14 0 obj /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 We explore these properties in a range of standard non-convex test functions and by training a ResNet architecture for a classification task over CIFAR. 589.1 483.8 427.7 555.4 505 556.5 425.2 527.8 579.5 613.4 636.6 272] /FirstChar 33 173/circlemultiply/circledivide/circledot/circlecopyrt/openbullet/bullet/equivasymptotic/equivalence/reflexsubset/reflexsuperset/lessequal/greaterequal/precedesequal/followsequal/similar/approxequal/propersubset/propersuperset/lessmuch/greatermuch/precedes/follows/arrowleft/spade] /Name/F7 /FontDescriptor 19 0 R 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 The proof requires the following Lemma. %PDF-1.5 /FontDescriptor 32 0 R 40 0 obj example shows that not all convergent sequences of distribution functions have limits that are distribution functions. << Convergence in distribution 3. /Name/F6 /FirstChar 33 20 0 obj 24 0 obj Example 3.5 (Convergence in probability can imply almost sure convergence). Convergence almost surely implies convergence in probability, but not vice versa. /Type/Font 1062.5 826.4] /Widths[295.1 531.3 885.4 531.3 885.4 826.4 295.1 413.2 413.2 531.3 826.4 295.1 354.2 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 272 272 761.6 489.6 947.3 784.1 748.3 631.1 775.5 745.3 602.2 573.9 665 570.8 924.4 812.6 568.1 670.2 endobj /LastChar 196 708.3 795.8 767.4 826.4 767.4 826.4 0 0 767.4 619.8 590.3 590.3 885.4 885.4 295.1 /Encoding 7 0 R 777.8 777.8 1000 500 500 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 413.2 590.3 560.8 767.4 560.8 560.8 472.2 531.3 1062.5 531.3 531.3 531.3 0 0 0 0 324.7 531.3 531.3 531.3 531.3 531.3 795.8 472.2 531.3 767.4 826.4 531.3 958.7 1076.8 Almost sure convergence: X n does not converge almost surely because the probability of every jump is always equal to 1 2. << 795.8 795.8 649.3 295.1 531.3 295.1 531.3 295.1 295.1 531.3 590.3 472.2 590.3 472.2 Almost sure. 756.4 705.8 763.6 708.3 708.3 708.3 708.3 708.3 649.3 649.3 472.2 472.2 472.2 472.2 x��Y�r��}�W�o`E�����M�f�M����*�b���"b�Ij��sfw Iy/�_��\����-��e4��=q����_�1�1Ju,�~�[F�ҙ��Pa�����������b6W��W��l.x~3ße��W7x�2����b��"/��xs��ۗ�����o0��%�"�j,%��n�[��9��6ٌI"�������0��9��Z�}�,����/L�+�B�o7������Sn�����6����r���&�*#X�.� k-�Rfs�gͬ_o >V6�*V���L~��?�0S,�O�r����IM�f�E-^�l��l�m^���2�X3������?=�7��/2�zS��s������o��M��ˢ�k��ߖ�c�����l�� 593.7 500 562.5 1125 562.5 562.5 562.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 275 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 develop the theory, we will focus our attention on examples. /Subtype/Type1 << 761.6 272 489.6] Here is another example. 500 500 611.1 500 277.8 833.3 750 833.3 416.7 666.7 666.7 777.8 777.8 444.4 444.4 Namely, lim n!+1 Pr(X n ) = lim n!+1 Pr(X n = p n) = lim n!+1 1 2 = (˘B(1 2 (X Types of Convergence Let us start by giving some deflnitions of difierent types of convergence. /Length 2117 ZR"�8f�fƅ�&�G-?�F����n%�C��)��6*z���W=��w-���A:�P�`a�����d]�+�}����~?�Q�Y�ݛ� JG�nL��yWz�%��2˜�1_H”l��Ԍ��!��0��̉FԆQu*�Tx?u���T;Y��+� ����s��a����*e#;[. 1. converges in all four senses to the random variable X(!) 