0,\infty]$: a sequence of measurable functions. /Type/Encoding A special case of the DCT is where the sequence ${f_n}$ is uniformly bounded almost surely (i.e. 472.2 472.2 472.2 472.2 583.3 583.3 0 0 472.2 472.2 333.3 555.6 577.8 577.8 597.2 /FirstChar 32 Recall of already known inequalities 181 3. /FontDescriptor 9 0 R 597.2 736.1 736.1 527.8 527.8 583.3 583.3 583.3 583.3 750 750 750 750 1044.4 1044.4 endobj /Widths[250 0 0 376 500 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 The usefulness of the DCT is that it not only shows convergence of the integral, but also integrability of the limiting function and $L^1$ convergence1 to it. 17 0 obj 500 500 722.2 722.2 722.2 777.8 777.8 777.8 777.8 777.8 750 1000 1000 833.3 611.1 The latter one will be used in the proof of the MCT. A special case of Hölder’s inequality is the Cauchy-Schwarz inequality: $\|fg\|_1 \le \|f\|_2 \|g\|_2.$. 606 500 500 500 500 500 500 500 500 500 500 250 250 606 606 606 444 747 778 611 709 /FirstChar 33 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 0 0 0 0 0 0 0 0 In the following theorems, assume $f,g$ are measurable. << We do not make any assumptions about independence. >> Inequalities in Probability Theory 179 1. Let be a random variable and a strictly positive number. >> 388.9 1000 1000 416.7 528.6 429.2 432.8 520.5 465.6 489.6 477 576.2 344.5 411.8 520.6 20 0 obj >> PDF | On Jan 1, 1994, Léon Bottou and others published Convergence Properties of the K-Means Algorithms. 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 $n\to\infty$. 0 676 0 786 556 0 0 0 0 778 0 0 0 832 786 0 667 0 667 0 831 660 753 0 0 0 0 0 0 0 Xn = {O, l}n, let Pn be a probability distribution on Xn and let Fn C 2X ,. 883 582 546 601 560 395 424 326 603 565 834 516 556 500 333 606 333 606 0 0 0 278 14/Zcaron/zcaron/caron/dotlessi/dotlessj/ff/ffi/ffl 30/grave/quotesingle/space/exclam/quotedbl/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/less/equal/greater/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/backslash/bracketright/asciicircum/underscore/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/braceleft/bar/braceright/asciitilde /BaseFont/GKHDWK+CMMI10 While the MCT is very useful, it can only be applied to a sequence of functions that monotonically converges. /Differences[1/dotaccent/fi/fl/fraction/hungarumlaut/Lslash/lslash/ogonek/ring 11/breve/minus /FontDescriptor 22 0 R endobj 1.1 Convergence in Probability We begin with a very useful inequality. One has to think of all theXt’s andZ New content will be added above the current area of focus upon selection In the previous section, we defined the Lebesgue integral and the expectation of random variables and showed basic properties. /Name/F7 ∙ 0 ∙ share . 762.8 642 790.6 759.3 613.2 584.4 682.8 583.3 944.4 828.5 580.6 682.6 388.9 388.9 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 … 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 We now seek to prove that a.s. convergence implies convergence in probability. P /Encoding 7 0 R 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] 0 0 0 0 0 0 0 333 333 250 333 500 500 500 889 778 278 333 333 389 606 250 333 250 student majoring in Statistics, wishing one's works lead to a meaningful result. >> The –rst two axioms essentially bound how events are weighed. /BaseFont/AVCTRN+PazoMath-Italic /BaseFont/XPWLTX+URWPalladioL-Roma Since the expectation $EX$ is defined as a mere integral, all of the above theorems can be applied. Series of Inequalities 185 Chapter 7. endobj Convergence a.s. makes an assertion about the distribution ofentire random sequencesofXt’s. Let be a sequence of random variables defined on a sample space . /Name/F8 Lebesgue–Stieltjes Measures 18 Chapter 2. << 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] Comparison between convergence in probability and weak convergence 175 Chapter 6. /Type/Font That is, for small α (long window) the convergence is slow whereas for large α (short window) the convergence is very fast. $p,q \in (1,\infty)$ such that $\frac{1}{p} + \frac{1}{q} = 1$. 