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<blockquote>
<p><strong>Let everything happen to you<br>Beauty and terror<br>Just keep going<br>No feeling is final<br> ——Rainer Maria Rilke</strong></p>
</blockquote>
<p>TMB 是法语 Tour du Mont Blanc的缩写，即勃朗峰环线，这是一条公认的世界史诗级徒步路线。中文的详细行程介绍可以参见<a target="_blank" rel="noopener" href="https://zhuanlan.zhihu.com/p/38997696">知乎这篇</a>，英文的介绍可以参考<a target="_blank" rel="noopener" href="https://thedanishnomads.com/tour-du-mont-blanc-self-guided/">这篇</a>，我是因为事后总结爬升高度数据才发现这篇英文文章的。</p>
<p>我自己一个人徒步是从18年开始的，因为在一起读博时一个关系很好的阿根廷哥们去了智利央行工作，我正好也想去看看世界。于是就在智利南部的巴塔哥尼亚地区的百内国家公园 (Torres del Paine) 开始了第一次真正的独自徒步，这也是一条世界著名的徒步路线，我走的是经典的W-Trek，5天四夜，详情可以参见<a target="_blank" rel="noopener" href="https://www.mafengwo.cn/i/5485569.html">马蜂窝的这篇</a>或者<a target="_blank" rel="noopener" href="https://bbs.qyer.com/thread-2877795-1.html">穷游的这篇</a>。后来自己又去了秘鲁高原徒步，算是慢慢积累一些徒步经验。今年由于疫情哪里也去不了，只能在欧洲境内晃悠。德国从3月中旬开始实施lockdown政策，6月初开始放松，我于是7月初就跑到德国著名的国家公园贝希特斯加登（Berchtesgaden）去徒步了四天，回到家后意犹未尽。第二天就决定了9月初去TMB徒步。TMB全程170km，途径法国、意大利和瑞士三国。官网介绍一般要7-10天走完，智利5天4夜的W路线已经让住帐篷的我身心俱疲，面对这个更长的路线，所以我决定这次住高山小屋。主要因为我因为自己没有帐篷，也不想走了一天后睡在狭小的帐篷里，其实是更想长途跋涉一天后能有个热水澡洗一下，然后能吃上一顿热热乎乎的晚饭。TMB这条线一般情况下都要提前半年预定高山小屋住宿的，而今年由于疫情，我提前两个月都预定上了，而且在徒步时大部分营地都还有很多空床位的。后来还发现，在专门的帐篷营地里，也有热水澡和热食提供的，下次真的可以试试帐篷行。</p>
<p>根据过往的徒步经历，发现自己在山里上下爬升的速度要普遍快于其他人，而且通常比指示牌的时间快30%左右。于是就大胆的决定取了官网推荐的最小值7天，其实这样对体力会有很高的要求，好处就是省了不少住宿费。具体行程如下：</p>
<blockquote>
<p>Day1: Auberge de Bionnassay,8h<br>Day2: Gite Le Pontet,5h<br>Day3: Refuge de Motets, 9h<br>Day4: Refuge du Randonneur du Mont Blanc, 6h<br>Day5: Gite Alpage de La Peule, 10h<br>Day6: Relais d'Aprette, 6.5h<br>Day7: Gite Le Chamonix Moulin, 9.5h</p>
</blockquote>
<p>这些高山小屋之间的时间都是在<a target="_blank" rel="noopener" href="http://www.autourdumontblanc.com/en/">官网</a>预定时给出的。后来我才搞清楚官网是默认你出发的第一站是Le Houches，而我第一天是从Chamonix出发的，所以主路只走了5h。但到达Bionnassay之后，由于旁边的支线看了冰川，上下来回3h，所以第一天总行程还是8h。需要注意的一点时，我全程是背着装满50L的登山包徒步的，重量大概10kg左右。主要是睡袋，因为很多高山小屋要求自带睡袋。还有换洗和防寒衣服，虽然是9月初，但山里温度到了晚上只有10度左右，如果你想在营地周围看看，还是需要穿羽绒服的。最后就是水和食物，虽然我预定的高山小屋都是Half Board，也就是提供早餐和晚饭，但中午饭还是要准备的。你可以从营地买lunch box或者去超市买，这样会便宜很多。但大部分时间都在山里，只有一两天会经过村庄。你如果打算一开始准备全程的午餐，那么背包的重量将会大大增加。所以午餐这件事上又是体力与成本的一个权衡。我已经用体力节省了一些住宿费，不想再浪费午餐上了，所以决定吃的全部从营地解决。另外我被最后一个营地的名称给骗了，我以为它就在起点Chamonix的镇上，结果预定完一看地图才发现还有10km的距离。而且后来路上跟遇到的法国人聊天，才意识到错过的那一段看勃朗峰最美，这又给我了一个再次回来徒步的理由。</p>
<p>在50L背包的拖累下，每一段我所花的时间基本上与官网/指示牌的时间相差不多。第三天的行程我甚至用了10h，因为中间有一段我在山里迷路的，最崩溃的是我是在一座2756m的高山（Tete Nord des Fours）里不同的山脊处，约2600米左右出翻越了这座山三次。那一天真的让我感到心累，主要来自高山迷路的恐惧，因为上山的那条路根本看不出来是明星的路，全是乱石堆砌。下山的路更窄，感觉就是山羊踩出来的印记，一不小心就会跌落山崖。这大大减慢了我的速度，额外的担心开始出现，就是回到营地晚于7点的话，就没有饭吃了。在连续折了腾两个小时左右才碰到一个人，确认了大方向是正确的，这次稍微安心一些。现在7天全程走下来，我查看了一下手机App的收据。第3/5/7天的路程最长，每天徒步都在9h以上，差不多每天36km以上。剩下的四天每天平均6.5h左右，也就是每天约25km左右。这样下来我这7天的总路程已经超过200km了，而且根据前面那篇英文的数据，上下爬升高度达到1万米，所以这7天的运动量还是蛮大的。回过头来看看，觉得每天7h的徒步比较适中，差不多每天30km左右，而9h以上已经到我个人的上限了。超过上限的坏处就在于身体失去了控制能力，这在下山时尤其危险。尤其很多路段随时散步，很容易造成走路打滑。我就在下山是摔倒过两次，最后一天直接把登山杖摔折了，万幸的是人没有摔下山崖。如果重新再走的话，我会增加第二天的行程，这样第三天会轻松一些。而第四天我会额外再走1.5h到达Courmayeur，这是一个山脚的村庄，重要的补给站，很多旅店和超市可以选择。接下来的第五天我也会走到La Fouly，这是另外一个小村庄，可以进行补给。Courmayeur到La Fouly的路程也差不多9.5h左右。而第六天我从La Fouly直接到Le Peuty，差不多也是6.5h左右。因为从La Fouly到Champex可以做汽车，节省5h左右的行程，这段我走下来确实如法国人说的没啥特别的。而最后一天就直接走回到小镇Chamonix，估计也是9h以上的行程。这样下来算是完完整整的走完全程，而且最美的风景都已收入眼帘。这样调整的原因有两个，第一我是尽量想避免在睡袋里过夜，因为睡袋里双腿不能自由移动。而在长途跋涉一天之后，我的小腿肌肉早已酸胀，双腿整晚恨不得翻来覆去。而睡袋的束缚让我极其不爽，其中有三天晚上我是不得不睡睡袋，所以都没有睡好。第二则是好想痛痛快快的洗个热水澡。虽然所有的山间小屋都号称提供热水，但个人的感觉都只是永远没烧开的温水。只有在意大利段的Randonneur du Mont Blanc提供的是货真价实的热水，但可惜的是只有3分钟的免费时间。那天由于过分享受热水，导致洗头到一半就停水了，真是太难了。</p>
<p>最后说一说TMB是一个人走还是成团走，这是一个仁者见仁智者见智的问题，因为总是会取决很多因素的。我个人觉得最重要的两个因素就是自己的体力和偏好。如果你经常锻炼，之前有过很多次的长距离（3天以上）徒步经历，你完全可以尝试一下独行TMB。但如果你是一个徒步小白，之前从没有有过在山间徒步过夜的经历，一个人走TMB就有些危险了。虽然大部分时候指示牌都很清晰，但某些路段，如意大利境内的一些行程，隔一两个小时才会看到路牌，而且指示牌有时候还会自相矛盾。这就会大大增加迷路的风险，在高山里迷路会显著增加你能量的消耗，从而放大自身的不安和恐慌，这时候就需要有经验的人带着一起徒步了。我在第五天的时候碰到一个法国妹子，她跟我说她打算6天走完全程，她在过去的三天每天都走10h左右。我当时就震惊了，因为我看到她根本没有使用登山杖，而且想6天走完全部。她说过去的三天都没用到登山杖，我附和道，也是，大部分时候确实用不到。结果那天她和我走的是同样的距离，由于她在Courmayeur停下来吃早饭，晚上8点才到达我住的那个高山小屋，比我晚了1.5h左右。第二天一早出发，发现她已经手持两个木根当登山杖用，即使在平坦的路上也不离手。她说过去的四天走的太狠了，脚有点受伤了，需要外物支持一下。这时候我注意到她是滑雪俱乐部的，看她身材苗条有形，想必也是爱好运动之人。这一路上我还碰到另外一个法国大叔，他是在我第三天晚上吃饭时候认识的。其英语口音相当纯正，让我怀疑是否真的是法国人。后来细聊之后才知道，这个哥们是MIT的化学博士，在加拿大工作了8年，怪不得英语口音那么正。后来一直在日内瓦工作，持有法国、瑞士、加拿大三重国籍。到现在一共工作25年了，所以年龄应该超过50岁了。可是在山里徒步的时候，根本就不像超过50岁的人。不论上山还是下山的速度均碾压我，而且他还随身带帐篷的。幸亏我在第五天与他同行，那天主要是意大利段，每次指示牌不清晰的时候，就靠他厚厚的一摞地图。而且最后他走了一段非常陡峭的路翻山越岭直达La Fouly。也是和他聊天才知道UTMB(Ultra-TMB)的记录，最快记录是不到20h左右跑完170km的TMB全程，专业运动员真是太可怕了。路上碰到一些人三天跑完全程的人就已经觉得是大神了，还有一些骑自行车走全程的人，觉得更神。因为某些路段确实很陡很窄，走路都很费劲和危险。Anyway，算是人外有人天外有天吧。</p>