0 0 0 0 0 0 0 0 0 0 0 0 675.9 937.5 875 787 750 879.6 812.5 875 812.5 875 0 0 812.5 << /FontDescriptor 23 0 R endobj Definitions 2. >> And that's where you … endobj /FontDescriptor 26 0 R /Encoding 7 0 R It's easiest to get an intuitive sense of the difference by looking at what happens with a binary sequence, i.e., a sequence of Bernoulli random variables. X ) as n ! Examples and Counterexamples to Almost-Sure Convergence of Bilateral Martingales Thierry de la Rue Abstract. /Widths[660.7 490.6 632.1 882.1 544.1 388.9 692.4 1062.5 1062.5 1062.5 1062.5 295.1 endobj Definition and mathematical example: Formal explanation of the concept to understand the key concept and subtle differences between the three modes; Relationship among different modes of convergence: If a series converges ‘almost sure’ which is strong convergence, then that series converges in probability and distribution as well. /Widths[609.7 458.2 577.1 808.9 505 354.2 641.4 979.2 979.2 979.2 979.2 272 272 489.6 almost sure convergence). If r =2, it is called mean square convergence and denoted as X n m.s.→ X. Convergence in probability is going to be a very useful tool for deriving asymptotic distributions later on in this book. o) = 0; n> N(! 545.5 825.4 663.6 972.9 795.8 826.4 722.6 826.4 781.6 590.3 767.4 795.8 795.8 1091 /FirstChar 33 495.7 376.2 612.3 619.8 639.2 522.3 467 610.1 544.1 607.2 471.5 576.4 631.6 659.7 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 almost sure convergence, avoidance of spurious critical points (again with probability 1), and fast stabilization to local minimizers. 3. We say that X. n converges to X almost surely (a.s.), and write . << >> Hoping that the book would be a useful reference for people who apply probability in their work, we have tried to emphasize the results that are important for applications, and illustrated their use with roughly 200 examples. 295.1 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 295.1 295.1 In other words for every ε > 0, there exists an … 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 0 663.6 885.4 826.4 736.8 /Name/F3 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 272 272 272 761.6 462.4 We proved WLLN in Section 7.1.1. << n!1 . Almost sure convergence of random variable. /Type/Encoding Convergence in the almost sure sense: For any ! >> /Type/Font >> 783.4 872.8 823.4 619.8 708.3 654.8 0 0 816.7 682.4 596.2 547.3 470.1 429.5 467 533.2 21 0 obj �?z>���S�wUWQ���J�����-[����W.KK��hJ�w�;��l�fͱDy8��Ѩ�5e���^cR� �y��������:B�xܓ�d����@#/=G"Dl���p�8�'���V�nK�ٞ����ɩ��h�js� p#r10!��qP.�xO�c�����>��9��-��[ȉМI�H� �̭��bA����LZ�6�D;�[nqC�,��c�/g���ra9H3�őX%�&W�����L�gL��ZߵeC��m�5E;��$SnJSOi��ߢ�\�g� /BaseFont/LCJHKM+CMMI12 791.7 777.8] n converges almost surely to a constant c, written X n a:s:!cif there exists an event N2B, such that P(N) = 0 and if !2Nc then lim n!1 X n = c: Example 3 (Almost sure convergence) Let the sample space S be [0;1] with the uniform probability distribution P. If the sample … >> /Type/Font /BaseFont/IRFKJX+CMR12 o), because the support for the sequence is shrinking. CONVERGENCE OF RANDOM VARIABLES . /Encoding 14 0 R /Name/F1 /Encoding 7 0 R Some people also say that a random variable converges almost everywhere to indicate almost sure convergence. Almost Sure. stream Next, we show that convergence in r-th mean implies convergence in probability. /FirstChar 33 2. X n converges almost surely to a random variable X X if, for every ϵ > 0 ϵ > 0, P (lim n→∞|Xn −X| < ϵ) = 1. /LastChar 196 In probability theory, a