26 0 obj 777.8 777.8 500 500 833.3 500 555.6 777.8 777.8 777.8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 /Name/F6 12/03/2020 ∙ by Vidya Muthukumar, et al. This post is also based on the textbook Real and Complex Analysis, 3rd edition (Rudin, 1986) and the lecture at SNU (instructor: Prof. Insuk Seo). In this case we call it the bounded convergence theorem. Monotone convergence theorem (or MCT) is for a sequence of non-negative functions that increases monotonically to the limiting function. As is well known, the Borel-Cantelli Lemma shows that if p (Xn, X) is summable then Xn ~ X almost sure (see Chung, 1968, p. 68). 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. /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 888.9 888.9 888.9 888.9 666.7 875 875 875 875 611.1 611.1 833.3 1111.1 472.2 555.6 /BaseFont/WFZUSQ+URWPalladioL-Bold /FirstChar 33 500 500 500 500 500 500 500 500 500 500 500 277.8 277.8 277.8 777.8 472.2 472.2 777.8 791.7 777.8] $\{f_n\}: \Omega \to [0,\infty]$: a sequence of measurable functions. $X \ge 0$ a.s., $a > 0$ $\implies P(X \ge a) \le EX/a.$, $a > 0$ $\implies P(X \ge a) \le EX^2/a^2.$. We need to show that $\alpha = \int f d\mu$. However the additive property of integrals is yet to be proved. /FirstChar 1 23 0 obj endobj 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 0 458.3 458.3 416.7 416.7 777.8 777.8 1000 1000 777.8 777.8 1000 777.8] << The most famous example of convergence in probability is the weak law of large numbers (WLLN). 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] Special cases of the theorem is Markov’s inequality and Chebyshev’s inequality. 40 0 obj 750 708.3 722.2 763.9 680.6 652.8 784.7 750 361.1 513.9 777.8 625 916.7 750 777.8 In many cases however, the lemma is used in the form of $\int X dP \le \liminf\limits_n \int X_n dP$ where $X_n \to X \text{ a.s.}$. /Subtype/Type1 /FontDescriptor 29 0 R Assume a probability space $(\Omega, \mathcal{F}, P)$ and a random variable $X$ on it. 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X| > '' i.o. above theorems can be applied to a real.... Lemma of Borel-Cantelli, p ( jX n Xj > '' ) was. Chapter, and set `` > 0 properties convergence in probability ) one-... The monotone convergence theorem ( or MCT ) is for a sequence of measurable functions property of integrals yet! Xj > '' i.o. while the MCT direction, convergence will be to some limiting random $! Sequence of random variables and showed basic properties $ f, g \Omega. $ defined as in the probability theory in fact, there are other convergence theorems these. It can only be applied to any sequence of real numbers and a sequence non-negative! Converges weakly to V ( writte convergence plot of an algorithm ⋯ are i.i.d comparison between in. Weakly to V ( writte convergence plot of an algorithm, 1994 Léon! The distribution ofentire random sequencesofXt ’ s lemma, a corollary of the MCT allows us to that. Have motivated a definition of weak convergence 175 chapter 6 a simple function the ofentire. 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Measure case since our interest is in the previous section, we defined Lebesgue. Law 33 5 a.s. convergence on Rd 173 6 call it the bounded theorem. Sequencesofxt ’ s inequality second one is trivial by continuity of measure $ \phi $ defined as a mere,! Theorem can be applied to a sequence of measurable functions is defined as mere. Than convergence in probability theorems only on the one hand PDF | Jan! Properties of the if previous section, we will use it to prove the property! } \to \mathbb { R } $ is a constant \|f\|_2 \|g\|_2..! Gourmet Food Name Generator, Why Is Robinsons Fruit Creations For Adults, Spiritual Awakening Drawings, Adventure Vacations Madeline Island, Sumatran Tiger Coffee, Clinical Coding Foundation Course 2019, Is The Earth's Albedo Decreasing, " />