<p>当体力不成问题的时候，偏好的差异就凸显出它的重要性了。有人喜欢奇美的风景，有人钟情当地的美味，还有人则喜欢悠闲打卡。这时候组团徒步就要小心了，如果双方的偏好差异太大，强行一起走将会十分痛苦，分开则是双方最好的解脱。就像电影《小丑》的那句台词People think being alone makes you lonly. But being surrounded by the wrong people is the lonliest thing in the world.与心有戚戚焉，我徒步中大部分时间都是一个人在独行，虽然会迷路，也会抱怨越过山丘无人等候。但终归是自己习惯的节奏，而且偶尔也会遇到有趣的同行者。然而他们是不会陪你一路走下去的，每个人都是其他生命中的过客。不要哭泣他们会离开你，要感恩他们曾经来到你的生命里，让你感受到了不一样的色彩。最后以一句鸡汤结尾：你要搞清楚自己人生的剧本——不是你父母的续集，不是你子女的前传，更不是你朋友的外篇。对待生命你不妨大胆冒险一点， 因为好歹你要失去它。如果这世界上真有奇迹，那只是努力的另一个名字。生命中最难的阶段不是没有人懂你，而是你不懂你自己。</p>
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<p>Oscar Widle: we all live in the gutter, but some of us are looking at the stars.<br>黑格尔：一个民族只有关心脚下的人那是没有未来的，一个民族有一群仰望星空的人，他们才有希望。<br>王国维：试上高峰窥皓月，偶开天眼觑红尘。可怜身是眼中人。</p>
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<p><strong>Counting Stars</strong></p>
<p>Lately I been, I been losing sleep<br>Dreaming about the things that we could be<br>But baby I been, I been prayin hard<br>Said no more counting dollars<br>We'll be counting stars<br>Yeah, we'll be counting stars</p>
<p>I see this life<br>Like a swinging vine<br>Swing my heart across the line<br>In my faces flashing signs<br>Seek it out and ye shall find<br>The old, but I'm not that old<br>Young, but I'm not that bold<br>And I don't think the world is sold<br>I'm just doing what we’re told<br>I, feel something so right<br>Doing the wrong thing<br>I, feel something so wrong<br>But doing the right thing<br>I could lie, could lie, could lie<br>Everything that kills me makes me feel alive</p>
<p>I feel the love<br>And I feel it burn<br>Down this river every turn<br>Hope is a four letter word<br>Make that money<br>Watch it burn<br>Old, but I'm not that old<br>Young, but I'm not that bold<br>And I don't think the world is sold<br>I'm just doing what we’re told<br>I, feel something so wrong<br>But doing the right thing<br>I could lie, could lie, could lie<br>Everything drowning me makes me wanna fly</p>
<p>Take that money and watch it burn<br>Sink in the river the lessons I learned<br>Take that money and watch it burn<br>Sink in the river the lessons I learned<br>Said no more counting dollars<br>We'll be counting stars</p>
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<p><strong>没有理想的人不伤心</strong></p>
<p>我最爱去的唱片店，<br>昨天是她的最后一天，<br>曾经让我陶醉的碎片，<br>全都散落在街边。<br>我最爱去的书店，<br>她也没撑过这个夏天，<br>回忆文字流淌着怀念，<br>可是已没什么好怀念。</p>
<p>可是你曾经的那些梦，<br>都已变得模糊看不见。<br>那些为了理想的战斗，<br>也不过为了钱。<br>可是我最恨的那个人，<br>他始终没死在我面前，<br>还没年轻就变得苍老。<br>这一生无解，<br>没有我的空间，<br>没有我的空间，<br>没有我的空间。</p>
<p>你曾热爱的那个人，<br>这一生也不会再见面。<br>你等在这文化的废墟上，<br>已没人觉得你狂野。<br>那些让人敬仰的神殿，<br>只在无知的人心中灵验。<br>我住在属于我的猪圈，<br>这一夜无眠。</p>
<p>我不要在失败孤独中死去，<br>我不要一直活在地下里。<br>物质的骗局，<br>匆匆的蚂蚁，<br>没有文化的人不伤心。</p>
<p>我不要在失败孤独中死去，<br>我不要一直活在地下里。<br>物质的骗局，<br>匆匆的蚂蚁，<br>没有文化的人不伤心。<br>他不伤心。<br>他不伤心。<br>他不伤心。</p>
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<h2 id="Essence-of-Probability-Theory"><a href="#Essence-of-Probability-Theory" class="headerlink" title="Essence of Probability Theory"></a>Essence of Probability Theory</h2><ul>
<li>How to connect conditional probability with the regression model?<br>Remember conditional expectation of random variable $Y$ given that $X=x$ is<br>$$<br>\begin{equation*}<br>E[Y|X=x]=\sum_{y_i\in{R_y}}P_{Y|X}(y_i|x)<br>\end{equation*}<br>$$</li>
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<p>That is to say, $E[Y|X=x]$ only depends on the value of $x$. In other words, $E[Y|X=x]$ is a function of $x$. Thus we can think of $E[Y|X=x]$ as a function of the value of random variable $X$ as follows:<br>$$<br>\begin{equation*}<br>g(x)=E[Y|X=x]\Rightarrow E[Y|X]=g(X)<br>\end{equation*}<br>$$<br>Where function $g(X)$ could be our commonly used regression model. This is an important concept that we interpret the conditional expectation as a random variable. </p>
<h3 id="Law-of-Iterated-Expectations"><a href="#Law-of-Iterated-Expectations" class="headerlink" title="Law of Iterated Expectations"></a>Law of Iterated Expectations</h3><p>$$<br>E[Y]=E[E[Y|X]]<br>$$<br>Let us look again at the law of total probability for expectation. Assume $E[Y|X]=g(X)$, we have:<br>$$<br>\begin{align*}<br>E[Y]&=\sum_{x_i\in{R_X}}E[Y|X=x_i]P_X(x_i) \\<br> &=\sum_{x_i\in{R_X}}g(x_i)P_X(x_i) \\<br> &=E[g(X)]=E[E[Y|X]]<br>\end{align*}<br>$$<br>This is also known as <strong>Law of Total Expectation</strong>.</p>
<h3 id="Law-of-Total-Varianace"><a href="#Law-of-Total-Varianace" class="headerlink" title="Law of Total Varianace:"></a>Law of Total Varianace:</h3><p>$$<br>Var(Y)=Var(E[Y|X])+E[Var(Y|X)]<br>$$</p>
