/Widths[791.7 583.3 583.3 638.9 638.9 638.9 638.9 805.6 805.6 805.6 805.6 1277.8 160/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] 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. 30 0 obj 687.5 312.5 581 312.5 562.5 312.5 312.5 546.9 625 500 625 513.3 343.7 562.5 625 312.5 For example, consider the following SDE for a process X. where Z is a given semimartingale and are fixed real numbers. /Widths[342.6 581 937.5 562.5 937.5 875 312.5 437.5 437.5 562.5 875 312.5 375 312.5 /Name/F7 CONVERGENCE OF RANDOM VARIABLES . /FontDescriptor 9 0 R (2) (2) P ( lim n → ∞ | X n − X | < ϵ) = 1. >> 1.1 Almost sure convergence Deﬁnition 1. endobj almost sure convergence, avoidance of spurious critical points (again with probability 1), and fast stabilization to local minimizers. Proposition7.4 Almost-sure convergence does not imply mean square conver-gence. Menu About; ... in many applications, it is necessary to weaken this condition a bit. /Subtype/Type1 Created Date: 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 >> /LastChar 196 Almost sure convergence of sum implies bounded sumands a.s./Proof of Kolmogorov's continuity theorem. /BaseFont/IRFKJX+CMR12 656.2 625 625 937.5 937.5 312.5 343.7 562.5 562.5 562.5 562.5 562.5 849.5 500 574.1 << �?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/KJKTTW+CMEX10 >> /Widths[295.1 531.3 885.4 531.3 885.4 826.4 295.1 413.2 413.2 531.3 826.4 295.1 354.2 813.9 813.9 669.4 319.4 552.8 319.4 552.8 319.4 319.4 613.3 580 591.1 624.4 557.8 m�U�6}ByP��1^���������)ۮz(��ŕ �4:����+����nh�F 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 fX 1;X almost sure convergence). In other words for every ε > 0, there exists an … example shows that not all convergent sequences of distribution functions have limits that are distribution functions. o), because the support for the sequence is shrinking. endobj Example 2.8 By any sensible deﬁnition of convergence, 1/n should converge to 0. << = 0. If an event happens almost surely, then it is called an almost sure event. /Name/F2 << J. /Encoding 7 0 R 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 /FontDescriptor 26 0 R Let be a sequence of random variables defined on a sample space.The concept of almost sure convergence … 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! /BaseFont/HOPQYU+CMCSC10 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 >> 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 {Xt} converges almost sure to µ, if there exists a set M ⊂ Ω, such that P(M) = 1 and for every ω ∈ N we have Xt(ω) → µ. 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 0 683.3 902.8 844.4 755.5 /Type/Encoding /Type/Font endobj /Type/Font X a.s. n → X, if there is a (measurable) set A ⊂ such that: (a) lim. Consider X1;X2;:::where X i » N(0;1=n). 675.9 1067.1 879.6 844.9 768.5 844.9 839.1 625 782.4 864.6 849.5 1162 849.5 849.5 << /FirstChar 33 20 0 obj >> >> 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. 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 1 . 777.8 777.8 1000 500 500 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 161/minus/periodcentered/multiply/asteriskmath/divide/diamondmath/plusminus/minusplus/circleplus/circleminus >> 531.3 826.4 826.4 826.4 826.4 0 0 826.4 826.4 826.4 1062.5 531.3 531.3 826.4 826.4 544 516.8 380.8 386.2 380.8 544 516.8 707.2 516.8 516.8 435.2 489.6 979.2 489.6 489.6 endobj << 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 /Subtype/Type1 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 21 0 obj 611.1 798.5 656.8 526.5 771.4 527.8 718.7 594.9 844.5 544.5 677.8 762 689.7 1200.9 Example (Almost sure convergence) Let the sample spaceSbe the closed interval [0,1] with the uniform probability distribution. /Subtype/Type1 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 /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 2. It remains to show that Xn → X almost-surely. The most famous example of convergence in probability is the weak law of large numbers (WLLN). /BaseFont/TQIKFG+CMMI8 Here is another example. << 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 /Type/Font endobj /FontDescriptor 36 0 R Let r > 0 be xed. 777.8 777.8 1000 1000 777.8 777.8 1000 777.8] Almost Sure Convergence of SGD on Smooth Non-Convex Functions. 767.4 767.4 826.4 826.4 649.3 849.5 694.7 562.6 821.7 560.8 758.3 631 904.2 585.5 380.8 380.8 380.8 979.2 979.2 410.9 514 416.3 421.4 508.8 453.8 482.6 468.9 563.7 295.1 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 295.1 888.9 888.9 888.9 888.9 666.7 875 875 875 875 611.1 611.1 833.3 1111.1 472.2 555.6 1. 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 Motivation 5.1 | Almost sure convergence (Karr, 1993, p. 135) Almost sure convergence | or convergence with probability one | is the probabilistic version of pointwise convergence known from elementary real analysis. An important example for almost sure convergence is the strong law of large numbers (SLLN). 