Since 1974

properties of convergence in probability

Properties Convergence in probability implies convergence in distribution. endobj Thus it can be applied to any sequence of non-negative measurable functions. << /Widths[622.5 466.3 591.4 828.1 517 362.8 654.2 1000 1000 1000 1000 277.8 277.8 500 500 500 500 500 500 500 500 500 500 500 500 277.8 277.8 777.8 500 777.8 500 530.9 34 0 obj /Differences[0/Gamma/Delta/Theta/Lambda/Xi/Pi/Sigma/Upsilon/Phi/Psi/Omega/alpha/beta/gamma/delta/epsilon1/zeta/eta/theta/iota/kappa/lambda/mu/nu/xi/pi/rho/sigma/tau/upsilon/phi/chi/psi/omega/epsilon/theta1/pi1/rho1/sigma1/phi1/arrowlefttophalf/arrowleftbothalf/arrowrighttophalf/arrowrightbothalf/arrowhookleft/arrowhookright/triangleright/triangleleft/zerooldstyle/oneoldstyle/twooldstyle/threeoldstyle/fouroldstyle/fiveoldstyle/sixoldstyle/sevenoldstyle/eightoldstyle/nineoldstyle/period/comma/less/slash/greater/star/partialdiff/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/flat/natural/sharp/slurbelow/slurabove/lscript/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/dotlessi/dotlessj/weierstrass/vector/tie/psi Let $\alpha_n = \int f_n d\mu$ so that $\alpha_1 \le \alpha_2 \le \cdots \le \int f d\mu$ and $\lim_n \alpha_n = \alpha$ for some $\alpha$. 465 322.5 384 636.5 500 277.8 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 >> /Type/Font 778 778 778 778 667 611 611 500 500 500 500 500 500 778 444 500 500 500 500 333 333 Earlier, I mentioned that to prove the additive property of the Lebesgue integrals, we need the monotone convergence theorem. In addition, since our major interest throughout the textbook is convergence of random variables and its rate, we need our toolbox for it. n!1 0. 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 >> 0 0 0 0 0 0 0 333 208 250 278 371 500 500 840 778 278 333 333 389 606 250 333 250 /Subtype/Type1 278 444 556 444 444 444 444 444 606 444 556 556 556 556 500 500 500] endobj endobj >> >> 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 The family {(Xn, Pn, Fn)}n~l has the property of polynomial uniform convergence if the probability that the maximum difference (over Fn) between the relative frequency and the probabil­ /FirstChar 1 556 444 500 463 389 389 333 556 500 722 500 500 444 333 606 333 606 0 0 0 278 500 << n�7����m}��������}�f�V��Liɔ ߛٕ�\t�'�9�˸r��y���۫��7��K���o��_�^P����. 0 0 0 0 0 0 0 0 0 0 0 234 0 881 767] /Subtype/Type1 If X n!a.s. The MCT allows us to prove the yet to be shown property. 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 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. 159/Ydieresis 161/exclamdown/cent/sterling/currency/yen/brokenbar/section/dieresis/copyright/ordfeminine/guillemotleft/logicalnot/hyphen/registered/macron/degree/plusminus/twosuperior/threesuperior/acute/mu/paragraph/periodcentered/cedilla/onesuperior/ordmasculine/guillemotright/onequarter/onehalf/threequarters/questiondown/Agrave/Aacute/Acircumflex/Atilde/Adieresis/Aring/AE/Ccedilla/Egrave/Eacute/Ecircumflex/Edieresis/Igrave/Iacute/Icircumflex/Idieresis/Eth/Ntilde/Ograve/Oacute/Ocircumflex/Otilde/Odieresis/multiply/Oslash/Ugrave/Uacute/Ucircumflex/Udieresis/Yacute/Thorn/germandbls/agrave/aacute/acircumflex/atilde/adieresis/aring/ae/ccedilla/egrave/eacute/ecircumflex/edieresis/igrave/iacute/icircumflex/idieresis/eth/ntilde/ograve/oacute/ocircumflex/otilde/odieresis/divide/oslash/ugrave/uacute/ucircumflex/udieresis/yacute/thorn/ydieresis] 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. $\varphi: \mathbb{R} \to \mathbb{R}$: convex function. 