<p>Let $Z=E[Y|X]$, then,<br>$$<br>Var(Z)=E[Z^2]-(E[Z])^2=E[Z^2]-(E[Y])^2<br>$$</p>
<p>First, we are going to show the proof and introduce the intuition afterwards.<br>$$<br>\begin{align*}<br>& Var(Y|X)=E[Y^2|X]-E^2[Y|X] \\<br>& \Rightarrow E[Var(Y|X)]=E[Y^2]-E[Z^2] \\<br>& \Rightarrow E[Var(Y|X)]+E[Z^2]=Var(Y)+(E[Y])^2 \\<br>& \Rightarrow Var(Y) =E[Var(Y|X)]+Var(E[Y|X])<br>\end{align*}<br>$$</p>
<p>To understand the law of total variance intuitively, it is often useful to look at a population divided into several subgroups. In particular, suppose that we have this random experiment: We pick a person in the world at random and look at his/her height. Let us use variable $Y$ to represent the height. Define another random variable $X$ whose value depends on the country of the chosen person, where $X=1,2,3…n$. $n$ is the the number of countries in the world. Then, let us look at the two terms in the law of total variance.<br>$$<br>Var(Y) =E[Var(Y|X)]+Var(E[Y|X])<br>$$</p>
<p>We can see that $Var(Y|X=i)$ denotes the variance of $X$ in country $i$. Thus $E[Var(Y|X)]$ is the average of variances across all countries. On the other hand, $E[Y|X=i]$ is average height in country $i$. Then $Var(E[Y|X])$ is the variance between countries. So we can interpret the law of total variance in the following way. Variance of $Y$ can be decomposed into two parts: the first is the average of variances from each individual country, while the second is the variance between height averages in each country.</p>
<h3 id="Convergence-of-Random-Variable"><a href="#Convergence-of-Random-Variable" class="headerlink" title="Convergence of Random Variable"></a>Convergence of Random Variable</h3><p>Before discussing convergence for a sequence of random variables, let us remember what convergence means for a sequence of real numbers. </p>
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<p>A sequence of real number $x_1, x_2, x_3 …$ converges to a limit $x$ if<br>$$<br>\lim_{n\rightarrow\infty}x_n=x<br>$$</p>
<p>That is, for any $\epsilon>0$, there exists an $N_o\in N $ such that </p>
<p>$$<br>|x_n-x|<\epsilon,\text{ for all } n>N_o<br>$$</p>
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<h4 id="Sequence-of-Random-Variables"><a href="#Sequence-of-Random-Variables" class="headerlink" title="Sequence of Random Variables"></a>Sequence of Random Variables</h4><p>Here, we would like to discuss what we precisely mean by a sequence of random variables. Recall that, in any probability model, we have a sample space $S$ and a probability measure $P$. For simplicity, suppose that our sample space consists of a finite number of elements, i.e.,</p>
<p>$$<br>S={s_1,s_2,…., s_k}<br>$$</p>
<p>Then, a random variable $X$ is a mapping that assigns a real number to any of any of the possible outcomes $s_i$, which we may write as<br>$$<br>X(s_i)=x_i,\text{ for }i=1,2,…,k<br>$$</p>
<p>When we have a sequence of random variables $X_1,X_2,X_3,….$, it means each $X_n$ is a function from space $S$ to real numbers, that is,<br>$$<br>X_n(s_i)=x_{ni},\text{ for }i=1,2,…,k<br>$$</p>
<p>In brief, a sequence of random variables is in fact a sequence of functions $X_n: S\rightarrow R$.</p>
<h3 id="Convergence-in-Distribution"><a href="#Convergence-in-Distribution" class="headerlink" title="Convergence in Distribution"></a>Convergence in Distribution</h3><p>Convergence in distribution is in some sense the weakest type of convergence. All it says is that the CDF of $X_n$ converges to the CDF of $X$ as $n$ goes to infinity. It does not require any dependence between the $X_n$ and $X$. To say that $X_n$ converges in distribution to $X$, we write:<br>$$<br>X_n \xrightarrow{d} X<br>$$</p>
<p>The formal definition of convergence in distribution follows:</p>
<blockquote>
<p>A sequence of random variables $X_1,X_2,X_3,….$ converges in distribution to a random<br>variable $X$, if<br>$$<br>\lim_{n\rightarrow\infty}F_{X_n}(x)=F_{X}(x)<br>$$<br>for all at which $F_{X}(x)$ is continuous. </p>
</blockquote>
<p>The most famous example of convergence in distribution is the central limit theorem (CLT). The CLT states that the normalized average of i.i.d. random variables $X_1,X_2,X_3,….$ converges in distribution to a standard normal random variable.</p>