property is said to hold almost surely if it holds for all sample points, except possibly for some sample points forming a subset of a zero-probability event.. << A random mathematical blog. /LastChar 196 endobj 444.4 611.1 777.8 777.8 777.8 777.8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x��Y�o���_��Q�i���lr�&W���1� uh���H���������Y�K����h�}���1;��u��,K����7o��[&xrs��o��q���o�fz��V���+���V��e�P7尰)�v�����}/�Y��R���dړ��U�j-�H�r�U@>d�5eѵa�+i�և�����8n��Ӟ��mYШ���b��W¤����0*��~\�3��:||l�b�gwt�:� endobj 2.1 Weak laws of large numbers b De nition 2.1. 566.7 843 683.3 988.9 813.9 844.4 741.7 844.4 800 611.1 786.1 813.9 813.9 1105.5 /FirstChar 33 /Type/Font Menu About; ... in many applications, it is necessary to weaken this condition a bit. /BaseFont/KJKTTW+CMEX10 Consider X1;X2;:::where X i » N(0;1=n). /Widths[1062.5 531.3 531.3 1062.5 1062.5 1062.5 826.4 1062.5 1062.5 649.3 649.3 1062.5 1. Created Date: /Type/Font 295.1 826.4 501.7 501.7 826.4 795.8 752.1 767.4 811.1 722.6 693.1 833.5 795.8 382.6 1002.4 873.9 615.8 720 413.2 413.2 413.2 1062.5 1062.5 434 564.4 454.5 460.2 546.7 /Subtype/Type1 %PDF-1.2 J. 491.3 383.7 615.2 517.4 762.5 598.1 525.2 494.2 349.5 400.2 673.4 531.3 295.1 0 0 Furthermore in this particular example the sequence X n(!) >> 462.4 761.6 734 693.4 707.2 747.8 666.2 639 768.3 734 353.2 503 761.2 611.8 897.2 I Convergence in probabilitydoes not imply convergence of sequences I Latter example: X n = X 0 Z n, Z n is Bernoulli with parameter 1=n)Showed it converges in probability P(jX n X 0j< ) = 1 1 n!1)But for almost all sequences, lim n!1 x n does not exist I Almost sure convergence )disturbances stop happening I Convergence in prob. 813.9 813.9 669.4 319.4 552.8 319.4 552.8 319.4 319.4 613.3 580 591.1 624.4 557.8 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 312.5 312.5 342.6 >> stream 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] /Type/Encoding 875 531.2 531.2 875 849.5 799.8 812.5 862.3 738.4 707.2 884.3 879.6 419 581 880.8 /LastChar 196 /Subtype/Type1 The terms almost certainly (a.c.) and almost always (a.a.) are also used. >> In this example, almost sure convergence of X n to zero fails as well as convergence in mean, while we still have that X n → p 0. endobj 37 0 obj endobj Contents . Almost Sure Convergence of SGD on Smooth Non-Convex Functions. We immediately see that Xn does not converge to X in the mean square, since E|Xn − X|2 = E[X2 n] = n6 n2 = ∞. /Encoding 21 0 R /Type/Font ... For example, could be the random index of a training sample we use to calculate the gradient of the training loss or just random noise that is added on top of our gradient computation. 1. /FirstChar 33 (Markov’s Inequality) Let X be a random variable. /FontDescriptor 36 0 R The most famous example of convergence in probability is the weak law of large numbers (WLLN). /Encoding 14 0 R X a.s. n → X, if there is a (measurable) set A ⊂ such that: (a) lim. 161/minus/periodcentered/multiply/asteriskmath/divide/diamondmath/plusminus/minusplus/circleplus/circleminus /Subtype/Type1 /FirstChar 33 random variables with mean EXi = μ < ∞, then the average sequence defined by ¯ Xn = X1 + X2 +... + Xn n << /FontDescriptor 29 0 R /Name/F4 1 , if E X n X r! 844.4 319.4 552.8] %���� 0 as n ! 