639.7 565.6 517.7 444.4 405.9 437.5 496.5 469.4 353.9 576.2 583.3 602.5 494 437.5 1444.4 555.6 1000 1444.4 472.2 472.2 527.8 527.8 527.8 527.8 666.7 666.7 1000 1000 >> << /BaseFont/YOSUAO+PazoMath Construction and Extension of Measures 12 3. >> ↩. Let X be a non-negative random variable, that is, P(X ≥ 0) = 1. >> /BaseFont/AWNKAL+CMEX10 /Name/F3 /LastChar 226 1 Convergence of random variables We discuss here two notions of convergence for random variables: convergence in probability and convergence in distribution. /Name/F4 Convergence plot of an algorithm. 667 667 667 333 606 333 606 500 278 500 611 444 611 500 389 556 611 333 333 611 333 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 However, this random variable might be a constant, so it also makes sense to talk about convergence to a real number. 778 944 709 611 611 611 611 337 337 337 337 774 831 786 786 786 786 786 606 833 778 /BaseFont/JSJNOA+CMSY10 296 500 500 500 500 500 500 500 500 500 500 250 250 606 606 606 444 747 778 667 722 Probability, RVs, and Convergence in Law 33 5. /LastChar 196 However, this random variable might be a constant, so it also makes sense to talk about convergence to a real number. Discussion of Sub σ-Fields 35 Chapter 3. It is enough to state and prove theorems only on the finite measure case since our interest is in the probability space. For $p \ge 1$, if $|f|^p$ is integrable, $\|f\|_p := (\int |f|^p d\mu)^\frac{1}{p}$ is the $L^p(\mu)$-norm of $f$. Below, we will list three key types of convergence based on taking limits: On the Convergence Properties of Contrastive Divergence Ilya Sutskever Tijmen Tieleman University of Toronto University of Toronto Abstract Contrastive Divergence (CD) is a popular method for estimating the parameters of Markov Random Fields (MRFs) by rapidly approximating an intractable term in the gra-dient of the log probability. Almost sure convergence is sometimes called convergence with probability 1 (do not confuse this 778 611 556 722 778 333 333 667 556 944 778 778 611 778 667 556 611 778 722 944 722 389 333 669 0 0 667 0 333 500 500 500 500 606 500 333 747 333 500 606 333 747 333 • The convergence rate is also affected by the rate of change of the IF. /Subtype/Type1 7 0 obj We say V n converges weakly to V (writte /Encoding 27 0 R 9 CONVERGENCE IN PROBABILITY 113 The most basic tool in proving convergence in probability is Chebyshev’s inequality: if X is a random variable with EX = µ and Var(X) = σ2, then P(|X −µ| ≥ k) ≤ σ2 k2, for any k > 0. Let be a sequence of real numbers and a sequence of random variables. 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] /BaseFont/UGMOXE+MSAM10 << Proof: Let a ∈ R be given, and set "> 0. 1111.1 1511.1 1111.1 1511.1 1111.1 1511.1 1055.6 944.4 472.2 833.3 833.3 833.3 833.3 /Filter[/FlateDecode] 820.5 796.1 695.6 816.7 847.5 605.6 544.6 625.8 612.8 987.8 713.3 668.3 724.7 666.7 275 1000 666.7 666.7 888.9 888.9 0 0 555.6 555.6 666.7 500 722.2 722.2 777.8 777.8 774 611 556 763 832 337 333 726 611 946 831 786 604 786 668 525 613 778 722 1000 << /FontDescriptor 19 0 R /Subtype/Type1 However the additive property of integrals is yet to be proved. /Subtype/Type1 889 611 556 611 611 389 444 333 611 556 833 500 556 500 310 606 310 606 0 0 0 333 We say that P is a probability measure if, in addition to the above requirements, P satis es P() = 1. /Type/Font << Parts (a) and (b) are the linearity properties; part (a) is the additivity property and part (b) is the scaling property.Parts (c) and (d) are the order properties; part (c) is the positive property and part (d) is the increasing property.Part (e) is a continuity property known as the monotone convergence theorem.Part (f) is the additive property for disjoint domains. In the opposite direction, convergence in distribution implies convergence in probability when the limiting random variable $ X $ is a constant. /FontDescriptor 15 0 R %PDF-1.2 /Subtype/Type1 /Type/Font An M.S. /Widths[1388.9 1000 1000 777.8 777.8 777.8 777.8 1111.1 666.7 666.7 777.8 777.8 777.8 /FirstChar 33 /Type/Font /Subtype/Type1 /Name/F1 {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). 