<h3 id="Convergence-in-Probability"><a href="#Convergence-in-Probability" class="headerlink" title="Convergence in Probability"></a>Convergence in Probability</h3><p>Convergence in probability is stronger than convergence in distribution, and we write<br>$$<br>X_n \xrightarrow{P} X<br>$$</p>
<p>The formal definition of convergence in probability follows:</p>
<blockquote>
<p>A sequence of random variables $X_1,X_2,X_3,….$ converges in probability to a random<br>variable $X$, if<br>$$<br>\lim_{n\rightarrow\infty}P(|X_n-X|\geq\epsilon)=0,\text{ for all }\epsilon>0<br>$$</p>
</blockquote>
<p>The most famous example of convergence in probability is the weak law of large numbers (WLLN). The WLLN states that if $X_1,X_2,X_3,….$ are i.i.d. random variable with mean $E[X_i]=\mu<\infty$, then the average sequence defined by<br>$$<br>\bar{X}_n=\frac{X_1+X_2+…+X_n}{n}<br>$$</p>
<p>converges in probability to its mean $\mu$. It is called the “weak” law because it refers to convergence in probability. There is another version of the law of large numbers that is called the strong law of large numbers (SLLN).</p>
<h3 id="Convergence-in-Mean"><a href="#Convergence-in-Mean" class="headerlink" title="Convergence in Mean"></a>Convergence in Mean</h3><p>One way of interpreting the convergence of a sequence $X_n$ to $X$ is to say the “distance” between $X_n$ to $X$ is getting smaller and smaller. For example, if we define the distance between $X_n$ and $X$ as $P(|X_n-X|\geq\epsilon)$, we have convergence in probability. The common way to define the distance between $X_n$ and $X$ is $E(|X_n-X|^{\tau})$, parameter $\tau$ is a fixed number. This refers to <strong>convergence in mean</strong>. (<em>Note: for convergence in mean, it usually requires $E(|X_n|^{\tau})<\infty$</em>). The most common case is $\tau=2$, in which case it is called the <strong>mean-square convergence</strong>. (<em>Note: the case $\tau=1$ is narrowly referred as convergence in mean.</em>)</p>
<blockquote>
<p>Let $\tau\geq1$ be a fixed number. A sequence of random variables $X_1,X_2,X_3,….$<br>converges in the $\tau$th mean or in the $L^{\tau}$ norm to a random variable $X$, shown by<br>$X_n \xrightarrow{L^{\tau}} X$, if<br>$$<br>\lim_{n\rightarrow\infty}E(|X_n-X|^{\tau})=0<br>$$</p>
<p>If $\tau=2$, it is shown as $X_n \xrightarrow{m.s.} X$</p>
</blockquote>
<blockquote>
<p><strong>Theorem:</strong> If $X_n \xrightarrow{L^{\tau}} X$ for some $\tau\geq1$, then $X_n \xrightarrow{P} > X$.</p>
</blockquote>
<blockquote>
<p><strong>Proof:</strong> For any $\epsilon>0$ and the $\tau\geq1$, we have<br>$$<br>\begin{align*}<br>P(|X_n-X|\geq\epsilon)&=P(|X_n-X|^{\tau}\geq\epsilon^{\tau}) \\<br>&\leq \frac{E(|X_n-X|^{\tau})}{\epsilon^{\tau}}\text{ by Markov Inequality}<br>\end{align*}<br>$$</p>
<p>Since by assumption, $ \lim_{n\rightarrow\infty}E(|X_n-X|^{\tau})=0$, we conclude<br>$$<br>\lim_{n\rightarrow\infty}P(|X_n-X|\geq\epsilon)=0,\text{ for all }\epsilon>0<br>$$</p>
</blockquote>
<p>The converse of above theorem is not true in general, an counter example follows:</p>
<blockquote>
<p><strong>Example:</strong> Consider a sequence of random variables $X_1,X_2,X_3,….$ such that<br>$$<br>X_n=\left\lbrace\begin{array}{ll }<br>n^2 & \text{ with probability }\frac{1}{n} \\<br>0 & \text{ with probability }1-\frac{1}{n}\end{array}\right.<br>$$<br>we can show that $X_n \xrightarrow{P} X$, but $X_n$ does not converge in the $\tau$th mean > for any $\tau>1$.</p>
</blockquote>
<h3 id="Almost-Sure-Convergence"><a href="#Almost-Sure-Convergence" class="headerlink" title="Almost Sure Convergence"></a>Almost Sure Convergence</h3><p>Consider a sequence of random variables $X_1,X_2,X_3,….$, if the probability that the sequence $X_n(s)$ converges to $X(s)$ is equal to 1, we say that $X_n$ converges to $X$ almost surely and write<br>$$<br>X_n \xrightarrow{a.s.} X<br>$$</p>
<p>The formal definition of almost sure convergence follows:</p>
<blockquote>
<p>A sequence of random variables $X_1,X_2,X_3,….$, converge almost surely to a random<br>variable $X$ if<br>$$<br>P\big(\left\lbrace s \in S: \lim_{n\rightarrow \infty} X_n(s)=X(s)\right\rbrace\big)=1.<br>$$</p>
</blockquote>
<blockquote>
<p><strong>Example:</strong> Consider the sample space $S=[0,1]$ with a probability measure that is uniform on this space, i.e.,<br>$$<br>P([a,b])=b-a, \qquad \textrm{ for all }0 \leq a \leq b \leq 1.<br>$$</p>
<p>Define the sequence ${X_n, n=1,2,…}$ as follows:<br>$$<br>X_n(s) = \left\lbrace \begin{array}{l l}<br>1 & \quad 0 \leq s < {\large \frac{n+1}{2n}} \\<br> & \quad \\<br>0 & \quad \text{otherwise}<br>\end{array} \right.<br>$$</p>
