A note on Spitzer identity for random walk

详细信息    查看全文

• 作者：Aimé ; Lachal
• 关键词：Spitzer identity ; Laplace– ; Fourier transform ; Erlang distribution
• 刊名：Statistics and Probability Letters
• 出版年：2008
• 出版时间：1 February 2008
• 年：2008
• 卷：78
• 期：2
• 页码：97-108
• 全文大小：204 K

文摘

Let rieve&_udi=B6V1D-4NYSX9V-1&_mathId=mml1&_user=10&_cdi=5672&_rdoc=2&_acct=C000050221&_version=1&_userid=10&md5=b337c9c242f6f49ca2640883dd3ea438"" title=""Click to view the MathML source"">(S_n)_{nrimg/entities/2a7e.gif"" alt=""greater-or-equal, slanted"" border=0>0} be a random walk evolving on the real line and introduce the first hitting time of the half-line rieve&_udi=B6V1D-4NYSX9V-1&_mathId=mml2&_user=10&_cdi=5672&_rdoc=2&_acct=C000050221&_version=1&_userid=10&md5=1d78c5f380eaf9fc61029123a4838049"" title=""Click to view the MathML source"">(a,+∞) for any real a: rieve&_udi=B6V1D-4NYSX9V-1&_mathId=mml3&_user=10&_cdi=5672&_rdoc=2&_acct=C000050221&_version=1&_userid=10&md5=ccbbd0fc73ed8397e5727b7905e7599b"" title=""Click to view the MathML source"">τ_a=min{nrimg/entities/2a7e.gif"" alt=""greater-or-equal, slanted"" border=0>1:S_n>a}. The classical Spitzer identity (1960) supplies an expression for the generating function of the couple rieve&_udi=B6V1D-4NYSX9V-1&_mathId=mml4&_user=10&_cdi=5672&_rdoc=2&_acct=C000050221&_version=1&_userid=10&md5=628d826b829ea2bb862a471b4e06495c"" title=""Click to view the MathML source"">(τ₀,S_τ0). In 1998, Nakajima [Joint distribution of the first hitting time and first hitting place for a random walk. Kodai Math. J. 21 (1998) 192–200.] derived a relationship between the generating functions of the random couples rieve&_udi=B6V1D-4NYSX9V-1&_mathId=mml5&_user=10&_cdi=5672&_rdoc=2&_acct=C000050221&_version=1&_userid=10&md5=bfb424a70762509439c8ca57e2b760df"" title=""Click to view the MathML source"">(τ₀,S_τ0) and rieve&_udi=B6V1D-4NYSX9V-1&_mathId=mml6&_user=10&_cdi=5672&_rdoc=2&_acct=C000050221&_version=1&_userid=10&md5=b377d9cd04aa0bc12b7281850650e193"" title=""Click to view the MathML source"">(τ_a,S_τa) for any positive number a. In this note, we propose a new and shorter proof for this relationship and complement this analysis by considering the case of an increasing random walk. We especially investigate the Erlangian case and provide an explicit expression for the joint distribution of rieve&_udi=B6V1D-4NYSX9V-1&_mathId=mml7&_user=10&_cdi=5672&_rdoc=2&_acct=C000050221&_version=1&_userid=10&md5=2bc353b798f201e407b433e73e81a67d"" title=""Click to view the MathML source"">(τ_a,S_τa) in this situation.