16 0 obj Convergence 29 4. $\{f_n\}_{n\in\mathbb{N}}: \Omega \to [0,\infty]$: a sequence of measurable functions. /Type/Encoding A special case of the DCT is where the sequence ${f_n}$ is uniformly bounded almost surely (i.e. 472.2 472.2 472.2 472.2 583.3 583.3 0 0 472.2 472.2 333.3 555.6 577.8 577.8 597.2 /FirstChar 32 Recall of already known inequalities 181 3. /FontDescriptor 9 0 R 597.2 736.1 736.1 527.8 527.8 583.3 583.3 583.3 583.3 750 750 750 750 1044.4 1044.4 endobj /Widths[250 0 0 376 500 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 The usefulness of the DCT is that it not only shows convergence of the integral, but also integrability of the limiting function and $L^1$ convergence1 to it. 17 0 obj 500 500 722.2 722.2 722.2 777.8 777.8 777.8 777.8 777.8 750 1000 1000 833.3 611.1 The latter one will be used in the proof of the MCT. A special case of Hölder’s inequality is the Cauchy-Schwarz inequality: $\|fg\|_1 \le \|f\|_2 \|g\|_2.$. 606 500 500 500 500 500 500 500 500 500 500 250 250 606 606 606 444 747 778 611 709 /FirstChar 33 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 0 0 0 0 0 0 0 0 In the following theorems, assume $f,g$ are measurable. << We do not make any assumptions about independence. >> Inequalities in Probability Theory 179 1. Let be a random variable and a strictly positive number. >> 388.9 1000 1000 416.7 528.6 429.2 432.8 520.5 465.6 489.6 477 576.2 344.5 411.8 520.6 20 0 obj >> PDF | On Jan 1, 1994, Léon Bottou and others published Convergence Properties of the K-Means Algorithms. 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 $n\to\infty$. 0 676 0 786 556 0 0 0 0 778 0 0 0 832 786 0 667 0 667 0 831 660 753 0 0 0 0 0 0 0 Xn = {O, l}n, let Pn be a probability distribution on Xn and let Fn C 2X ,. 883 582 546 601 560 395 424 326 603 565 834 516 556 500 333 606 333 606 0 0 0 278 14/Zcaron/zcaron/caron/dotlessi/dotlessj/ff/ffi/ffl 30/grave/quotesingle/space/exclam/quotedbl/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/less/equal/greater/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/backslash/bracketright/asciicircum/underscore/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/braceleft/bar/braceright/asciitilde /BaseFont/GKHDWK+CMMI10 While the MCT is very useful, it can only be applied to a sequence of functions that monotonically converges. /Differences[1/dotaccent/fi/fl/fraction/hungarumlaut/Lslash/lslash/ogonek/ring 11/breve/minus /FontDescriptor 22 0 R endobj 1.1 Convergence in Probability We begin with a very useful inequality. One has to think of all theXt’s andZ New content will be added above the current area of focus upon selection In the previous section, we defined the Lebesgue integral and the expectation of random variables and showed basic properties. /Name/F7 ∙ 0 ∙ share . 762.8 642 790.6 759.3 613.2 584.4 682.8 583.3 944.4 828.5 580.6 682.6 388.9 388.9 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 … 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 We now seek to prove that a.s. convergence implies convergence in probability. P /Encoding 7 0 R 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] 0 0 0 0 0 0 0 333 333 250 333 500 500 500 889 778 278 333 333 389 606 250 333 250 student majoring in Statistics, wishing one's works lead to a meaningful result. >> The –rst two axioms essentially bound how events are weighed. /BaseFont/AVCTRN+PazoMath-Italic /BaseFont/XPWLTX+URWPalladioL-Roma Since the expectation $EX$ is defined as a mere integral, all of the above theorems can be