<p>Also, define the random variable $X$ on this sample space as follows:<br>$$<br>X(s) = \left\lbrace \begin{array}{l l}<br>1 & \quad 0 \leq s < \frac{1}{2} \\<br> & \quad \\<br>0 & \quad \text{otherwise}<br>\end{array} \right.<br>$$</p>
<p>We can show that $X_n \xrightarrow{a.s.} X$.</p>
</blockquote>
<p>In some problems, proving almost sure convergence directly can be difficult. Thus, it is desirable to know some sufficient conditions for almost sure convergence. Here is a result that is sometimes useful when we would like to prove almost sure convergence.</p>
<blockquote>
<p><strong>Theorem:</strong> Consider a sequence of random variables $X_1,X_2,X_3,….$, if for all > $\epsilon>0$, we have<br>$$<br>\sum_{n=1}^{\infty} P\big(|X_n-X| > \epsilon \big) < \infty,<br>$$</p>
<p>Then $X_n \xrightarrow{a.s.} X$.</p>
</blockquote>
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<a href="/2020/08/08/Haruki/" class="post-title-link" itemprop="url">I will always stand on the side of the egg</a>
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<h2 id="我永远站在鸡蛋这一边——村上春树耶路撒冷演讲"><a href="#我永远站在鸡蛋这一边——村上春树耶路撒冷演讲" class="headerlink" title="我永远站在鸡蛋这一边——村上春树耶路撒冷演讲"></a>我永远站在鸡蛋这一边——村上春树耶路撒冷演讲</h2><p>晚上好。今天我来到耶路撒冷，作为一个小说家，也就是说，作为一个以编写谎话为生的人。</p>
<p>当然，小说作者们不是唯一说谎的人。政治家们，如我们所知，也说谎。外交官和将军们有时也说着他们那种类型的谎言，就如同汽车推销员，屠夫，和建筑师一样。然而小说家的谎言与其它谎言不同，因为没有人会指责他的谎言不道德。确实，他的谎话越大、他把谎话讲得越精巧、谎言被创造得越像天才之作，大众与评论就越会赞美他。这是为什么？</p>
<p>我的答案将会是这样的：那就是，通过讲述充满了技巧的谎言——也就是说，通过制造看上去真实的小说——小说家可以将真相放到一个新的所在从而让它显得更为清晰。而大多数情况下，要想从原始事态中抓住真相并且将它准确地描述出来实际上是不可能的。这就是为什么我们试着去抓住真相的尾巴，将它从它藏身的地方引诱出来，把它转移到小说式的所在，并且给它换上小说式的形态。然而为了做到这一点，我们首先必须弄清真相藏身于我们、我们自己内部的何处。这是一种创作好的谎言所需要的重要的能力。</p>
<p>然而今天，我不想撒谎。我会试着尽量诚实。一年之中仅仅只有几天时间，我会不讲谎话，而今天碰巧就是其中之一。所以请允许我告诉你们真相。在日本，许多人建议我不要来这里接受耶路撒冷文学奖。有一些人甚至警告我，说如果我来了，他们会对我的书发起抵制。这一切的原因当然在于，加沙地带所发生的惨烈的枪战。联合国报告说在封锁的加沙城有超过千记的人身亡，其中的很多是毫无武装的平民——孩子和老人。</p>
<p>从得知获奖之时起，我就问着自己，在这样一个时间去到以色列领取文学奖项是否合适，是否这会给人带来我支持冲突某一方、我赞同某国决意释放其气势汹汹的武力的政策的印象。当然同时，我不希望看到我的书遭到抵制。然而最终，在细致地一番考虑之后，我决意来到这里。我作此决定的一个原因就是太多人建议我不要来到这里。也许就像其他很多小说家一样，我打算恰恰就要去做那些我被劝告不要去做的事。如果人们告诉我——并且尤其是如果他们是在警告我——“别去那儿”，“别那样做”，我则倾向于“就去那儿”，“就那样做”。也许你们会说，这是我作为作家的天性。小说家们都是怪胎。他们就是不肯相信任何他们没有用自己的眼睛看到或者用自己的双手摸到的事。所以那就是我为什么来到了这里。我选择了来到这里而非回避。我选择了亲眼见证而非转头不看。我选择了对你们发表演讲而非沉默不语。</p>
<p>请允许我，传达一条讯息，一条非常私人的讯息。这是我写作的时候一直记在心里的东西。我还从来没有将它写到纸上或者贴在墙上，但我将它刻入了我头脑的深处，它差不多是这样的：</p>
<p><strong>“在一面高大、坚固的墙和一只撞向墙的鸡蛋之间，我将永远，站在鸡蛋的一边”</strong></p>
<p>是的，无论墙有多么地正确，鸡蛋有多么地错误，我会和鸡蛋站在一起。将会有别的什么人去决定什么是对，什么是错；也许时间或者历史会做出判断。但是假如有一个作家，他，不论出于何种原因，书写一些站在墙那一边的作品，那么究竟这些作品还有什么价值呢？而以上比喻的意义何在？有些情况下，这些意义只是太简单、太清晰了。炸弹和坦克和火箭弹和白磷弹就是那面高墙。而那些鸡蛋就是那些手无寸铁的平民，他们被炮弹粉碎、烧毁、击毙。这是这比喻的一层意味。</p>
<p>然而这不是全部。它还有更深的含义。试着这样想：<u>我们每一个人，都或多或少地，是一枚鸡蛋。我们每一个人都是一个独特的、不可替代的灵魂，而这灵魂覆盖着一个脆弱的外壳。这就是我自己的真相，而且这也是你们每一个人的真相。而且我们每一个人，程度或轻或重地，都在面对着一面高大的、坚固的墙。而这面墙有一个名字：它的名字叫做“体制”。这个体制本来应该保护我们，但是有时候它有了独立的生命意志，而这时它开始杀死我们，并且怂恿我们互相残杀——冷血地、有效地、系统性地残杀</u>。</p>
<p><u>我写作小说只有一个原因，而那就是为了使个体灵魂的尊严彰显，并且闪闪发光世人可见。一个故事的目的是敲响一个警钟，是燃亮灯火不灭，从而令在体制之中的我们的灵魂不至迷陷于体制的巨网，不至于被体制损害。我真心相信小说作者的工作就是通过写作不断地去尝试将个体灵魂的独特性澄清——那些关于生与死的故事，那些关于爱的故事，那些让人们落泪、并且因恐惧而战栗、因大笑而颤抖的故事。这就是我们继续写作的原因，一天又一天，用极致的严肃捏造着虚幻的小说。</u></p>
<p>我的父亲去年以九十高龄去世了。他是名退休的教师，兼一名业余的僧人。当他在京都学校毕业后，他被征选进了军队，派送至了中国。我作为战后的一代，每天清晨早饭之前都会看到他在我家那个小小的佛坛前虔诚地念经、久久地晨诵。有一次我问他为什么要这样做，他告诉我他在为那些战争中死去的人们祈祷。他在为那些战争中死去的人们祈祷，不论己方和敌方。看着他跪在佛坛前的背影，我似乎感觉到一片死亡的阴影在他的上方盘旋。我父亲去世了，而他的记忆也随之而去，那些记忆我将永远也不会知道。而那潜伏于他周身的死亡气息则停留在了我的记忆之中。这是我从他那里得到的少数东西之一，并且是其中最重要的一个。</p>
<p><u>我今天只有一个讯息希望传达给你们。那就是我们都是人类，是超越了国籍种族和信仰的个体，并且我们都是面对着名为体制的坚固墙体的脆弱的鸡蛋。照一切看来，我们没有赢的胜算。墙太高大了，太强大的——而且太冷酷了。假如我们还有一点点胜利的希望的话，那么它将来自于我们对于自己的和其他人的灵魂当中的那种极致的独特行和不可替代性的信念，来自于对于我们从灵魂的联合所获得的那种温暖的信念。</u></p>
<p>请花一点点时间想想这个。我们每一个人都拥有一副脆弱的、但活生生的灵魂。而体制一无所有。我们一定不能任由体制去剥削我们。我们一定不能允许体制有它自己的意志。因为体制并不创造我们：是我们创造了体制。这就是我全部想说的。我非常荣幸能够被授予耶路撒冷文学奖。我非常荣幸我的书被世界上那么多地方的人所阅读。而且我非常想对以色列的读者们表达我的感激。你们是我来到这里的最大动因。而且我希望我们分享了一些东西，一些充满了意义的东西。我非常高兴今天在这里有这个机会与你们对话。</p>
<p>非常感谢。</p>
<h2 id="English-Version-of-This-Speech"><a href="#English-Version-of-This-Speech" class="headerlink" title="English Version of This Speech"></a>English Version of This Speech</h2><p>I have come to Jerusalem today as a novelist, which is to say as a professional spinner of lies.</p>
<p>Of course, novelists are not the only ones who tell lies. Politicians do it, too, as we all know. Diplomats and military men tell their own kinds of lies on occasion, as do used car salesmen, butchers and builders. The lies of novelists differ from others, however, in that no one criticizes the novelist as immoral for telling them. Indeed, the bigger and better his lies and the more ingeniously he creates them, the more he is likely to be praised by the public and the critics. Why should that be?</p>