694.5 295.1] 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 /Subtype/Type1 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] << /Type/Encoding 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 0 458.3 458.3 416.7 416.7 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 272 272 761.6 489.6 Convergence in distribution 3. 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 /FirstChar 33 531.3 531.3 413.2 413.2 295.1 531.3 531.3 649.3 531.3 295.1 885.4 795.8 885.4 443.6 413.2 590.3 560.8 767.4 560.8 560.8 472.2 531.3 1062.5 531.3 531.3 531.3 0 0 0 0 /Name/F8 /Encoding 7 0 R So when$P(\textrm{$A_n$ infinitely often})=0$for all$\epsilon>0$you have $$Z_n\stackrel{a.s.}{\to} 0$$ Conversely, if there is an$\epsilon>0$such that$P(\textrm{$A_n$ infinitely often})\neq 0\$ then you don't have almost sure convergence. X. n 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 /Type/Font Almost sure convergence of a sequence of random variables. /FontDescriptor 16 0 R The hierarchy of convergence concepts 1 DEFINITIONS . converges in all four senses to the random variable X(!) /FontDescriptor 12 0 R /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 1. In comparison stochastic convergence: lim n → ∞ P ( | X n − X | > ϵ) = 0 ∀ ϵ > 0. m�� dI��sH�y�\��>����D�v�I�o� ry��)�_��*��"�>�\���ۗ��sms+�UAY�s��sl��羧����j͍�������X�#gt~�?v����cy�)�O�OfL�#�2G�a�EU\,�Y d��8���Gu�]נ]��*�MX����{��1�|���y��-t+�@�����nN���ސ0Pl!�!|W~a_���Z�. 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 >> 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 Convergence in probability is going to be a very useful tool for deriving asymptotic distributions later on in this book. /Encoding 14 0 R 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 We say that X. n converges to X almost surely (a.s.), and write . 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 /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 >> /Subtype/Type1 33 0 obj Furthermore in this particular example the sequence X n(!) 24 0 obj ... 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. >> /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 << >> 1062.5 1062.5 826.4 288.2 1062.5 708.3 708.3 944.5 944.5 0 0 590.3 590.3 708.3 531.3 2.1 Weak laws of large numbers b De nition 2.1. 761.6 272 489.6] A random mathematical blog. 3. 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 /FontDescriptor 19 0 R 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] 37 0 obj >> o 2, X n(! We immediately see that Xn does not converge to X in the mean square, since E|Xn − X|2 = E[X2 n] = n6 n2 = ∞. /LastChar 196 /FirstChar 33 %PDF-1.5 stream 1. Alongside convergence in distribution it will be the most commonly seen mode of convergence. 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 13 0 obj Convergence almost surely implies convergence in probability, but not vice versa. /FirstChar 33 /FontDescriptor 29 0 R Almost sure convergence example. << /Length 2103 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 312.5 312.5 342.6 593.7 500 562.5 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 /Subtype/Type1 7 0 obj 3. The notation X n a.s.→ X is often used for al- /Differences[0/minus/periodcentered/multiply/asteriskmath/divide/diamondmath/plusminus/minusplus/circleplus/circleminus/circlemultiply/circledivide/circledot/circlecopyrt/openbullet/bullet/equivasymptotic/equivalence/reflexsubset/reflexsuperset/lessequal/greaterequal/precedesequal/followsequal/similar/approxequal/propersubset/propersuperset/lessmuch/greatermuch/precedes/follows/arrowleft/arrowright/arrowup/arrowdown/arrowboth/arrownortheast/arrowsoutheast/similarequal/arrowdblleft/arrowdblright/arrowdblup/arrowdbldown/arrowdblboth/arrownorthwest/arrowsouthwest/proportional/prime/infinity/element/owner/triangle/triangleinv/negationslash/mapsto/universal/existential/logicalnot/emptyset/Rfractur/Ifractur/latticetop/perpendicular/aleph/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/union/intersection/unionmulti/logicaland/logicalor/turnstileleft/turnstileright/floorleft/floorright/ceilingleft/ceilingright/braceleft/braceright/angbracketleft/angbracketright/bar/bardbl/arrowbothv/arrowdblbothv/backslash/wreathproduct/radical/coproduct/nabla/integral/unionsq/intersectionsq/subsetsqequal/supersetsqequal/section/dagger/daggerdbl/paragraph/club/diamond/heart/spade/arrowleft /Type/Font by bremen79. 708.3 708.3 826.4 826.4 472.2 472.2 472.2 649.3 826.4 826.4 826.4 826.4 0 0 0 0 0 812.5 875 562.5 1018.5 1143.5 875 312.5 562.5] 1062.5 826.4] /LastChar 196 Convergence in probability: X n!p: 0 for the same reasons as Example 5. LARGE SAMPLE THEORY Limits and convergence concepts: almost sure, in probability and in mean Letfa n: n= 1;2;:::gbeasequenceofnon-randomrealnumbers. /Type/Font /Subtype/Type1 /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 /Widths[319.4 552.8 902.8 552.8 902.8 844.4 319.4 436.1 436.1 552.8 844.4 319.4 377.8 444.4 611.1 777.8 777.8 777.8 777.8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 833.3 1444.4 1277.8 555.6 1111.1 1111.1 1111.1 1111.1 1111.1 944.4 1277.8 555.6 1000 826.4 295.1 531.3] 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 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 … /BaseFont/KTIQCF+CMR8 /Name/F6 Almost sure convergence is sometimes called convergence with probability 1 (do not confuse this with convergence in probability). 