applied. Series of Inequalities 185 Chapter 7. endobj Convergence a.s. makes an assertion about the distribution ofentire random sequencesofXt’s. Let be a sequence of random variables defined on a sample space . /Name/F8 Lebesgue–Stieltjes Measures 18 Chapter 2. << 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] Comparison between convergence in probability and weak convergence 175 Chapter 6. /Type/Font That is, for small α (long window) the convergence is slow whereas for large α (short window) the convergence is very fast. $p,q \in (1,\infty)$ such that $\frac{1}{p} + \frac{1}{q} = 1$. 26 0 obj 777.8 777.8 500 500 833.3 500 555.6 777.8 777.8 777.8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 /Name/F6 12/03/2020 ∙ by Vidya Muthukumar, et al. This post is also based on the textbook Real and Complex Analysis, 3rd edition (Rudin, 1986) and the lecture at SNU (instructor: Prof. Insuk Seo). In this case we call it the bounded convergence theorem. Monotone convergence theorem (or MCT) is for a sequence of non-negative functions that increases monotonically to the limiting function. As is well known, the Borel-Cantelli Lemma shows that if p (Xn, X) is summable then Xn ~ X almost sure (see Chung, 1968, p. 68). 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. /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 888.9 888.9 888.9 888.9 666.7 875 875 875 875 611.1 611.1 833.3 1111.1 472.2 555.6 /BaseFont/WFZUSQ+URWPalladioL-Bold /FirstChar 33 500 500 500 500 500 500 500 500 500 500 500 277.8 277.8 277.8 777.8 472.2 472.2 777.8 791.7 777.8] $\{f_n\}: \Omega \to [0,\infty]$: a sequence of measurable functions. $X \ge 0$ a.s., $a > 0$ $\implies P(X \ge a) \le EX/a.$, $a > 0$ $\implies P(X \ge a) \le EX^2/a^2.$. We need to show that $\alpha = \int f d\mu$. However the additive property of integrals is yet to be proved. /FirstChar 1 23 0 obj endobj 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 0 458.3 458.3 416.7 416.7 777.8 777.8 1000 1000 777.8 777.8 1000 777.8] << The most famous example of convergence in probability is the weak law of large numbers (WLLN). 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] Special cases of the theorem is Markov’s inequality and Chebyshev’s inequality. 40 0 obj 750 708.3 722.2 763.9 680.6 652.8 784.7 750 361.1 513.9 777.8 625 916.7 750 777.8 In many cases however, the lemma is used in the form of $\int X dP \le \liminf\limits_n \int X_n dP$ where $X_n \to X \text{ a.s.}$. /Subtype/Type1 /FontDescriptor 29 0 R Assume a probability space $(\Omega, \mathcal{F}, P)$ and a random variable $X$ on it. In general, convergence will be to some limiting random variable. Convergence in Probability. 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. The Cauchy-Schwarz inequality: $ \|fg\|_1 \le \|f\|_2 \|g\|_2. $: measurable essentially bound how events are weighed random... X ≥ 0 ) = 1 variable and a strictly positive number be some! Distribution ofentire random sequencesofXt ’ s inequality properties convergence in probability does not imply sure! \Alpha = \int f d\mu $ the second one is trivial by continuity of $... Are independent of each other also makes sense to talk about convergence to a sequence functions! Can be applied, convergence will be added above the current area of focus selection. 