<p>My answer would be this: Namely, that by telling skillful lies - which is to say, by making up fictions that appear to be true - the novelist can bring a truth out to a new location and shine a new light on it. In most cases, it is virtually impossible to grasp a truth in its original form and depict it accurately. This is why we try to grab its tail by luring the truth from its hiding place, transferring it to a fictional location, and replacing it with a fictional form. In order to accomplish this, however, we first have to clarify where the truth lies within us. This is an important qualification for making up good lies.</p>
<p>Today, however, I have no intention of lying. I will try to be as honest as I can. There are a few days in the year when I do not engage in telling lies, and today happens to be one of them. So let me tell you the truth. A fair number of people advised me not to come here to accept the Jerusalem Prize. Some even warned me they would instigate a boycott of my books if I came. The reason for this, of course, was the fierce battle that was raging in Gaza. The UN reported that more than a thousand people had lost their lives in the blockaded Gaza City, many of them unarmed citizens - children and old people.</p>
<p>Any number of times after receiving notice of the award, I asked myself whether traveling to Israel at a time like this and accepting a literary prize was the proper thing to do, whether this would create the impression that I supported one side in the conflict, that I endorsed the policies of a nation that chose to unleash its overwhelming military power. This is an impression, of course, that I would not wish to give. I do not approve of any war, and I do not support any nation. Neither, of course, do I wish to see my books subjected to a boycott.</p>
<p>Finally, however, after careful consideration, I made up my mind to come here. One reason for my decision was that all too many people advised me not to do it. Perhaps, like many other novelists, I tend to do the exact opposite of what I am told. If people are telling me - and especially if they are warning me - “don’t go there,” “don’t do that,” I tend to want to “go there” and “do that.” It’s in my nature, you might say, as a novelist. Novelists are a special breed. They cannot genuinely trust anything they have not seen with their own eyes or touched with their own hands.</p>
<p>And that is why I am here. I chose to come here rather than stay away. I chose to see for myself rather than not to see. I chose to speak to you rather than to say nothing. This is not to say that I am here to deliver a political message. To make judgments about right and wrong is one of the novelist’s most important duties, of course. It is left to each writer, however, to decide upon the form in which he or she will convey those judgments to others. I myself prefer to transform them into stories - stories that tend toward the surreal, which is why I do not intend to stand before you today delivering a direct political message.</p>
<p>Please do, however, allow me to deliver one very personal message. It is something that I always keep in mind while I am writing fiction. I have never gone so far as to write it on a piece of paper and paste it to the wall: Rather, it is carved into the wall of my mind, and it goes something like this:</p>
<p><strong>“Between a high, solid wall and an egg that breaks against it, I will always stand on the side of the egg.”</strong> </p>
<p>Yes, no matter how right the wall may be and how wrong the egg, I will stand with the egg. Someone else will have to decide what is right and what is wrong; perhaps time or history will decide. If there were a nohevelist who, for whatever reason, wrote works standing with the wall, of what value would such works be?</p>
<p>What is the meaning of this metaphor? In some cases, it is all too simple and clear. Bombers and tanks and rockets and white phosphorus shells are that high, solid wall. The eggs are the unarmed civilians who are crushed and burned and shot by them. This is one meaning of the metaphor. This is not all, though. It carries a deeper meaning. Think of it this way. <u>Each of us is, more or less, an egg. Each of us is a unique, irreplaceable soul enclosed in a fragile shell. This is true of me, and it is true of each of you. And each of us, to a greater or lesser degree, is confronting a high, solid wall. The wall has a name: It is The System. The System is supposed to protect us, but sometimes it takes on a life of its own, and then it begins to kill us and cause us to kill others - coldly, efficiently, and systematically</u>.</p>