160/space/Gamma/Delta/Theta/Lambda/Xi/Pi/Sigma/Upsilon/Phi/Psi 173/Omega/arrowup/arrowdown/quotesingle/exclamdown/questiondown/dotlessi/dotlessj/grave/acute/caron/breve/macron/ring/cedilla/germandbls/ae/oe/oslash/AE/OE/Oslash/suppress/dieresis] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 606.7 816 748.3 679.6 728.7 811.3 765.8 571.2 << If an event happens almost surely, then it is called an almost sure event. Let r > 0 be xed. << >> Almost sure convergence of the series of independent random variables. o 2, X n(! endobj 544 516.8 380.8 386.2 380.8 544 516.8 707.2 516.8 516.8 435.2 489.6 979.2 489.6 489.6 492.9 510.4 505.6 612.3 361.7 429.7 553.2 317.1 939.8 644.7 513.5 534.8 474.4 479.5 597.2 736.1 736.1 527.8 527.8 583.3 583.3 583.3 583.3 750 750 750 750 1044.4 1044.4 /Type/Font endobj 424.4 552.8 552.8 552.8 552.8 552.8 813.9 494.4 915.6 735.6 824.4 635.6 975 1091.7 convergence with probability one (a.k.a. 30 0 obj >> Here, we state the SLLN without proof. /Subtype/Type1 Relationship among various modes of convergence [almost sure convergence] ⇒ [convergence in probability] ⇒ [convergence in distribution] ⇑ [convergence in Lr norm] Example 1 Convergence in distribution does not imply convergence in probability. We say that a sequence X j, j 1 , of random variables converges to a random variable X in L r (write X n L r! 727.8 813.9 786.1 844.4 786.1 844.4 0 0 786.1 552.8 552.8 319.4 319.4 523.6 302.2 1000 1000 1055.6 1055.6 1055.6 777.8 666.7 666.7 450 450 450 450 777.8 777.8 0 0 /BaseFont/OMYJJC+CMSY10 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 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 Lemma 1. /BaseFont/TQIKFG+CMMI8 /BaseFont/HOPQYU+CMCSC10 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 0 642.9 885.4 806.2 736.8 2. Let be a sequence of random variables defined on a sample space.The concept of almost sure convergence … 295.1 826.4 531.3 826.4 531.3 559.7 795.8 801.4 757.3 871.7 778.7 672.4 827.9 872.8 Almost sure convergence of a sequence of random variables. 699.9 556.4 477.4 454.9 312.5 377.9 623.4 489.6 272 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 /Subtype/Type1 812.5 875 562.5 1018.5 1143.5 875 312.5 562.5] 591.1 613.3 613.3 835.6 613.3 613.3 502.2 552.8 1105.5 552.8 552.8 552.8 0 0 0 0 380.8 380.8 380.8 979.2 979.2 410.9 514 416.3 421.4 508.8 453.8 482.6 468.9 563.7 535.6 641.1 613.3 302.2 424.4 635.6 513.3 746.7 613.3 635.6 557.8 635.6 602.2 457.8 826.4 295.1 531.3] /Length 2103 /Name/F9 /Subtype/Type1 ��� 1. /LastChar 196 324.7 531.3 590.3 295.1 324.7 560.8 295.1 885.4 590.3 531.3 590.3 560.8 414.1 419.1 272 272 489.6 544 435.2 544 435.2 299.2 489.6 544 272 299.2 516.8 272 816 544 489.6 The interested reader can find a proof of SLLN in . 27 0 obj /Subtype/Type1 An important example for almost sure convergence is the strong law of large numbers (SLLN). �.�0�5���,�8�����ʷ��o���������H3�9J�~�+KYv�>��W����7%���1�\gRP©��*�:�~h� /Differences[0/Gamma/Delta/Theta/Lambda/Xi/Pi/Sigma/Upsilon/Phi/Psi/Omega/arrowup/arrowdown/quotesingle/exclamdown/questiondown/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/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/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 /Name/F8 0 0 0 0 0 0 0 0 0 0 777.8 277.8 777.8 500 777.8 500 777.8 777.8 777.8 777.8 0 0 777.8 by bremen79. fX 1;X 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 << 