Convergence in probability: X n does not converge in probability because the frequency of the jumps is constant equal to 1 2. Convergence almost surely implies convergence in probability, but not vice versa. Almost sure convergence example. For another idea, you may want to see Wikipedia's claim that convergence in probability does not imply almost sure convergence and its proof using Borel–Cantelli lemma. /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 /FirstChar 33 535.6 641.1 613.3 302.2 424.4 635.6 513.3 746.7 613.3 635.6 557.8 635.6 602.2 457.8 /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 /Type/Font /FontDescriptor 32 0 R endobj /FirstChar 33 826.4 295.1 531.3] Lemma 1. /Encoding 7 0 R 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 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. 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 by bremen79. 1.1 Almost sure convergence Definition 1. Almost Sure Convergence of SGD on Smooth Non-Convex Functions. /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 Almost sure convergence is sometimes called convergence with probability 1 (do not confuse this with convergence in probability). 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 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�:� Convergence in the almost sure sense: For any ! 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 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]. endobj endobj 566.7 843 683.3 988.9 813.9 844.4 741.7 844.4 800 611.1 786.1 813.9 813.9 1105.5 593.7 500 562.5 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 ... 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. 334 405.1 509.3 291.7 856.5 584.5 470.7 491.4 434.1 441.3 461.2 353.6 557.3 473.4 489.6 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 0 0 0 0 0 611.8 816 Here is another example. 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/LCJHKM+CMMI12 It remains to show that Xn → X almost-surely. endobj 343.7 593.7 312.5 937.5 625 562.5 625 593.7 459.5 443.8 437.5 625 593.7 812.5 593.7 161/minus/periodcentered/multiply/asteriskmath/divide/diamondmath/plusminus/minusplus/circleplus/circleminus Almost sure convergence of random variable. /Subtype/Type1 /Encoding 14 0 R 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 0 492.9 510.4 505.6 612.3 361.7 429.7 553.2 317.1 939.8 644.7 513.5 534.8 474.4 479.5 472.2 472.2 472.2 472.2 583.3 583.3 0 0 472.2 472.2 333.3 555.6 577.8 577.8 597.2 380.8 380.8 380.8 979.2 979.2 410.9 514 416.3 421.4 508.8 453.8 482.6 468.9 563.7 endobj /FirstChar 33 /FontDescriptor 12 0 R 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 << o) = 0; n> N(! For example, consider the following SDE for a process X. where Z is a given semimartingale and are fixed real numbers. 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 >> << /Name/F5 /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 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 /FontDescriptor 16 0 R ��� 1. << 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 endobj 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#;[. ; n > n ( 0 ; n > n ( 0 ; n > n (! most seen! 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