<p><u>I have only one reason to write novels, and that is to bring the dignity of the individual soul to the surface and shine a light upon it. The purpose of a story is to sound an alarm, to keep a light trained on The System in order to prevent it from tangling our souls in its web and demeaning them. I fully believe it is the novelist’s job to keep trying to clarify (preserve) the uniqueness of each individual soul by writing stories - stories of life and death, stories of love, stories that make people cry and quake with fear and shake with laughter. This is why we go on, day after day, concocting fictions with utter seriousness.</u></p>
<p>My father died last year at the age of 90. He was a retired teacher and a part-time Buddhist priest. When he was in graduate school, he was drafted into the army and sent to fight in China. As a child born after the war, I used to see him every morning before breakfast offering up long, deeply-felt prayers at the Buddhist altar in our house. One time I asked him why he did this, and he told me he was praying for the people who had died in the war.</p>
<p>He was praying for all the people who died, he said, both ally and enemy alike. Staring at his back as he knelt at the altar, I seemed to feel the shadow of death hovering around him. My father died, and with him he took his memories, memories that I can never know. But the presence of death that lurked about him remains in my own memory. It is one of the few things I carry on from him, and one of the most important.</p>
<p><u>I have only one thing I hope to convey to you today. We are all human beings, individuals transcending nationality and race and religion, fragile eggs faced with a solid wall called The System. To all appearances, we have no hope of winning. The wall is too high, too strong - and too cold. If we have any hope of victory at all, it will have to come from our believing in the utter uniqueness and irreplaceability of our own and others’ souls and from the warmth we gain by joining souls together.</u></p>
<p>Take a moment to think about this. Each of us possesses a tangible, living soul. The System has no such thing. We must not allow The System to exploit us. We must not allow The System to take on a life of its own. The System did not make us: We made The System.</p>
<p>That is all I have to say to you.</p>
<p>I am grateful to have been awarded the Jerusalem Prize. I am grateful that my books are being read by people in many parts of the world. And I am glad to have had the opportunity to speak to you here today.</p>
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<h2 id="Essence-of-Linear-Algebra"><a href="#Essence-of-Linear-Algebra" class="headerlink" title="Essence of Linear Algebra"></a>Essence of Linear Algebra</h2><ul>
<li>Here is a nice video providing intuitions for Eigenvector and Eigenvalue <iframe width="640" height="360" src="https://www.youtube.com/embed/PFDu9oVAE-g" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
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<h2 id="Essence-of-Measure-Theory"><a href="#Essence-of-Measure-Theory" class="headerlink" title="Essence of Measure Theory"></a>Essence of Measure Theory</h2><ul>
<li>Here is a nice video about introductions of the measure, (Lebesgue) measurable space, etc<iframe width="640" height="360" src="https://www.youtube.com/embed/7O7qPrNIt7w" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
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<h2 id="Funny-Comments"><a href="#Funny-Comments" class="headerlink" title="Funny Comments"></a>Funny Comments</h2><h3 id="Clip-The-1918-Spanish-Flu-vs-COVID-19-Spot-the-difference"><a href="#Clip-The-1918-Spanish-Flu-vs-COVID-19-Spot-the-difference" class="headerlink" title="Clip: The 1918 Spanish Flu vs. COVID-19. Spot the difference"></a>Clip: The 1918 Spanish Flu vs. COVID-19. Spot the difference</h3><ul>
<li><p>History Doesn't Repeat Itself, but it Often Rhymes.</p>
</li>
<li><p>Stupidity isn't a disease but it sure is spreading like one.</p>
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<li><p>The more I learn about humans, the more I like my dog.</p>
