exponential bound for sum of geometric variables
$begingroup$
I am writing my bachelor thesis and I am using the book "Brownian motion" by Mörters and Perez. I have troubles with the following lemma:
Let $X_j$, $j in mathbb{N}$, geometrically distributed on ${1,2,...}$ with mean $2$. For $epsilon$ sufficiently small, for every $m in mathbb{N}$ and all $k leqslant m$ the following inequality holds.
begin{align*}
mathbb{P}left({|sum_{j=1}^k (X_j-2)| geqslant epsilon m}right) leqslant 4 exp{-tfrac{1}{5} epsilon^2 m}
end{align*}
I already showed, using the hints in the book, (without Hoeffdings inequality) the inequality
begin{align*}
mathbb{P}left({sum_{j=1}^k (X_j-2) geqslant epsilon m}right) leqslant exp{-tfrac{1}{5} epsilon^2 m}
end{align*}
Now I want to proof the other part of the inequality via the one I already showed. In his book he states that it is obvious. I cannot see that and my endless computation did not really lead anywhere.
probability probability-theory
asked Jan 15 at 17:43
user631620user631620
363
$endgroup$
add a comment |
$begingroup$
I am writing my bachelor thesis and I am using the book "Brownian motion" by Mörters and Perez. I have troubles with the following lemma:
Let $X_j$, $j in mathbb{N}$, geometrically distributed on ${1,2,...}$ with mean $2$. For $epsilon$ sufficiently small, for every $m in mathbb{N}$ and all $k leqslant m$ the following inequality holds.
begin{align*}
mathbb{P}left({|sum_{j=1}^k (X_j-2)| geqslant epsilon m}right) leqslant 4 exp{-tfrac{1}{5} epsilon^2 m}
end{align*}
I already showed, using the hints in the book, (without Hoeffdings inequality) the inequality
begin{align*}
mathbb{P}left({sum_{j=1}^k (X_j-2) geqslant epsilon m}right) leqslant exp{-tfrac{1}{5} epsilon^2 m}
end{align*}
Now I want to proof the other part of the inequality via the one I already showed. In his book he states that it is obvious. I cannot see that and my endless computation did not really lead anywhere.
probability probability-theory
asked Jan 15 at 17:43
user631620user631620
363
$endgroup$
add a comment |
$begingroup$
I am writing my bachelor thesis and I am using the book "Brownian motion" by Mörters and Perez. I have troubles with the following lemma:
Let $X_j$, $j in mathbb{N}$, geometrically distributed on ${1,2,...}$ with mean $2$. For $epsilon$ sufficiently small, for every $m in mathbb{N}$ and all $k leqslant m$ the following inequality holds.
begin{align*}
mathbb{P}left({|sum_{j=1}^k (X_j-2)| geqslant epsilon m}right) leqslant 4 exp{-tfrac{1}{5} epsilon^2 m}
end{align*}
I already showed, using the hints in the book, (without Hoeffdings inequality) the inequality
begin{align*}
mathbb{P}left({sum_{j=1}^k (X_j-2) geqslant epsilon m}right) leqslant exp{-tfrac{1}{5} epsilon^2 m}
end{align*}
Now I want to proof the other part of the inequality via the one I already showed. In his book he states that it is obvious. I cannot see that and my endless computation did not really lead anywhere.
probability probability-theory
asked Jan 15 at 17:43
user631620user631620
363
$endgroup$
I am writing my bachelor thesis and I am using the book "Brownian motion" by Mörters and Perez. I have troubles with the following lemma:
Let $X_j$, $j in mathbb{N}$, geometrically distributed on ${1,2,...}$ with mean $2$. For $epsilon$ sufficiently small, for every $m in mathbb{N}$ and all $k leqslant m$ the following inequality holds.
begin{align*}
mathbb{P}left({|sum_{j=1}^k (X_j-2)| geqslant epsilon m}right) leqslant 4 exp{-tfrac{1}{5} epsilon^2 m}
end{align*}
I already showed, using the hints in the book, (without Hoeffdings inequality) the inequality
begin{align*}
mathbb{P}left({sum_{j=1}^k (X_j-2) geqslant epsilon m}right) leqslant exp{-tfrac{1}{5} epsilon^2 m}
end{align*}
Now I want to proof the other part of the inequality via the one I already showed. In his book he states that it is obvious. I cannot see that and my endless computation did not really lead anywhere.
probability probability-theory
probability probability-theory
asked Jan 15 at 17:43
user631620user631620
363
asked Jan 15 at 17:43
user631620user631620
363
asked Jan 15 at 17:43
user631620user631620
363
asked Jan 15 at 17:43
user631620user631620
363
asked Jan 15 at 17:43
user631620user631620
363
363
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Apply the same proof to $-sum_{j=1}^k (X_j-2)$ and notice that
$$
mathsf{E}e^{-lambda(X_j-2)}=frac{e^{2lambda}}{2e^lambda-1}le 1+lambda^2le e^{lambda^2}.
$$
answered Jan 16 at 6:09
d.k.o.d.k.o.
10.6k730
$endgroup$
add a comment |
Your Answer
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3074699%2fexponential-bound-for-sum-of-geometric-variables%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Apply the same proof to $-sum_{j=1}^k (X_j-2)$ and notice that
$$
mathsf{E}e^{-lambda(X_j-2)}=frac{e^{2lambda}}{2e^lambda-1}le 1+lambda^2le e^{lambda^2}.
$$
answered Jan 16 at 6:09
d.k.o.d.k.o.
10.6k730
$endgroup$
add a comment |
$begingroup$
Apply the same proof to $-sum_{j=1}^k (X_j-2)$ and notice that
$$
mathsf{E}e^{-lambda(X_j-2)}=frac{e^{2lambda}}{2e^lambda-1}le 1+lambda^2le e^{lambda^2}.
$$
answered Jan 16 at 6:09
d.k.o.d.k.o.
10.6k730
$endgroup$
add a comment |
$begingroup$
Apply the same proof to $-sum_{j=1}^k (X_j-2)$ and notice that
$$
mathsf{E}e^{-lambda(X_j-2)}=frac{e^{2lambda}}{2e^lambda-1}le 1+lambda^2le e^{lambda^2}.
$$
answered Jan 16 at 6:09
d.k.o.d.k.o.
10.6k730
$endgroup$
Apply the same proof to $-sum_{j=1}^k (X_j-2)$ and notice that
$$
mathsf{E}e^{-lambda(X_j-2)}=frac{e^{2lambda}}{2e^lambda-1}le 1+lambda^2le e^{lambda^2}.
$$
answered Jan 16 at 6:09
d.k.o.d.k.o.
10.6k730
answered Jan 16 at 6:09
d.k.o.d.k.o.
10.6k730
answered Jan 16 at 6:09
d.k.o.d.k.o.
10.6k730
answered Jan 16 at 6:09
d.k.o.d.k.o.
10.6k730
10.6k730
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3074699%2fexponential-bound-for-sum-of-geometric-variables%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