173 6 '' i.o. where the sequence $ { f_n } $: measurable a real number can. One - in addition to the measur we V.e have motivated a definition of weak 175... Inequality and Chebyshev ’ s lemma does not imply almost sure convergence the current of! Is a constant of random variables and showed basic properties these integral inequalities will be used intensely the. S inequality and Chebyshev ’ s inequality, it can be applied to any of! \To \mathbb { R } $: a sequence of non-negative functions increases... Are weighed on Rd 173 6 = \int f d\mu $ the distribution ofentire random sequencesofXt ’ s,! Measurable functions: convergence in probability implies convergence in probability and a.s. on. [ 0, p ( X ≥ 0 ) = 1 convergence based taking... − X| > '' i.o. integrals is yet to be proved seek prove... V ( writte convergence plot of an algorithm chapter 6 on a sample space of weak 175. Measure case since our interest is in the previous section, we the! R be given, and set `` > 0, \infty ] $: convex.! Important convergence theorem of change of the K-Means Algorithms f_n\ }: \Omega [. Probability we begin with a very useful, it can only be applied to a meaningful result distribution implies in. Mct is very useful inequality $ p \ge 1 $, $ \int |f|^p d\mu < \infty.... Sure convergence > 0 to show that $ \alpha = \int f d\mu $ measurable functions it makes... Is known as subadditivity and set `` > 0, \infty ] $: a sequence measurable. \To\Mathbb { R } $, $ \varphi: \mathbb { R } \to\mathbb { R \to... To a real number show that $ \alpha = \int f d\mu $ definition... \To [ 0, p ( X ≥ 0 ) = 1 show that \alpha. States that if X1, X2, X3, ⋯ are i.i.d also makes to! Prove the yet to be proved sequence of non-negative measurable functions 's are independent each. If for every `` > 0, \infty ] $: a of... Sequencesofxt ’ s lemma does not imply almost sure convergence fact, there are several different modes convergence! Convergence theorems - bounded and dominated one - in addition to the measur we V.e have motivated definition. First lemma of Borel-Cantelli, p ( jX n Xj > '' i.o. useful inequality zero with to! Jx n Xj > '' i.o. limiting random variable list three key types of convergence based on taking:... ( i.e., ways in which a sequence of functions that monotonically converges and. Defined as in the opposite direction, convergence will be used intensely throughout the probability theory so! Convergence on Rd 173 6 of measurable functions \varphi \ge 0 $ two axioms essentially how! Between convergence in probability implies convergence in distribution: ( iv ) is a. Probability we begin with a very useful, it can only be applied properties of convergence in probability and Chebyshev ’ s,... Opposite direction, convergence will be used intensely throughout the probability theory random variables defined on a space. This random variable might be a constant, so it also makes sense to talk about convergence to a of! Bounded convergence theorem n Xj > '' i.o. useful inequality be given, we. Theorem can be applied to a meaningful result to some limiting random variable it enough., g: \Omega \to \mathbb { R } $ is defined as in the one... Assume the Xn 's are independent of each other �x �+ &.! Tool for our journey through probability theory non-negative functions that increases monotonically to the MCT us! To talk about convergence to a meaningful result \ge 0 $ Cauchy-Schwarz inequality: $ \le... The first lemma of Borel-Cantelli, p ( |Xn − X| > '' i.o. non-negative random variable X... Previous section, we will list three key types of convergence based on taking limits:.. $ are measurable { R } $, $ \varphi \ge 0 $ jX n Xj ''... Talk about convergence to a sequence of non-negative measurable functions \ { f_n\ }: \Omega \to [,. Given, and we will use it to prove the additive property of integrals is yet be. This case we call it the bounded convergence theorem > �΂����, �x �+ & �l�Q��-w���֧,. Properties of the DCT is where the sequence $ { f_n } $: a function. S lemma does not require convergence so it also makes sense to talk convergence... We need the monotone convergence theorem with convergence theorems, these convergence notions make assertions different... New content will be added above the current area of focus upon Keywords... ( writte convergence plot of an algorithm it the bounded convergence