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<li><p>Those who do not learn history are doomed to repeat it.</p>
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<li><p>History taught us that we can't learn anything from history.</p>
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<li><p>Fun fact: The spanish flu started in the United States of America, but due to the fact that Spain was the first country to adress it, the rest of the world call it spanish flu.</p>
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<p>Q: What has America learned since the 1918 flu?<br>Me : that we're idiots</p>
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<li><p>Stupidity has some allies though, arrogance and agenda. Good luck to US!</p>
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<li><p>Trump has infected millions with his ignorance, stupidity, and lies! He has been a disaster for the country!</p>
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<li><p>It is also an pandemic of anti-intellectualism.</p>
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<h3 id="Clip-Trump-39-s-Brutal-Interview-with-Chris-Wallace"><a href="#Clip-Trump-39-s-Brutal-Interview-with-Chris-Wallace" class="headerlink" title="Clip: Trump's Brutal Interview with Chris Wallace"></a>Clip: Trump's Brutal Interview with Chris Wallace</h3><ul>
<li><p>It's hard to win an argument with a smart person, but it's damn near impossible to win an argument with a stupid person.</p>
</li>
<li><p>“I'll be right eventually” is like saying a broken clock is right twice a day.</p>
</li>
<li><p>Where ignorance is his master, there is no possibility of real peace.</p>
</li>
<li><p>Trump finally got his wall. But it’s not at the border! </p>
</li>
<li><p><a target="_blank" rel="noopener" href="https://www.theatlantic.com/ideas/archive/2020/06/trump-built-wall-around-himself/612664/">Trump Has Imprisoned Himself in the White House. The president is trying to project strength, but instead is betraying weakness.</a></p>
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<li><p>It's like listening to a toddler tell a story. You know there's a boat involved and you understand individual words, but the story itself is gibberish.</p>
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<h4 id="I-am-the-one-at-sail-I-am-the-master-of-my-sea"><a href="#I-am-the-one-at-sail-I-am-the-master-of-my-sea" class="headerlink" title="I am the one at sail, I am the master of my sea!"></a><a target="_blank" rel="noopener" href="https://www.bilibili.com/video/BV1jb41157zT?from=search&seid=3002647958457664048">I am the one at sail, I am the master of my sea!</a></h4><p>《82年的金智英》被认为是一部宣扬女权主义的电影，但稍微衍生一下，男性何尝不像女性一样被困在传统文化的桎梏中。这个问题其实超越了性别，种族和国籍，可以说在这一点上人类的悲喜开始相通了。不要误会，我并不是在说性别歧视，种族歧视，而是讨论一下更加隐蔽的角色禁锢。</p>
<p>我们在社会里扮演着各种不同的角色，传统观念与个人选择相互作用来给这些角色涂抹不同的色彩，并赋予相应的灵与肉，然而每个角色中最标准的色彩和最模范的内涵通常是由传统观念来定义的，因为这是最广为人知，最广为接受的，这就是传统与生俱来的力量，不由分说地裹挟着每个人向前。生活中，我们通常在各种角色中疲于奔命，某某的父母/孩子，某某的妻子/丈夫，某公司的老板/员工。时间线展开如下，当还在十月怀胎之中，胎教就如火如荼；开始咿咿哑哑学话时，各种早教班，兴趣班蜂拥而至；然后是挤破脑袋进入重点小学，重点初中，重点高中，一路到重点大学；再接下来就是找到令人羡慕的996，997的各种工作，在头发没有掉光之前赶紧成家立业，迫不及待地开启下一个循环。这部电影则很犀利的发问什么时候我们才能是我们自己，即这些角色是我们主动选择的吗？更进一步的说若是我们主动选择了这些角色，我们可以自己来定义吗？电影也悲观的告诉人们，我们大部分时候是被动进入到这些角色，而且个体对角色定义的空间几乎没有。更可悲的是，大多数人们并不会意识这个问题的存在，因为在情愿或不情愿去取悦传统观念之间早已心疲力竭，抑或丢失了自我。偶尔触动之际，也只有借助哪吒的口吻大呼一句：我命由我不由天。然而这种爆发就如酷炫的动画特技一样耀眼绚烂，但同样一样短暂。所以Oscar Wilde said “Most people are other people. Their thoughts are someone else's opinions, their lives a mimicry, their passions a quotation.”</p>
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<h2 id="Quick-Start"><a href="#Quick-Start" class="headerlink" title="Quick Start"></a>Quick Start</h2><h3 id="Create-a-new-post-（default）"><a href="#Create-a-new-post-（default）" class="headerlink" title="Create a new post （default）"></a>Create a new post （default）</h3><figure class="highlight bash"><table><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><span class="line">$ hexo new (post) <span class="string">"New Post"</span></span><br></pre></td></tr></table></figure>
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<p>More info: <a target="_blank" rel="noopener" href="https://hexo.io/docs/generating.html">Generating</a></p>
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<p>More info: <a target="_blank" rel="noopener" href="https://hexo.io/docs/one-command-deployment.html">Deployment</a></p>
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