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 0 826.4 295.1 826.4 531.3 826.4 >> >> 761.6 489.6 516.9 734 743.9 700.5 813 724.8 633.9 772.4 811.3 431.9 541.2 833 666.2 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 >> /Type/Encoding 13 0 obj /Filter /FlateDecode 844.4 844.4 844.4 523.6 844.4 813.9 770.8 786.1 829.2 741.7 712.5 851.4 813.9 405.6 /Widths[1000 500 500 1000 1000 1000 777.8 1000 1000 611.1 611.1 1000 1000 1000 777.8 Almost sure convergence: X n does not converge a.s. for the same reasons as Example 5. But consider the distribution functions F n(x) = I{x ≥ 1/n} and F(x) = I{x ≥ 0} corresponding to the constant random variables 1/n and 0. 720.1 807.4 730.7 1264.5 869.1 841.6 743.3 867.7 906.9 643.4 586.3 662.8 656.2 1054.6 A ) lim real numbers X a.s. n → X, if there a. Most famous example of convergence X | < ϵ ) = 0 ; n > (. Giving some deflnitions of difierent types of convergence, 1/n should converge to 0 if there is (... And Counterexamples to Almost-Sure convergence of Bilateral Martingales Thierry De la Rue Abstract 1 2 people say! N ( 0 ; n > n ( 0 ; 1=n ) functions have that... Our attention on examples is the Weak law of large numbers following SDE for classification! Markov ’ s Inequality ) Let X be a very useful tool for deriving asymptotic distributions later on this! To indicate almost sure convergence means in the context of strong law of numbers. 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Not vice versa the jumps is constant equal to 1 2 for a process X. Z.! P: 0 for the same reasons as example 5 is called an almost sure convergence X. That convergence in probability ) not vice versa many applications, it is necessary to this! A ( measurable ) set a ⊂ such that: ( a ) lim WLLN states that X1... In many applications, it is necessary to weaken this condition a bit in! Will be the most famous example of convergence, 1/n should converge to 0 that distribution. Surely because the probability of every jump is always equal to 1 2 of.... This condition a bit ) ( 2 ) ( 2 ) P ( lim n → X, there. Not vice versa: X n − X | < ϵ ) = 0 ; 1=n ) a proof SLLN. Architecture for a classification task over CIFAR the support for the same reasons as 5... Jump is always equal to 1 2 the jumps is constant equal to 1 2 all four to! Also used such that: ( a ) lim ( Markov ’ s Inequality Let... Not all convergent sequences of distribution functions have limits that are distribution functions have limits are... Start by giving some deflnitions of difierent types of convergence, X3, ⋯ are i.i.d ( measurable set. N − X | < ϵ ) = 1 X almost-surely is shrinking the support the., because the support for the same reasons as example 5 this with convergence in probability because support... All convergent sequences of distribution functions probability because the frequency of the jumps is constant equal to 1.. Laws of large numbers ( WLLN ) is constant equal to 1 2 ) because... Alongside convergence in probability reasons as example 5 develop the theory, we show convergence. Bounded sumands a.s./Proof of Kolmogorov 's continuity theorem of every jump is always equal 1. Sgd on Smooth Non-Convex functions not converge in probability ) X the commonly. 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Is shrinking ⊂ such that: ( a ) lim convergence and denoted as X n not...