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. 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 endobj /Subtype/Type1 /LastChar 196 Almost Sure. 39 0 obj Intuitively, X n is very concentrated around 0 for large n. But P(X n =0)= 0 for all n. The next section develops appropriate methods of discussing convergence of random variables. 2. endobj 5.2. endobj 1. endobj The WLLN states that if X1, X2, X3, ⋯ are i.i.d. 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 endobj 334 405.1 509.3 291.7 856.5 584.5 470.7 491.4 434.1 441.3 461.2 353.6 557.3 473.4 947.3 784.1 748.3 631.1 775.5 745.3 602.2 573.9 665 570.8 924.4 812.6 568.1 670.2 /Widths[609.7 458.2 577.1 808.9 505 354.2 641.4 979.2 979.2 979.2 979.2 272 272 489.6 795.8 795.8 649.3 295.1 531.3 295.1 531.3 295.1 295.1 531.3 590.3 472.2 590.3 472.2 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 o) = 0; n> N(! endobj endobj << /Encoding 21 0 R /BaseFont/LCJHKM+CMMI12 Next, we show that convergence in r-th mean implies convergence in probability. The interested reader can find a proof of SLLN in . stream /FontDescriptor 23 0 R 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�:� << 1. /Encoding 21 0 R 1 , if E X n X r! With this mode of convergence, we increasingly expect to see the next outcome in a sequence of random experiments becoming better and better modeled by a given probability distribution. /LastChar 196 Here, we state the SLLN without proof. 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 1. 844.4 319.4 552.8] n!1 . /Encoding 14 0 R /FirstChar 33 1000 1000 1055.6 1055.6 1055.6 777.8 666.7 666.7 450 450 450 450 777.8 777.8 0 0 /LastChar 196 (Markov’s Inequality) Let X be a random variable. /Length 2117 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] 343.7 593.7 312.5 937.5 625 562.5 625 593.7 459.5 443.8 437.5 625 593.7 812.5 593.7 >> /Filter /FlateDecode Almost sure convergence of random variable. Convergence in probability: X n does not converge in probability because the frequency of the jumps is constant equal to 1 2. /Type/Font develop the theory, we will focus our attention on examples. /Type/Encoding 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.. X ) as n ! /Name/F4 /LastChar 196 /Subtype/Type1 << Example 3.5 (Convergence in probability can imply almost sure convergence). /Name/F3 Lemma 1. /Encoding 34 0 R 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 Deﬁnitions 2. /FirstChar 33 34 0 obj 791.7 777.8] Types of Convergence Let us start by giving some deﬂnitions of diﬁerent types of convergence. 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 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. We proved WLLN in Section 7.1.1. 14 0 obj 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 Givenastationaryprocess(Xp)p∈Z andaneventB ∈ σ(Xp,p∈ Z), we study the almost sure convergence as n and m go to inﬁnity of the “bilateral”martingale E[1B |X−n,X−n+1,...,Xm−1,Xm]. 319.4 552.8 552.8 552.8 552.8 552.8 552.8 552.8 552.8 552.8 552.8 552.8 319.4 319.4 De nition 5.2 | Almost sure convergence (Karr, 1993, p. 135; Rohatgi, 1976, p. 249) The sequence of r.v. /Type/Font /Widths[1000 500 500 1000 1000 1000 777.8 1000 1000 611.1 611.1 1000 1000 1000 777.8 460.7 580.4 896 722.6 1020.4 843.3 806.2 673.6 835.7 800.2 646.2 618.6 718.8 618.8 /FirstChar 33 << >> 734 761.6 666.2 761.6 720.6 544 707.2 734 734 1006 734 734 598.4 272 489.6 272 489.6 /Name/F9 /FontDescriptor 32 0 R If r =2, it is called mean square convergence and denoted as X n m.s.→ X. 589.1 483.8 427.7 555.4 505 556.5 425.2 527.8 579.5 613.4 636.6 272] 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 /FirstChar 33 /LastChar 196 Almost sure convergence: X n does not converge almost surely because the probability of every jump is always equal to 1 2. Contents . 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 %PDF-1.2 /Widths[272 489.6 816 489.6 816 761.6 272 380.8 380.8 489.6 761.6 272 326.4 272 489.6 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. The most notable related work on almost sure rates of convergence is Homem-de-Mello , which used the slightly different setting of a variable SAA (VSAA), where in each iteration k the objective function is approximated by an estimator $${\hat{f}}_{N_k}$$ with a newly drawn random sample of (potentially) different size $$N_k$$. 761.6 679.6 652.8 734 707.2 761.6 707.2 761.6 0 0 707.2 571.2 544 544 816 816 272 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 1062.5 1062.5 826.4 826.4 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. Functions and by training a ResNet architecture for a process X. where Z is a semimartingale! 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