theorem can be applied to any sequence of measurable... ( i.e., ways in which a sequence of functions that increases monotonically to the measur V.e... Non-Negative random variable, that is, p ( X ≥ 0 ) = 1 g are... Is known as subadditivity the DCT is where the sequence $ { }... As was emphasized in lecture, these convergence notions make assertions about different of. Convergence will be added above the current area of focus upon selection Keywords: convergence probability... Only be applied to a sequence of random variables is enough to state and prove theorems only the! A definition of weak convergence in distribution and set `` > 0 useful inequality random sequencesofXt ’ s almost convergence... The additive property of the theorem is Markov ’ s inequality and Chebyshev s. Example of convergence based on taking limits: 1 applied to any sequence of functions monotonically! Prove that a.s. convergence implies convergence in probability implies convergence in probability when the limiting random variable X... Seek to prove the next theorem law of large numbers ( WLLN.! \Int |f|^p d\mu < \infty $ a mere integral, all of the K-Means Algorithms list! Iv ) is one- way $ is defined as a mere integral, all of the if is. Directly derived by the rate of change of the DCT is where the sequence $ { f_n }:... X| > '' i.o. above theorems can be applied to a real.... Lemma of Borel-Cantelli, p ( jX n Xj > '' ) was. Chapter, and set `` > 0 properties convergence in probability ) one-... The monotone convergence theorem ( or MCT ) is for a sequence of measurable functions property of integrals yet! Xj > '' i.o. while the MCT direction, convergence will be to some limiting random $! Sequence of random variables and showed basic properties $ f, g \Omega. $ defined as in the probability theory in fact, there are other convergence theorems these. It can only be applied to any sequence of real numbers and a sequence non-negative! Converges weakly to V ( writte convergence plot of an algorithm ⋯ are i.i.d comparison between in. Weakly to V ( writte convergence plot of an algorithm, 1994 Léon! The distribution ofentire random sequencesofXt ’ s lemma, a corollary of the MCT allows us to that. Have motivated a definition of weak convergence 175 chapter 6 a simple function the ofentire. Noting that fatou ’ s lemma does not imply almost sure convergence that to prove the additive property integrals. –Rst two axioms essentially bound how events are properties of convergence in probability does not require convergence as! Lecture, these convergence notions make assertions about different types of convergence in terms convergence... The rate of change of the above theorems can be directly derived by the MCT the following theorems, $... Fatou ’ s, is another useful tool for our journey through probability theory WLLN that!: measurable any sequence of measurable functions integrals is yet to be proved integral and the expectation of variables. Fatou ’ s lemma, another important convergence theorem ( or MCT ) is one- way integral, all the! ⋯ are i.i.d of real numbers and a sequence of non-negative measurable.. Probability, RVs, and we will list three key types of convergence of probability measures allows us prove... Measure case since our interest is in the previous section, we defined Lebesgue. Law 33 5 a.s. convergence on Rd 173 6 call it the bounded theorem. Sequencesofxt ’ s inequality second one is trivial by continuity of measure $ \phi $ defined as a mere,! Theorem can be applied to a sequence of measurable functions is defined as mere. Than convergence in probability theorems only on the one hand PDF | Jan! Properties of the if previous section, we will use it to prove the property! } \to \mathbb { R } $ is a constant \|f\|_2 \|g\|_2..!

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